# How to make motion of the Sun more apparent at seconds scale?

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Each time I look at a shadow cast by a sunlit object it seems as if the shadow doesn't move at all. But if I look at it after say 5 minutes, I can notice that the shadow has actually moved a bit. This allows to say "yes, the Sun has moved in the sky". But this doesn't make it all that obvious that "the Sun is continuously moving all the time".

So I'd like to make some optical configuration which would allow me to watch a spot of sunlight or a shadow moving much faster - say, 1 mm/s - still being from the Sun, not from an artificial light source. I've tried doing this with a cylindrical silvery surface like that of a kitchen rail system, sticking black tape on it and leaving only a lateral strip of reflecting surface, then then letting Sun rays to hit it almost tangentially, and to observe the reflection. But either this doesn't give good enough magnification to make Sun's motion apparent, or the reflected spot just becomes too faint for the naked eye.

So, what can I do to effectively magnify angular motion of the Sun to observe it (almost) directly at the scale of seconds? I's OK if it could be visible for e.g. a minute and required me to adjust my instrument to view the next piece of motion. But I'd like to do it as simply as possible, without too expensive hardware.

The obvious answer is to use a telephoto lens -- or any other magnification scheme. As an extreme example, if you had a post 1 km tall, the shadow of its top (think sundial) would move pretty clearly. If you re-image a short post, or just re-image the sun itself, which enough magnification, you'll see the motion.

This is kind of the same as trying to observe a celestical object with a high-power telescope. If you don't have an equatorial mount, everything in your field of view will move quite noticeablly.

## LECTURE 2: DISTANCES AND LUMINOSITIES OF STARS

The apparent daily path of a star is a circle in the sky, reflecting the rotation of the earth on its axis. Some stars rise and set, so we don't see the whole circle from one point on the earth.

We can't see stars during the day because the sun is too bright. Over the year, the earth moves around the sun, and different parts of the sky are visible.

The relative positions of stars on the sky are nearly fixed they barely change over a day or a year. But we expect to see parallax of nearby stars as the earth moves around the sun (illustration in Figure 9-4).

## How to make motion of the Sun more apparent at seconds scale? - Astronomy

That part of a star's motion which is not toward or away from us is called its tangential motion, and is perpendicular to its radial motion. The tangential motion is measured by the change in the star's direction in space over a period of time. If the star is far away or its tangential motion is small it may take millennia to observe any change in its position but if the star is very close to us, so that any motion it has looks relatively large, or if it is at a more moderate distance but is moving sideways at a rapid rate we may be able to observe a change in its position and calculate its tangential motion in just a few decades.
The change in a star's position, measured in seconds of arc per year or per century, is referred to as proper motion. As explained above proper motion cannot be measured for all stars -- only for stars that are unusually close or moving unusually fast relative to the Sun, or both -- and even then it takes decades or centuries to measure it, whereas radial velocity may be measured as quickly as a spectrum of the star can be obtained and analyzed. As a result most stars (and galaxies) have known radial velocities, whereas for most stars (and all galaxies) the proper motion is unknown.

Faster Than A Speeding Bullet -- Barnard's Star
Since stars that are close to the Sun can have relatively small velocities and still have noticeable proper motions, one of the easiest ways to recognize nearby stars is to look for stars with large proper motions. In fact the star with the fastest proper motion, Barnard's Star, with a proper motion of 10.3 seconds of arc per year, is the second closest stellar system to the Sun, and because it is rapidly approaching the Sun will be the closest star to the Sun, with a distance of less than four light years, in less than ten thousand years.

(2007 note: The following material was written before the preceding material, and will eventually be changed to fit the new layout)

In the case of the stars shown below the reason for the larger than normal proper motion is partly their distance (Barnard's Star is one of the closest stars to the Sun, and Mira, though more distant, is closer than most stars), and partly a larger than usual motion of the stars relative to the other stars in their vicinity. Barnard's Star is moving at about the same speed as the stars near it, but they are moving around the Galaxy in more nearly circular paths than not, while Barnard's Star is moving almost perpendicular to that direction, more nearly toward the center of the Galaxy. The stars moving around the Galaxy with the Sun have small motions relative to us, and small proper motions. Barnard's Star, moving in a very different direction, has a motion relative to us which is a combination of its and our overall motion, so it appears to move much faster relative to the background of stars than any other star.

An animation showing the motion of Barnard's Star from 1985 to 2005, at about the same scale as the Palomar Survey images, but taken with a considerably smaller telescope, so the fainter stars visible in the Palomar images are not visible. Most stars hardly seem to move over periods as long as a millennium but Barnard's Star moves a distance equal to the diameter of the full moon in less than two centuries.

The rapid apparent movement of Barnard's Star is not due to its having an exceptionally large velocity. It just happens to be moving in a different direction from the Sun and most of its neighbors. Most stars near the Sun are moving in the same direction with nearly the same speed, and have relative motions that are only about 10 to 15 percent of their actual motion, while Barnard's Star is moving in a different direction, so although its actual speed is about the same, its speed relative to the Sun is larger (about 85 mi/sec, compared to 15 to 25 mi/sec for other stars). Even this motion wouldn't be particularly noticeable, however, if not for the fact that Barnard's Star is one of the closest stars to the Sun. At just over 5 light years distance it is only a light year further than the nearest star, Alpha Centauri, and is the closest star north of the Celestial Equator. Because it is approaching the Sun, Barnard's Star will pass less than 4 light years from us about ten thousand years from now, and for some time before and after that will be the closest star to the solar system.

Mira -- A Whale of a Tail
Mira's motion is not as different from that of its neighbors as Barnard's Star, but has recently attracted attention because of its spectacular, thirteen-light-year long "tail". (Mira is in the constellation of Cetus, the Whale, hence the pun commonly used to refer to its tail.)
Mira is a red giant near the end of its life as an easily visible star. Because of its large size it has a low surface gravity, and gas is easily ejected from its outer layers into space. If it were stationary relative to the material surrounding it, the gas leaving Mira would gradually expand in all directions, forming a "pre-planetary" nebula. But it is moving through space relative to nearby stars and interstellar gas at about 80 miles per second (most such motions are only a quarter that speed), and as the star and the gas leaving it plow through the interstellar medium a "bow shock" is formed, like the bow wave in front of a speedboat, and a long "wake" or "tail" has formed behind the star. The collision of the stellar and interstellar gas heats them to very high temperatures, and as the gas streams behind the star the hydrogen which makes up most of the gas emits radiation (almost entirely in the ultraviolet), as shown below.

Tidal lock means the rotation and orbit are synchronous in a 1:1 relationship. That is, it takes the same amount of time to complete one rotation and one orbit.
The moon on the left is tidally-locked and rotating once per orbit, the moon on the right is not tidally-locked and is not rotating.

The situation on the right is possible, but is not stable. All systems of pairs of orbiting bodies will tend to synchronise in a full tidal lock over time, or crash into one another if certain initial conditions are met.

Lastly, my co-worker is convinced the moon on the left, which always has the same side pointed to the center, is not rotating.

To me it seems the one on the right is not rotating. But ,like I asked before, is the right side even possible?

Tidal locking doesn't work that way. The spin north pole would have the same direction as the orbit north pole.

But from what you asked about, it's the substellar point and its antipode. To lowest order, they have the same size of acceleration of gravity, though in opposite directions.

That's a counterintuitive feature of tides. The Moon and the Sun each make 2 evenly-spaced tides a day, and not just 1 tide, as one would naively expect. That's because the ocean away from the Earth is left behind as the Moon and the Sun pull on the Earth. Remember that the acceleration of gravity is independent of mass and composition. That's the Equivalence Principle, and there are some strong upper limits to violations of it.

Expanding to the next order in (planet radius) / (distance from its star), one does find an effect, but its size is about (tide size) * (that factor), and that is teeny teeny teeny tiny except for a star in a close binary-star system or something similar for planets.

I think you have to determine the difference, between acceleration due to gravity from the planet and star.

and the acceleration required to maintain circular motion.

the difference would be supplied by the surface of the planet pushing up against the bottom of the object, which would be experienced as weight

D = distance from CMstar to CMplanet
M = mass of planet
M = mass of star

at sub stellar point, gravities pull in opposite directions, but the required centripetal force is lower

at ant stellar point, gravities pull in concert, but more centripetal force is required

note that W is the orbital frequency

so to lowest order, objects would have the same weight at both poles

In explaining the above graphic, you say the moon on the left has a typical 1:1 sidereal spin rate:

You also say that the moon on the right is possible, but not stable. By "not stable," I assume you mean that the image on the right still has one remaining clockwise (CW) polar axial rotation (since that moon's surface is still moving in relation to its primary's surface), thus it appears that the moon on the right has a 0:1 sidereal spin rate:

Though not common, an astronomical body with a clockwise (CW) rotation can also be spun down to 1:1, as (CW) rotating Venus proves, so is Venus the situation you had in mind by saying the graphic on the right is possible, but not stable and wouldn't last for very long?

If so, I agree that a 0:1 sidereal spin rate wouldn't be stable for very long, since once Venus spins down to that 0:1 rate (1 CW polar axial rotation per each CCW orbit), Venus at that point will still have one last CW polar axial rotation left to lose before the planet is fully despun at 1:1, so Venus would relatively quickly pass thru that 0:1 sidereal spin rate.

That's almost what I meant.

The lack of stability simply means that in the presence of tidal forces, a non-rotating (in an inertial frame) moon will inevitably end up as rotating. I.e., it will have some angular acceleration around its rotational axis. Conversely, a full (1:1) tidal lock means no angular acceleration of the moon, so the rotation remains constant over time.

A 0:1 rotation is then not merely 'not stable for very long', but not stable at all - at no point in the evolution of Venus' rotation prior to reaching a 1:1 lock (shouldn't happen before the Sun engulfs it) will the angular acceleration equal zero.
This is assuming a simplified, circular orbit-case with no outside influences.

These two bits are confusing, though. I think you're using a rotating frame here, as otherwise the underlined parts make no sense. This is not what is normally used to describe synchronous motion. Whenever you see a rotation : orbit ratio, it means rotation in an inertial frame of reference.

That's almost what I meant.

The lack of stability simply means that in the presence of tidal forces, a non-rotating (in an inertial frame) moon will inevitably end up as rotating. I.e., it will have some angular acceleration around its rotational axis. Conversely, a full (1:1) tidal lock means no angular acceleration of the moon, so the rotation remains constant over time.

I think I understand your viewpoint. What's confusing is that the sidereal versus the center-point perspectives always differ by one apparent rotation.

E.g., from the sidereal perspective Earth has 366.25 CCW rotations, each 23 hours, 56 minutes and a few seconds long. But from the center-point perspective, Earth has one less rotation, 365.25 CCW rotations, each 24 hours long, which is one less apparent rotation than from the sidereal perspective.

From the sidereal perspective, Venus currently has .92 CW rotations, but since the sidereal and the center-point perspectives always differ by one apparent rotation, Venus actually has nearly two full solar days per orbit (1.92 CW polar axial rotations per orbit).

Today, since Venus has slightly less than one apparent sidereal rotation (only .92 sidereal rotations), it appears that Venus has recently passed thru a 1:1 sidereal spin rate as that planet is in the process of being despun – how else could Venus currently have a .92:1 sidereal spin rate if Venus didn't recently pass thru a 1:1 rate?

If so, then Venus is now slowing to that mythical 0:1 sidereal spin rate, before finally stopping at a final (and 2nd) 1:1 sidereal spin rate as it becomes tidally locked.

If our moon spun down from the CWW direction, then from our center-point perspective, our moon spun down like this (starting randomly with 4 polar axial rotations, but there may have been more):

If our moon spun down from the CW direction (which it may have done), then from our center-point perspective, our moon spun down like this (the same):

No matter which direction our moon spun down from, from Earth's center-point perspective, our moon spun down to ZERO polar axial rotations.

From from the sidereal (outside) perspective, our moon *APPEARS* to have spun down like this from the CCW direction:

Using the ROTATION:ORBIT convention, for a CCW spinning moon, you would write the above like this, with our moon stopping at one (1) *apparent* rotation per each 360˚ orbit of the moon:

From the center-point perspective (i.e., from the sun), Venus would appear to spin down similarly to our moon if our moon had spun down from the CW direction:

From the above center-point perspective, currently at 1.92 CW polar axial rotations, Venus recently had 2 full CW rotations, and is currently heading towards 1 CW rotation, before stopping at zero, thereafter with one face pointing inwards towards the sun in a tidal lock.

Using the ROTATION:ORBIT convention again, you would write Venus' above spin-down like this, with Venus stopping at one (1) sidereal rotation, per each 360˚ CCW orbit around the sun:

Remember, Venus currently has nearly two full CW polar axial rotations left (1.92), so Venus must have already passed thru one 1:1 sidereal spin rate. How else could Venus bleed off nearly two more CW rotations and get to that final 1:1 rate, if Venus doesn't need to pass thru a 0:1 sidereal spin rate, as depicted on the right side in the above graphic?

In my opinion, as Venus passes thru that transitory 0:1 sidereal spin rate, Venus will still have one (1) full CW polar axial rotation left to lose, so as Venus passes thru that transitory 0:1 rate, Venus would still have angular momentum around its polar axis at that point.

I.e., the 0:1 spin rate is an optical illusion. Look at the moon on the right in the upper image, and while many people see a zero-rotating moon, I see a moon with one (1) CW polar axial rotation as it orbits in the opposite CCW direction, and two 360˚ spins in the opposite direction, around two different axes (one axis its barycenter, and the 2nd axis around its polar axis), these two 360˚ spins in opposite directions merely cancel each other out . MOMENTARILY, since a 0:1 sidereal spin rate is transitory.

As for Venus being engulfed by our sun going red giant before Venus becomes tidally locked, Venus only has 1.92 polar axial rotations left to lose, and Magellan's cloud piercing radar found that Venus had lost 6 minutes of rotation during the 16 years Magallan observed Venus back in the 1990s:

I think I understand your viewpoint. What's confusing is that the sidereal versus the center-point perspectives always differ by one apparent rotation.

E.g., from the sidereal perspective Earth has 366.25 CCW rotations, each 23 hours, 56 minutes and a few seconds long. But from the center-point perspective, Earth has one less rotation, 365.25 CCW rotations, each 24 hours long, which is one less apparent rotation than from the sidereal perspective.

This one rotation of difference is exactly the same difference between the 0:1 and 1:1 rotation : orbit relations.
It would probably be less confusing if you dropped this switching of reference frames. Stick to the inertial frame (i.e. the one used for the sidereal rather than the tropical year).

From the sidereal perspective, Venus currently has .92 CW rotations, but since the sidereal and the center-point perspectives always differ by one apparent rotation, Venus actually has nearly two full solar days per orbit (1.92 CW polar axial rotations per orbit).

Today, since Venus has slightly less than one apparent sidereal rotation (only .92 sidereal rotations), it appears that Venus has recently passed thru a 1:1 sidereal spin rate as that planet is in the process of being despun – how else could Venus currently have a .92:1 sidereal spin rate if Venus didn't recently pass thru a 1:1 rate?

If Venus has 0.92 clockwise rotations per year, then it has -0.92 counter-clockwise rotations per year. As a convention, the counter-clockwise direction is the direction used for positive values in orbital and rotational periods. So, current rotation : orbit relation for Venus is -0.92:1, and it had passed -1:1 rather than 1:1.

edit: here's me not checking the facts. Venus' rotational period is longer than the orbital period, so it's definitely not -0.92:1. but -1.08:1.

Using the ROTATION:ORBIT convention again, you would write Venus' above spin-down like this, with Venus stopping at one (1) sidereal rotation, per each 360˚ CCW orbit around the sun:

So this is wrong for the aforementioned reason.

No. When its rotation in an inertial frame of reference is 0, it necessarily means that its rotational angular momentum is also 0. As tidal interactions continue to accelerate the planet, it gains rotational angular momentum until it settles in a stable 1:1 synchronous rotation.

The moon on the right rotates only in a non-inertial, rotating reference frame.
There's nothing wrong in using a non-inertial frame to describe motion, as long as you clearly state if you're using it and are consistent about it. The default for most applications in astrodynamics is the inertial frame.
It's strongly advisable not to switch frames as you analyse motion to avoid confusion and wrong results - for example, while the moon on the right rotates in any rotating reference frame, and in particular rotates at 1 rotation per orbit in a frame rotating with the angular velocity equal to orbital angular velocity of the moon, in the same frame the moon is not orbiting - it is hovering in one place above the observer, supported by a pseudo-force, and has no orbital angular momentum as a result.

I think the bottom line here is: stick to the inertial frames, or at least don't switch frames, and watch the signs. Otherwise you seem to get it right.

By the by, I wasn't aware of the discrepancy in Venus' rotation period. That's interesting. I wouldn't jump to conclusions, though.

In order to fully understand the sidereal perspective, then it's imperative to compare the sidereal (OUTSIDE) perspective with the center-point perspective (such as from Earth or from the sun, as the case may be).

Let's review your claim that there's only a difference of one (1) rotation between -0:1 and +1:1 (NOTE - as you suggested, I'm now adding a neg "-" sign for retrograde CW orbits, and a "+" sign for regular CCW prograde orbits).

Clearly, the sidereal (view from the stars) and center-point perspectives (view from Earth, or other primary), they always differ by ONE (1) apparent rotation:

The word synodic derives from the Greek word for meeting or assembly. It is motion relative to a conjunction or alignment of sorts. A synodic or solar day is the time it takes the sun to successively pass the meridian (astronomical noon). A mean solar day is 24 hours (the “mean” is there to average over the effect of the analemma). The earth has to rotate more than 360° for the sun to come back to “noon”.

A synodic year is the time it takes for a planet-sun alignment to reoccur. For the case of the sun, it is the time it takes the sun to come to the same place on the ecliptic (equinox to equinox) and is called a Tropical Year. A tropical year is 365.242 mean solar days (366.242 sidereal days. It is just over 20 minutes shorter than a sidereal year (again, the effect of precession).

If Earth has 366.242 sidereal days, then we need to subtract ONE (1) to determine Earth's actual CCW polar axial rotations, as explained here:

Another way to see this difference is to notice that, relative to the stars, the Sun appears to move around the Earth once per year. Therefore, there is one fewer solar day per year than there are sidereal days. This makes a sidereal day approximately 365.24 ⁄ 366.24 times the length of the 24-hour solar day, giving approximately 23 hours, 56 minutes, 4.1 seconds (86,164.1 seconds).

Venus rotates retrograde with a sidereal day lasting about 243.0 earth-days, or about 1.08 times its orbital period of 224.7 earth-days hence by the retrograde formula its solar day is about 116.8 earth-days, and it has about 1.9 solar days per orbital period.

The reason that the sidereal perspective always has one (1) more apparent 360° rotation per CCW orbit, is because typically from the sidereal perspective the orbiting body's polar axial rotations (if any) are aggregated in with the orbiting body's 360° orbit.

Conversely, for astronomical bodies with a retrograde CW rotation, such as Venus, you instead need to ADD one (1) to their sidereal rotation rate to determine the actual number of their polar axial rotations:

"All the planets of the Solar System orbit the Sun in a counter-clockwise direction as viewed from above the Sun’s north pole. Most planets also rotate on their axis in a counter-clockwise direction, but Venus rotates clockwise (called “retrograde” rotation) once every 243 Earth days—the slowest rotation period of any planet. To an observer on the surface of Venus, the Sun would rise in the west and set in the east.

"A Venusian sidereal day thus lasts longer than a Venusian year (243 versus 224.7 Earth days). Because of the retrograde rotation, the length of a solar day on Venus is significantly shorter than the sidereal day, at 116.75 Earth days (making the Venusian solar day shorter than Mercury’s 176 Earth days) one Venusian year is about 1.92 Venusian (solar) days long."

Venus has 1.92 actual CW polar axial rotations per each orbit, which is slightly less than two full solar days. However, Venus only has -0.92 sidereal rotations (224.7 divided by 243 = 0.92). The reason why you need to ADD one (1) for CW rotating bodies, is the same reason for subtracting one (1) for CCW bodies, because the sidereal perspective likewise aggregates together a retrograde body's CW polar axial rotations with the orbiting body's single CCW 360° orbit.

An astronomical body with a prograde 360° CCW orbit with a retrograde 360° CW rotation, these two 360° spins in opposite directions (around two different axes) merely cancel each other out, which is why you need to add one (1) to Venus' -0.92 sidereal rotations to determine Venus' actual 1.92 CW polar axial rotations – it's as simple as that.

Can we now agree that the sidereal (outside) perspective differs from the center-point perspective, by one (1) APPARENT rotation?

As the below orrery spins, if you were to view it from above, the moon in this orrery would behave exactly like the moon does (on the left) in the above graphic that the poster had asked about:

Let's compare the original poster's above graphic with the orrery:

In normal operation the orrery's faux-moon would match the moon on the left. However, if you reached out and forced the orrery's faux-moon to point at one wall as you cranked that orrery around, then the faux-moon would necessarily be forced to spin CW around its supporting metal spindle (360° one time), as the moon on the right is doing. If you reached out and spun the faux-moon CW 0.92 times per revolution, then that is what Venus is now doing, -0.92 from the sidereal perspective, but 1.92 360° CW rotations from the sun's center-point perspective.

NOTE – When viewing either the orrery or any graphic such as the graphic above, you are necessarily placed in the outside sidereal position, so you need to either add one (1) or subtract one (1) to determine the moon's actual polar axial rotations in these models.

The moon's surface on the right is clearly moving in relation to the planet in the center (and would thus still be experiencing tidal braking forces). Imagine yourself standing on the planet on the right, and clearly you would observe the moon on the right making one 360° CW polar axial rotation per each CCW orbit, so while from the sidereal perspective the moon appears to be pointing in one direction and not rotating, ADD one (1) to that sidereal zero rotation rate, and the moon is actually making one (1) CW polar axial rotation per orbit.

That many people see a zero-rotating moon is just an optical illusion from the sidereal perspective since that moon is actually making TWO 360° spins in opposite directions.

When you're dealing with bodies in motion there are many possible optical illusions arising from the nature of the orbits. E.g., from Earth's perspective, there are times when Mars appears to be traveling backwards compared to the "fixed stars”:

Mars Retrograde Happens Every Two Years:

If Venus has 0.92 clockwise rotations per year, then it has -0.92 counter-clockwise rotations per year. As a convention, the counter-clockwise direction is the direction used for positive values in orbital and rotational periods. So, current rotation : orbit relation for Venus is -0.92:1, and it had passed -1:1 rather than 1:1.

edit: here's me not checking the facts. Venus' rotational period is longer than the orbital period, so it's definitely not -0.92:1. but -1.08:1.

Your 1.08 rotation claim is wrong since 1.08 is what you get by dividing Venus' 225 day orbit into Venus' 'APPARENT' sidereal rotation rate of 243 days:

Venus sluggishly rotates on its axis once every 243 Earth days, while it orbits the Sun every 225 days - its day is longer than its year! Besides that, Venus rotates retrograde, or "backwards," spinning in the opposite direction of its orbit around the Sun. From its surface, the Sun would seem to rise in the west and set in the east.

However, Venus' 225 day orbit divided by its 243 day apparent sidereal rotation rate, instead gives the correct answer of 0.92 sidereal rotations per orbit.

1.08 is instead how many times Venus needs to rotate around its polar axis per each sidereal day in relation to a Venusian year (i.e., Venus' sidereal day is a tad longer than a Venusian year):

Another way to see this difference is to notice that, relative to the stars, the Sun appears to move around the Earth once per year. Therefore, there is one fewer solar day per year than there are sidereal days. This makes a sidereal day approximately 365.24 ⁄ 366.24 times the length of the 24-hour solar day, giving approximately 23 hours, 56 minutes, 4.1 seconds (86,164.1 seconds).

Venus rotates retrograde with a sidereal day lasting about 243.0 earth-days, or about 1.08 times its orbital period of 224.7 earth-days hence by the retrograde formula its solar day is about 116.8 earth-days, and it has about 1.9 solar days per orbital period.

We still have the same dilemma, if Venus currently has -0.92 CW sidereal rotations per orbit (-0.92:1), then Venus must have recently passed thru -1:1 to now be at -0.92:1.

Since Venus also clearly has nearly two (2) actual CW polar axial rotations per orbit (and nearly two solar days remaining), that means Venus has also passed thru a 2 full solar day rotation to wind down to only 1.92 solar days that it has today, which means that Venus has nearly two full polar axial rotations left to lose before being fully despun at your sidereal +1:1 tidally locked rate.

I already explained, from the SIDEREAL perspective, how it would be possible for Venus to lose nearly two (2) actual CW polar axial rotations and go from its current -0.92:1 sidereal rotation rate to a tidally locked rotation rate of +1:1:

If you can explain how Venus can bleed off nearly two full CW polar axial rotations, starting at its current -0.92:1 rate without passing thru that zero sidereal rotation rate, then I'd like to see that done?

Also, if you believe that a tidally locked body has one (1) CCW rotation per orbit (once despun at your +1:1), then you may want to explain how Venus will not only lose nearly two CW polar axial rotations starting from -0.92:1, but then once Venus is fully despun at +1:1, please do explain how Venus will suddenly start rotating one (1) full time in the opposite CCW direction after Venus becomes tidally locked at +1:1?

The sidereal perspective has its practical uses, but as Venus proves, the sidereal perspective has its quirks, too.

Are you claiming tidal interactions can accelerate a planet's polar axial rotation? If you really do mean its rate of rotation, then do you have a citation for that? Yes, tidal braking can increase an orbiting body's orbital speed, but we're talking about decelerating Venus' rotational speed from -0.92:1 down to +1:1. Increased orbital speed is just where the lost rotational energy is transferred to (along with some loss to heating), but that isn't what you appear to be claiming?

As far as I know, whether a planet or moon has either a prograde or retrograde axial spin, tidal braking forces would slow down either type of rotating body. Of course, with a retrograde CW rotating (and CCW orbiting) body, the primary's gravity would be pulling the planet's tidal bulge in the opposite direction of its orbital direction, which may decrease its orbital speed a tad, but not sure about that and I'm just throwing out the possibility? Even so, a planet the size of Venus likely wouldn't have its orbital speed affected too much as it loses only two more CW polar axial rotations.

The moon on the right rotates only in a non-inertial, rotating reference frame.

There's nothing wrong in using a non-inertial frame to describe motion, as long as you clearly state if you're using it and are consistent about it. The default for most applications in astrodynamics is the inertial frame.

Let's try to keep this simple and explain it in a way a layman can understand. By a "non-inertial frame" you mean from Earth's perspective.

The seemingly zero-rotating moon on the right in the above graphic would NOT be hovering in one place above an observer standing on the planet in the center, since such an observer would clearly see that moon both orbiting CCW as well as rotating around its polar axis CW, ONCE per each orbit. The only thing not stable about a seemingly zero-rotating moon (from the sidereal perspective), is that such a moon would still have one last CW rotation left to lose, so it would merely pass thru -0:1, then go to -0.01:1, then -0.02:1, then -0.03:1, then -0.04:1, etc, until stopping at a final +1:1 sidereal rate. As that moon passes thru that -0:1 rate, the background stars would slowly begin to move again in relation to its surface, and gradually increase their movement until the moon reached its final +1:1 rate.

It's only from the sidereal perspective that such a body would seemingly pass thru that transitional -0:1 rate as it was being despun. On planet Venus, its days will just get gradually longer and longer (going from 1.92 days per orbit to zero days), until Venus eventually runs out of steam and locks, at which point Venus will only show one face inwards towards the sun, just as our moon now only shows one face inwards towards Earth.

Unlike our own moon, Venus' backside will be in perpetual darkness one it locks to the sun.

Further, if you stood on that seemingly zero-rotating moon's surface on the right, then the stars wouldn't appear to move much (the stars would move a tad due to librations), but the Earth would rise in the West and set in the East each orbit. If you instead stood on the other moon's surface (the left moon in the graphic), as astronauts have already done, then Earth would hang nearly motionless in the sky, except for minor movements again due to librations, caused by the moon's elliptical orbit, as well as by the tilt of the moon's orbit in relation to Earth's orbital plane.

I feel I have it exactly right, and it's also necessary to switch between reference frames to help explain which frame better represents the reality of the movements that we're discussing here, as I hope I'm now doing with you.
:)

Despite being our closest planetary neighbor, since Venus has been shrouded in clouds, it was only relatively recently by using radar that Venus's axial rotation could even be studied, and there doesn't appear to be much published on the planet's unique rotation. In my opinion, to understand what our own moon is doing, then you need to understand what Venus is doing.

For example, as you try to harmonize how Venus can lose nearly two full polar axial rotations, and be despun down from its current -0.92:1 to a tidally locked +1:1, then it may help you to understand what your +1:1 final sidereal rate actually means. The tip to that I already posted above:

Likewise, if our moon has a sidereal spin rate of one (1), then subtract one (1) from our moon's apparent sidereal spins, just as you need to do for Earth's 366 sidereal spin total to find the actual polar axial rotations of both:

If you consider that our fully despun +1:1 moon now has zero polar axial rotations, then everything makes sense as Venus must lose nearly two rotations to go from -0.92 down to +1:1 sidereal rotations:

I realize what I'm saying here may sound like heresy to you at first blush, but synchronous theory has been disputed for hundreds of years by many notable scientists and astronomers.

If you need further proof, then please try and explain our moon's longitudinal librations within your synchronous theory framework?

Most websites claim our moon has a steady CCW rotation rate that gets out of sync as our moon's orbital velocity varies as it passes thru its apogee and perigee. However, our moon's two maximum longitudinal librations actually occur midway between apogee and perigee!

That our moon's maximum longitudinal librations occur midway between apogee and perigee can be easily explained if you consider the possibility that our moon is today fully despun –– simply, midway between apogee and perigee, our fully despun moon is merely facing the empty focus of its elliptical orbit.

The sidereal perspective has its practical uses, but it's also not some 'God's Eye' view of reality!

#### Attachments

Back in the late 1880s synchronous rotation was thoroughly debated, and the majority of English astronomers back then eventually voted against the concept.

There are many arguments PRO & CON, but I like this one the best, especially his #5 reason below. Even though this letter was written in 1864, these were obviously very intelligent men. Sir John Herschel mentioned in this letter was one of the main proponents of synchronous rotation theory back then. There were actually two Sir Herschel astronomers back then, a father and a son, and they were both lunatic lunar rotators:

---snip---
THE MOON CONTROVERSY,

TO THE EDITOR OF THE ASTRONOMICAL REGISTER.

"Sir,—On reviewing the whole controversy respecting the moon's rotation, or non-rotation, on her own axis, it will be observed that mathematicians are ranged on one aide, and practical mechanics and engineers on the other, and that theory is at variance with practice. "Mathematicians have adopted a quasi definition of rotation, which leads to innumerable absurdities, e.g. that every atom of a spinning- top rotates on its own axis, that the ball on St. Paul's cathedral rotates on its own axis once in 23h, 56m,. etc, etc..

"They use the terms of "rotation" and "rotation on its own axis" as though they were synonymous in meaning, quite overlooking the distinction between axial and radial rotation. It would surely be quite possible for them to be more concise in the use of language without calling in question any single mathematical deduction or truth of any sort or kind. *It would be better to understand the term "rotation" in the sense used by mechanics, engineers, and men of plain common sense.*

"It is really astonishing how one writer after another, like Mr. Bird in the last No. of the Register, can persuade themselves that "turning round" necessarily involves "rotation," either axial or radial.

"Before closing the correspondence on this most interesting subject it may be as well to notice Sir John Herschel's illustration of axial rotation. He "plants a staff in the ground, and, grasping it in both hands, walks round it, keeping as close to it as possible, with his face always turned towards it, "when the unmistakable sensation of giddiness effectually satisfies him of the fact that he has rotated on his own axis!"

"Mr. Bertram Mifford, of Cheltenham, in a letter addressed to the Practical Mechanics' Journal, Dec. 1, 1859, thus effectually disposed of this most transparent fallacy by the following arguments—

"1. Because giddiness may be produced in various other ways, by looking over a precipice, etc.

"2. Because (mechanically speaking) no man can walk round his own axis in any way, much less turn upon it when holding on to any fixed object, which must be external to his body. He may walk his axis round any object he pleases, but he cannot in any sense walk round his own axis.

"3. Because a man holding by a stick and revolving round it turns on the axis of the stick and not on his own axis, the axis of the stick in this case becoming the common axis of both man and stick.

"4. The alteration of the relative position of the objects round him is no proof that the man has turned on his own axis, he has merely dragged his axis after him, and has not turned upon it.

"5. "Keeping as close to it as possible" is certainly the nearest approach to turning on his own axis, but until he is impaled on the stick and turned on it he does not revolve (mechanically speaking) on his own axis.​

"I have the honour to be, very faithfully yours,

John G-Wolbach Library, Harvard-Smithsonian Center for Aslrophysics • Provided by the NASA Astrophysics— The above letter (starting on page 19) and other old pro & con letters on the issue can be found here: http://articles.adsabs.harvard.edu//full/seri/AReg./0003//0000034.000.html​

Lastly, my co-worker is convinced the moon on the left, which always has the same side pointed to the center, is not rotating.

Even from the point of view on Earth, the Moon is still observed to be rotating.

As the Moon moves across the sky, it must turn, otherwise you'd see one side, then the front, then the other side (like a car passing you, you see the front, then the side, then the back). The only way you can see the same face of the Moon from the Earth, is if the Moon is rotating to match your changing angle of observation.

Unless, the co-worker is CORRECT and you're WRONG?

The last I looked at our moon, it certainly didn't appear to be rotating when viewed from my backyard?

Where are you observing our moon from?

Anything in particular I should ingest before seeing this claimed rotational movement of our moon that you claim to see?
:)

If you sit in the stands at a car race, then you will see both sides of each race car as they turn a lap. HOWEVER, if you stand in the infield, then you will only observe the left door of all the race cars.

Can't you see that this issue involves your viewing angle?

Are the race-cars revolving 360° around their center-mass as they complete each 360° lap (synchronous rotation), or are the cars merely TURNING about their center-mass as they complete a lap?

As the cars complete a lap, EXACTLY where is their turn axis?

A – is the axis around each car's center-mass or

B – are the cars instead turning around an axis in the center of the racetrack or

C – turning around BOTH axes at the same time?​

He's not wrong. The Moon is rotating.

As the cars complete a lap, EXACTLY where is their turn axis?

A – is the axis around each car's center-mass or

B – are the cars instead turning around an axis in the center of the racetrack or

C – turning around BOTH axes at the same time?

The cars rotate about an axis through the middle of each car, halfway between the front and rear axles. They also rotate revolve around an axis passing through the center of this track. So the answer is C, both.

Terrific, a simplistic RULE that seemingly solves everything!

However, we were discussing this graphic:

Consistent with your stated RULE, the image (our current Earth & moon) on the left does REVOLVE CCW around Earth with a sidereal rotation rate of 27.3, but is the moon in that graphic both REVOLVING around the Earth 360° and ROTATING around its center-mass (polar axis) 360°, as well?

Synchronous rotation theory holds that Earth's moon does indeed have two synchronous rotations, a 360° REVOLUTION around its barycenter, as well as a 2nd 360° ROTATION around its polar axis, and the issue here is, since both types of 360° spins would cause the stars to move while standing on such a moon, is it our moon's REVOLUTION or ROTATION, or both types of spin in conjunction which would cause the stars to move?

Can REVOLUTION & ROTATION in conjunction cause the starfield to stop (or at least, to slow down), as appears to be happening with the moon on the right?

Let's now examine the supposedly zero-rotating moon on the right. I see that moon as having two 360° spins, a 360° CCW REVOLUTION around its barycenter, as well as a 360° CW ROTATION around its polar axis. In relation to Earth in the center, from the outside perspective, that moon would not appear to have a rotation in relation to the Earth in the center, but from Earth's perspective, that moon would be rotating since its surface is moving.

If the surface of the moon on the right is still moving in relation to the Earth's surface, then tidal braking forces would still be slowing the moon on the right down – can you agree that the surface of the moon on the right is still turning when viewed from the center?

Clearly, if you were to stand on that Earth, then you would clearly see that moon both REVOLVING (orbiting) 360° CCW each orbit, as well as you would see the entire surface of that moon each orbit spinning 360° in the opposite CW direction.

If our own moon had spun down from the CW direction (as Venus is now doing), then our moon would have passed thru that same so-called zero-spin rate, which is a -0:1 sidereal spin rate. It is clearly possible for an astronomical body to pass thru a -0:1 in the process of being despun to a +1:1 sidereal spin rate. Actually, it would be impossible for a CW rotating body to be despun without passing thru that -0:1, as depicted on the right in the above graphic.

So, let's apply your RULE to that supposedly zero-rotating moon. If you were to stand on that supposedly zero-rotating moon's surface, then the stars would still move about its sky, but not in 27.3 days, but instead in 365 days since both the Earth and moon are together in orbit around the sun. 365 is not 27.3 days, but the stars would indeed still move while standing on that moon's surface!

If our moon was despun from the CW direction (as is now happening to Venus), then that -0:1 rate wouldn't have lasted for very long, perhaps only a matter of days, so the starfield would at most appear to slow down as our moon passed thru that -0:1 rate, before speeding up again. Of course, once our moon was fully tidally locked to Earth, since that happened many billions of years ago when our moon was much closer to Earth with a much faster orbital period, so the 27.3 day part of your rule wouldn't apply to our moon until relatively recently after our moon evolved into its current orbit, which continues to change to this day.

BTW – since no two lunar orbits are exactly the same, the "27.321582" day portion is merely an average (mean) sidereal rotation rate, so don't engrave that into stone.

Yep, it's surely as simple as that!
:)

Using your RULE, would you care to explain how CW rotating Venus, which now has a -0.92:1 sidereal rotation rate (but 1.92 solar days per orbit), how Venus can bleed off nearly two full CW polar axial spins to arrive at the classic +1:1 of a despun body?

As viewed from the sidereal perspective, my explanation is the only possible way to count Venus' spin down:

Since Venus now has slightly less than a single CW sidereal rotation (-0.92:1), venus has recently passed thru that first -1:1 sidereal rotation rate, and is now heading towards that -0:1 rate before stopping at a final +1:1 sidereal spin rate. While passing thru that -0:1 sidereal spin rate, Venus will match that supposedly zero-spinning moon on the right side of the graphic under discussion here, so Venus would still have one final CW polar axial rotation left to lose before becoming tidally locked with the sun.

Of course, when viewing Venus' spin down from the center-point perspective, things get far less confusing.

The simple center-point is followed by the sidereal perspective, as Venus loses nearly 2 full CW polar axial rotations as that planet is despun – (in parenthesis is Venus' current spin rates from both perspectives):

## New advances in solar cell technology

With the high environmental cost of conventional energy sources and the finite supply of fossil fuels, the importance of renewable energy sources has become much more apparent in recent years. However, efficiently harnessing solar energy for human use has been a difficult task. While silicon-based solar cells can be used to capture sunlight energy, they are costly to produce on an industrial scale. Research from the Energy Materials and Surface Sciences Unit at the Okinawa Institute of Science and Technology Graduate University (OIST), led by Prof. Yabing Qi, has focused on using organo-metal halide perovskite films in solar cells. These perovskite films are highly crystalline materials that can be formed by a large number of different chemical combinations and can be deposited at low cost. Recent publications from Prof. Qi's lab cover three different areas of innovation in perovskite film research: a novel post annealing treatment to increase perovskite efficiency and stability, a discovery of the decomposition products of a specific perovskite, and a new means of producing perovskites that maintains solar efficiency when scaled up.

In order to be useful as solar cells, perovskite films must be able to harvest solar energy at a high efficiency that is cost-effective, be relatively easy to manufacture, and be able to withstand the outdoor environment over a long period of time. Dr. Yan Jiang in Prof. Qi's lab has recently published research in Materials Horizons that may help increase the solar efficiency of the organo-metal halide perovskite MAPbI3. He discovered that the use of a methylamine solution during post-annealing led to a decrease in problems associated with grain boundaries. Grain boundaries manifest as gaps between crystalline domains and can lead to unwanted charge recombination. This is a common occurrence in perovskite films and can reduce their efficiency, making the improvement of grain boundary issues essential to maintain high device performance. Dr. Jiang's novel post annealing treatment produced solar cells that had fused grain boundaries, reduced charge recombination, and displayed an outstanding conversion efficiency of 18.4%. His treated perovskite films also exhibited exceptional stability and reproducibility, making his method useful for industrial production of solar cells.

One of the biggest disadvantages to the use of perovskites when compared to silicon in solar cells is their relatively short lifespan. In order to create a solar cell that can withstand the outdoor environment over a long period of time, it is crucial to determine the major products of perovskite decomposition. Previous research on MAPbI3 perovskite films led to the conclusion that the gas products of thermal degradation of this material were methylamine (CH3NH2) and hydrogen iodide (HI). However, exciting new research from Dr. Emilio J. Juarez-Perez, also in Prof. Qi's lab, published in Energy & Environmental Science, shows that major gas products of degradation are methyliodide (CH3I) and ammonia (NH3) instead. Dr. Juarez-Perez used a combination of thermal gravimetric differential thermal analysis (TG-DTA) and mass spectrometry (MS) to correctly determine both the mass loss and chemical nature of these products. Because the products of decomposition have now been correctly identified, researchers can look for ways to prevent degradation of the material, leading to more stable materials for use in the future.

A pervasive problem in academic research is often the inability to scale up experiments for use in industry. While perovskite films can be made with relative ease on a small scale in the laboratory, they can be difficult to replicate on the large scale needed for mass production. New research from Dr. Matthew Leyden in the Journal of Materials Chemistry A has the potential to make industrial production of perovskites much easier. His work uses chemical vapor deposition, a cost-effective process commonly used in industry, to create large solar cells and modules of FAPbI3 perovskites. This is one of the first demonstrations of perovskite solar cells and modules fabricated by a method widely employed in industry, making the mass production of perovskite films more feasible. The solar cells and modules produced are significantly larger, e.g., 12 cm 2 , than those commonly studied in academia, typically <0.3cm 2 . These solar modules show enhanced thermal stability and relatively high efficiencies, which is impressive as many perovskite solar cells lose efficiency drastically as they are scaled up, making this type of research useful for commercial purposes.

Research from Prof. Qi's research unit has brought perovskite solar cells one step closer to mass production by providing solutions to problems of efficiency, life-span, and scalability. With more exciting research on the horizon, the unit is bringing the dream of utilizing cost-effective renewable energy resources into reality.

## Abraham's Three Truths of Astronomy

Contents
 1. Law and Order 1.1 An Organized System 1.2 Kepler's Three Laws 1.3 Bode's Law 2. Abraham's Truths 2.1 Abraham's First Truth 2.2 Abraham's Second Truth 2.3 Abraham's Third Truth Notes
Abraham's great vision on astronomy revealed the design of the heavens and can be summarized as three truths.

Is the universe ruled by intelligence or chance? Modern science is currently enamored with the theory that chance rules the universe, from the tiny atomic scales of quantum mechanics to our entire universe of the supposed "Big Bang." The Lord, however, has made it clear that intelligence governs. Even for believers, however, the question arises, "Just how much of a role does chance play?" For example, did our solar system form naturally according to God's laws from a condensing cloud of interstellar matter with planets having more or less random periods of rotation and revolution, or was it designed as a precision timepiece? This article proposes that Abraham's vision on astronomy (Abraham 3) is the key to understanding that the Lord designed our solar system to be a Great Timepiece.

When the anti-Christ Korihor refused to believe in God, Alma used the high degree of order found in the solar system as a proof of the existence of a Creator. Alma refuted Korihor's agnostic teachings by declaring, "all the planets which move in their regular form do witness that there is a Supreme Creator" (Alma 30:44). Atheists today would not be convinced by Alma's argument because most scientists believe that our solar system formed according to natural law and that all can be explained without God. Indeed, scientists have discovered many wonderful natural laws such as Kepler's laws and Newton's laws which the planets all faithfully follow. Computer studies have shown how a random collection of particles could form into a solar system, with a sun and planets very much like our own. So how would Alma's argument fare today, with new age atheists claiming that the solar system simply created itself, according to these apparently self existent laws of physics? Should the witness of the planets be relegated to an unenlightened age during which Korihor was forced to remain mute only because he didn't have today's scientific responses at the tip of this tongue? If Korihor had lived today, and had quoted all the modern laws of physics and principles of self-organization, could Alma have had an answer for him?

Alma argued for a Creator from the orderly planetary motion.
This article proposes that Alma could respond today that the solar system is far more orderly than the known laws of physics would predict, and hence a Creator is still required to explain it. This order leads to the proposal that the solar system is indeed a Great Timepiece, designed to display time very much like an ordinary clock. This huge clock in the sky is a testimony that there is indeed a Watchmaker who created it. Moreover, as we have seen in past articles, it appears that the Watchmaker is using his Timepiece to schedule key events of history.

The Lord has prophesied that

This passage makes it clear that the times of revolutions of the sun, moon, and stars have not only been appointed, but that their "set times" shall be revealed in this current dispensation. Moreover, even the timing of when those set times would be revealed was planned in the great Council before this world was created. Modern astronomers have discovered the periods of rotation and revolution of the planets in our solar system to a very high precision, which seems to qualify as at least partially fulfilling this prophecy that their times of revolutions would be revealed in our days. The keys to discovering the incredible order and precision of the solar system, however, and each of the "set times," are given in the great revelation on astronomy in the Book of Abraham.

Let us begin by discussing some of the ways that scientists recognize intelligent forces in action, review some examples of order found in our solar system, and then begin to explore in detail how the Book of Abraham unfolds the workings of the Lord's timepiece.

## 1. Law and Order

Of course, when scientists make the distinction between an intelligence and a natural law, they usually avoid the question, "Who wrote the laws of nature?" The scriptures tell us: "God . hath given a law unto all things, by which they move in their times and their seasons" (D&C 88:41-42). Thus, in reality, at least two of the three explanations of order actually involve intelligence because God wrote the laws of nature.

Let us now discuss some of the higher order found in the solar system, and decide for ourselves whether or not it should be attributed to "natural laws," to mere chance, or to the hand of the Creator (D&C 59:21).

### 1.1 An Organized System

Planets orbit the sun in ellipses.
The solar system is composed of the sun, a number of planets which revolve around it, their moons, and other miscellaneous objects including comets and asteroids. One of the greatest signs of order in the solar system is that the sun and all of the principal planets are located nearly in a plane and all revolve around the sun in the same direction. [1] A similar phenomenon is seen throughout our universe: there are many flattened disks of stars, which apparently are nearly all revolving in orbits in the same direction around a center. Statistically, that is so incredibly unlikely to have happened by chance that either a law or an intelligent ordering is clearly indicated.

Science and scripture agree on the explanation of this ordering: orbiting bodies all follow a law (D&C 88:42). Newton's laws of motion and his law of gravity explain all but the tiniest deviations in their orbits, and Einstein's modifications account for the rest. Thus, these laws greatly simplify God's creative work, because his law includes self-organization. That is, even if one starts with a random accumulation of particles, there will almost always be some amount of spinning of the group as a whole. As the particles interact with each other, they will tend to fall into the plane of rotation and stay there in an orbit as the gravitational force toward the center is balanced by the inertial reaction away from it.

### 1.2 Kepler's Three Laws

Planets move faster when near the sun.
Three other regular motions discovered in the solar system were articulated by Kepler. His first law is that the planets do not move in perfect circles around the sun, but in ellipses. His second law is that the speed of the planet revolving around the sun is slower when it is farther away from the sun and faster when it is closer, according to a precise mathematical law. His third law is that a planet's orbital period is determined solely by an average distance from the sun, again according to a mathematical law, and not by its size, shape or mass. When Newton arrived on the scene, he showed that all three of these laws were derivable in turn from his three laws of motion and his law of gravity. These self-organizing natural laws provide for a very orderly universe.

### 1.3 Bode's Law

Planetary periods depend on half the long axis of the ellipse.
The order of the solar system just reviewed summarizes most of the ordering which has been discovered by scientists in the solar system. [3] It can all be achieved by beginning with a random set of particles governed by the known laws of physics. Hence, no compelling evidence for including a Creator in the history of the formation of the solar system has been scientifically required.

There is one important area, however, where most scientists have not noticed any order and therefore have not needed to explain it. That is in the precise periods of revolution of the planets around the sun. Both the earth's period of rotation (the day) and of revolution (the year) have been assumed to be simply random periods of time, determined by whatever chance initial angular momentum our planet happened to have when the solar system formed. Similarly, the periods of revolution of all of the planets are thought to be essentially random numbers, except for being within approximate whole-number ratios. Interestingly, that is precisely the area where the Lord revealed to Abraham that there is a high degree of order designed into our system. Let us now look in detail at what is contributed to this subject by the Book of Abraham.

## 2. Abraham's Truths

Abraham's vision occurred at night.
Even as we have Kepler's laws, Newton's Laws, and Bode's Law, the Lord revealed to Abraham what could be called "Abraham's Truths." These three truths differ somewhat from the other laws in that they each have to do with design principles for the solar system, rather than deduced mathematical relationships. I refer to them as "truths" rather than laws because they enjoy a privileged status not common in science. Science deals with theories, and makes no claim to have absolute truth. Most scientists have not had the privilege of being invited to see the design of the entire system, as taught by the Designer himself. Let us review just what the Lord showed Abraham in that marvelous vision, and restate the results as Abraham's Three Truths.

### 2.1 Abraham's First Truth

Here we are given a great truth, directly from the Creator, who told us what would otherwise be incredibly hard to discover by ourselves. The period of one earth year (365.242 days) is not a random number at all, but was designed to be such that 1,000 of our years is equal to a day of the Lord. Scientists may not have even discovered Kolob yet, much less its period of revolution. And if we had, how could we know that the earth's orbit was designed such that 1,000 years is one revolution of Kolob, rather than it just being a chance coincidence? The place where the Lord's revelations are the most important seems to be in the areas where man could never discover those truths by himself.

Although we have been told elsewhere that a day unto the Lord is as a thousand years (2 Peter 3:8), one could get the impression that a "day of the Lord" might be a vague term, meaning simply a "long time." When Abraham actually saw Kolob, and heard the Lord explain the design, he must have clearly understood that one revolution of Kolob precisely determines one day of the Lord, even as one rotation of the earth determines one day on earth.

Kolob, which governs all of the stars of the order to which we belong, apparently also is the master clock of the entire order. It is "first in government, the last pertaining to the measurement of time" (Fac. 2, Fig. 1). As an example of Kolob being a time standard, we are told that, "Oliblish, which is the next grand governing creation" has a time period "which is equal with Kolob in its revolution and in its measuring of time" (Fac. 2, Figs. 2, 4). Thus, Kolob is like the heart of the order of stars to which we belong. It both governs the order, and its heartbeat rate is the time regulator even as a quartz crystal regulates our watches.

Thus, let us summarize this information as Abraham's First Truth:

One revolution of Kolob is one day unto the Lord, being 1,000 earth years.

The significance is that the length of the year is not a random unit of time, but was designed to be 1/1,000th of one day of the Lord. We are not told the precision of this statement, so it is not clear whether a revolution of Kolob is 365,242 earth days, or perhaps 365,000. The important point here is that the length of our year is not a random number but was carefully designed.

### 2.2 Abraham's Second Truth

The Lord showed Abraham more of the workings of our solar and stellar systems. Abraham was given to know the "set times" of the sun, moon, and earth:

Let us restate this as Abraham's Second Truth:

What does "set time" mean? Grant Athay, one of the first L.D.S. astronomers to attempt any interpretation at all of the Book of Abraham, tentatively proposed a meaning for the set times of the earth, moon and sun. He said he suspected that the set time of the earth refers to the day, the set time of the moon to the lunar month, in which the moon completes its cycle of phases, and the set time of the sun to the tropical year, in which the sun makes its annual excursion through the north and south parts of the sky, causing the seasons. However, he took that interpretation no further, noting that, "Those parts of the Book of Abraham that discuss set periods of time for the sun, moon, and planets do not invoke a strong interest from astronomers." [4] He must be correct, because very few other professional astronomers have commented at all on the Book of Abraham.

The earth and moon have set times.
The Hebrew calendar also includes the time interval of the week of seven days built into it. [5] Mathematically, the 7-day week is the best interval for measuring both the month and the year because a month is nearly 4 weeks and a year is nearly 52 weeks. Again, this shows the consistency of the Lord in telling Moses to use 7-day weeks and also to use a lunisolar calendar.

The importance of Abraham's Second Truth is that no one needs to apologize for the Hebrew calendar being more complicated than our simple (Gregorian) solar calendar. It is the result of design, not chance. Moreover, the Designer seems to be using such a lunisolar calendar, as we would expect. That is, many key events in religious history have occurred on days of the Hebrew calendar which the Lord has designated as holy, as I have discussed at length in my other articles. [6]

The set time of the Earth. The set time of the earth apparently refers to the mean solar day. The day is a very stable unit of time because the rotating earth is such a good clock that it only slows down by about 1.5 milliseconds (thousandths of a second) per century. [7]

The Hebrew value for the "set time" of the moon is super accurate.
The set time of the moon. The Lord stated that Abraham knew the set time of the moon. The value for the average length of the lunar month on which the Hebrew calendar has long been based is 29.530594 days. [8] That value is far better than any other used in antiquity, and today's calculation of the average value (29.530593 days) only differs by 0.000001 day, which is less than a tenth of a second. The Hebrew value is so phenominally good that I've believed for years that it must have been revealed and that the lunar orbit was designed to come out even in Hebrew time units. [9] This revelation to Abraham might explain the origin of this super-accurate value. It is also possible, however, that the value had been known by Enoch, and was contained in the records in Abraham's possession (Abr. 1:31). The revelation states only that he knows it, not that it was being revealed at that time.

The set time of the sun. Abraham also knew the set time of the sun, which apparently means the length of the seasonal year. Our best modern estimate for a historical average is 365.2425 days, [10] which is the value used in our modern Gregorian calendar.

The sun's set time was known to Abraham.
The length of the year is not nearly as fixed as the length of the lunar month, although it is still a very precise unit of time. The current value for the length of the year is 365.2422 days, which shows a variation of 0.0003 days from the 7,000-year average value of 365.2425. In contrast, the current value for the length of the month is 29.530588 days, which shows a deviation of only 0.000005 days from the historical average value of 29.530593. That means that a calendar based on a fixed length for the cycle of the moon, as is the Hebrew calendar, is a much more accurate calendar than one based on a fixed length for the sun's seasons, like our Gregorian calendar.

The reason for this difference in stability might be worth noting because it again suggests design in the solar system. Various frictional effects, especially the tides, cause the earth to slow down in its rotational speed very slightly, causing the day to lengthen by about 0.0015 seconds every century. That effect causes the length of the year as measured in days slowly to decrease because the number of rotations of the earth is less in every orbital revolution, meaning a smaller number of days per year. There is also another effect equally as important to the length of the year. As the earth encounters frictional drag from encountering particles in its path as it speeds around the sun, it loses energy and the distance to the sun sightly decreases (by less than an inch per year), which also causes the length of the year to shorten by about the same amount caused by the lengthening of the day. Thus the two effects combine to double the shortening rate of the length of the year as measured in mean solar days.

On the other hand, as the tides slosh around, they tend always to rotate ahead of the moon which is causing them, and this effect actually accelerates the moon in its orbit around the earth, causing the moon slowly to recede from the earth. This effect slowly increases the length of the month and tends to compensate for the lengthening of the day. In other words, even though the length of the month is getting longer, so also is the length of the day, so that the number of days in a month is nearly constant. Hence, the month is a much better length of time on which to base a calendar than is the sun, which again points to the Hebrew lunisolar calendar as being inspired by the Designer. [11] So far, however, the atheist will not be impressed with the "set times" of the earth, moon and sun, because they still appear to be random numbers. It is as we explore the Third Truth that the design of the "set times" will become more apparent.

### 2.3 Abraham's Third Truth

Let us restate this concept more briefly as Abraham's Third Truth:

### Earth-based View

The Lord's Timepiece is to be viewed from earth.
One key point from the revelation is that the progression starts from the earth, from which it was clearly designed to be viewed. What use would a great clock in the sky be to man if it had to be viewed from the sun? When the Lord told Abraham to begin at the earth and to count the moon as second in the progression, he was not implying that the earth is fixed nor that is in the center of the universe. He was simply explaining the design of the solar system, namely that there is a progression of planets designed to keep time, as seen from the earth. In that progression of "one planet above another" the moon "standeth above the earth" because it "moveth in order more slow," meaning the earth is first and the moon, in the slot above it, being second in the order of increasing periods. The Lord implied that the sun stands above the moon in this progression because its period (the year) is greater than the moon's (the lunar month).

The year can be thought of two different ways. It can be thought of as the period of the sun's annual motion through the sky which causes the seasons, which is the view of the Book of Abraham. We moderns tend to think of the year as the period of the earth's orbit around the sun, rather than the sun's motion relative to the earth. These are simply two ways at looking at the same relative motion. The Lord is explaining that the system was set up to be viewed from the earth, and that the period of one earth year is referred to as the "set time" of the sun.

Abraham was shown the governing stars near to the throne of God, and they were a long way away from the earth. Clearly, Abraham did not think that the earth was the center of the universe, for it was dwarfed by the brilliant stars near to the throne of God. It is Kolob, which is near the center of government and of time-keeping, which is most likely also near the center of the Lord's creations. The prophets have long known that "it is the earth that moveth and not the sun" (Hel. 12:15). The Lord was merely explaining that the Great Timepiece of the solar system was designed to be viewed from the earth.

Next month's article will take a detailed look at the planetary progression in our solar system, referred to in Abraham's Third Truth, which progression also continues with certain stars, until we come nigh unto Kolob.

## Modern Redefinitions of the Meter

In 1960, the official definition of the meter was changed again. As a result of improved technology for generating spectral lines of precisely known wavelengths (see the chapter on Radiation and Spectra), the meter was redefined to equal 1,650,763.73 wavelengths of a particular atomic transition in the element krypton-86. The advantage of this redefinition is that anyone with a suitably equipped laboratory can reproduce a standard meter, without reference to any particular metal bar.

In 1983, the meter was defined once more, this time in terms of the velocity of light. Light in a vacuum can travel a distance of one meter in 1/299,792,458.6 second. Today, therefore, light travel time provides our basic unit of length. Put another way, a distance of one light-second (the amount of space light covers in one second) is defined to be 299,792,458.6 meters. That’s almost 300 million meters that light covers in just one second light really is very fast! We could just as well use the light-second as the fundamental unit of length, but for practical reasons (and to respect tradition), we have defined the meter as a small fraction of the light-second.

## Detection of cosmic effect may bring universe's formation into sharper focus

The first observation of a cosmic effect theorized 40 years ago could provide astronomers with a more precise tool for understanding the forces behind the universe's formation and growth, including the enigmatic phenomena of dark energy and dark matter.

A large research team from two major astronomy surveys reports in a paper submitted to the journal Physical Review Letters that scientists detected the movement of distant galaxy clusters via the kinematic Sunyaev-Zel'dovich (kSZ) effect, which has never before been seen. The paper was recently posted on the arXiv preprint database, and was initiated at Princeton University by lead author Nick Hand as part of his senior thesis. Fifty-eight collaborators from the Atacama Cosmology Telescope (ACT) and the Baryon Oscillation Spectroscopic Survey (BOSS) projects are listed as co-authors.

Proposed in 1972 by Russian physicists Rashid Sunyaev and Yakov Zel'dovich, the kSZ effect results when the hot gas in galaxy clusters distorts the cosmic microwave background radiation -- which is the glow of the heat left over from the Big Bang -- that fills our universe. Radiation passing through a galaxy cluster moving toward Earth appears hotter by a few millionths of a degree, while radiation passing through a cluster moving away appears slightly cooler.

Now that it has been detected, the kSZ effect could prove to be an exceptional tool for measuring the velocity of objects in the distant universe, the researchers report. It could provide insight into the strength of the gravitational forces pulling on galaxy clusters and other bodies. Chief among these forces are the still-hypothetical dark energy and dark matter, which are thought to drive the universe's expansion and the motions of galaxies.

In addition, the strength of the kSZ effect's signal depends on the distribution of electrons in and around galaxies. As a result, the effect also can be used to trace the location of atoms in the nearby universe, which can reveal how galaxies form.

The benefits of the kSZ effect stem from a unique ability to pinpoint velocity, said Hand, a 2011 Princeton graduate who is now a graduate student in astronomy at the University of California-Berkeley. The researchers detected the motion of galaxy clusters that are several billion light years away moving at velocities of up to 600 kilometers (372 miles) per second.

"Traditional methods of measuring velocities require very precise distance measurements, which is difficult. So, these methods are most useful when objects are closer to Earth," Hand said.

"One of the main advantages of the kSZ effect is that its magnitude is independent of a galaxy cluster's distance from us, so we can measure the velocity of an object's motion toward or away from Earth at much larger distances than we can now," Hand said. "In the future, it can provide an additional statistical check that is independent of our other methods of measuring cosmological parameters and understanding how the universe forms on a larger scale."

Pedro Ferreira, an astrophysics professor at the University of Oxford, called the paper a "beautiful piece of work" that neatly demonstrates an accurate method for studying the evolution of the universe and the distribution of matter in it. Ferreira had no role in the research but is familiar with it.

"This is the first time the kSZ effect has been unambiguously detected, which in and of itself is a really important result," Ferreira said.

"By probing how galaxies and clusters of galaxies move around in the universe, the kSZ effect is directly probing how objects gather and evolve in the universe," he said. "Therefore it is hugely dependent on dark matter and dark energy. You can then think of the kSZ effect as a completely new window on the large-scale structure of the universe."

Combining fundamentally different data

To find the kSZ effect, the researchers combined and analyzed data from the ACT and BOSS projects. The kSZ effect is so small that it is not visible from the interaction with an individual galaxy cluster with the cosmic microwave background (CMB), but can be detected by compiling signals from several clusters, the researchers discovered.

ACT is a custom-designed 6-meter telescope in Chile built to produce a detailed map of the CMB using microwave frequencies. The ACT collaboration involves a dozen universities with leading contributions from Princeton and the University of Pennsylvania, and includes important detector technology from NASA's Goddard Space Flight Center, the National Institute of Standards and Technology, and the University of British Columbia.

BOSS, a visible-light survey based at the Apache Point Observatory in New Mexico, has captured spectra of thousands of luminous galaxies and quasars to improve understanding of the large-scale structure of the universe. BOSS is a part of the Sloan Digital Sky Survey III, the third phase of the most productive astronomy project in history, and a joint effort among 27 universities and institutions from around the world.

For the current project, researchers from ACT compiled a catalog of 27,291 luminous galaxies from BOSS that appeared in the same region of sky mapped by ACT between 2008 and 2010. Because each galaxy likely resides in a galaxy cluster, their positions were used to determine the locations of clusters that would distort the CMB radiation that was detected by ACT.

Hand used the 7,500 brightest galaxies from the BOSS data to uncover the predicted kSZ signal produced as galaxy clusters interacted with CMB radiation. ACT collaborator Arthur Kosowsky, an associate professor of physics and astronomy at the University of Pittsburgh, suggested a particular mathematical average that reflects the slight tendency for pairs of galaxy clusters to move toward each other due to their mutual gravitational attraction, which made the kSZ effect more apparent in the data.

The overlap of data from the two projects was essential because the amplitude of the signal from the kSZ effect is so small, said ACT collaborator David Spergel, professor and department chair of astrophysical sciences at Princeton, as well as Hand's senior thesis adviser. By averaging the ACT's CMB maps with thousands of BOSS galaxy locations, the kSZ signal got stronger in comparison to unrelated signals and measurement errors, Spergel said.

"The kSZ signal is small because the odds of a microwave hitting an electron while passing through a galaxy cluster are low, and the change in the microwave's energy from this collision is slight," said Spergel, the Charles A. Young Professor of Astronomy on the Class of 1897 Foundation. "Including several thousand galaxies in the dataset reduced distortion and we were left with a strong signal."

In fact, if analyzed separately, neither the ACT nor the BOSS data would have revealed the kSZ effect, Kosowsky said. "This result is a great example of an important scientific discovery relying on the rich data from more than one large astronomy survey," he said. "The researchers of the ACT and BOSS collaborations did not have this in mind when they first designed their experiments."

That is because the ACT and BOSS projects are fundamentally different, which makes the researchers' combination of data unique, said SDSS-III scientific spokesman Michael Wood-Vasey, an assistant professor of physics and astronomy at the University of Pittsburgh. The projects differ in the cosmic objects studied, the method of gathering data, and even the wavelengths in which they operate -- microwaves for ACT, visible-light waves for BOSS.

"Collaborations between projects of this scale aren't common in my experience," Wood-Vasey said. "This also was a collaboration after the fact in the sense that the data-acquisition strategies for these projects was already set without thinking about this possibility. The insight of the key researchers on this project allowed them to combine the two datasets and make this measurement."

## How to make motion of the Sun more apparent at seconds scale? - Astronomy

Ptolemy's aim in the Almagest is to construct a kinematic model of the solar system, as seen from the earth. In other words, the Almagest outlines a relatively simple geometric model which describes the apparent motions of the sun, moon, and planets, relative to the earth, but does not attempt to explain why these motions occur (in this respect, the models of Copernicus and Kepler are similar). As such, the fact that the model described in the Almagest is geocentric in nature is a non-issue, since the earth is stationary in its own frame of reference. This is not to say that the heliocentric hypothesis is without advantages. As we shall see, the assumption of heliocentricity allowed Copernicus to determine, for the first time, the ratios of the mean radii of the various planets in the solar system.

We now know, from the work of Kepler, that planetary orbits are actually ellipses which are confocal with the sun. Such orbits possess two main properties. First, they are eccentric : i.e. , the sun is displaced from the geometric center of the orbit. Second, they are elliptical : i.e. , the orbit is elongated along a particular axis. Now, Keplerian orbits are characterized by a quantity, , called the eccentricity , which measures their deviation from circularity. It is easily demonstrated that the eccentricity of a Keplerian orbit scales as , whereas the corresponding degree of elongation scales as . Since the orbits of the visible planets in the solar system all possess relatively small values of ( i.e. , ), it follows that, to an excellent approximation, these orbits can be represented as eccentric circles : i.e. , circles which are not quite concentric with the sun. In other words, we can neglect the ellipticities of planetary orbits compared to their eccentricities. This is exactly what Ptolemy does in the Almagest. It follows that Ptolemy's assumption that heavenly bodies move in circles is actually one of the main strengths of his model, rather than being the main weakness, as is commonly supposed.

Kepler's second law of planetary motion states that the radius vector connecting a planet to the sun sweeps out equal areas in equal time intervals. In the approximation in which planetary orbits are represented as eccentric circles, this law implies that a typical planet revolves around the sun at a non-uniform rate. However, it is easily demonstrated that the non-uniform rotation of the radius vector connecting the planet to the sun implies a uniform rotation of the radius vector connecting the planet to the so-called equant : i.e. , the point directly opposite the sun relative to the geometric center of the orbit--see Fig. 1. Ptolemy discovered the equant scheme empirically, and used it to control the non-uniform rotation of the planets in his model. In fact, this discovery is one of Ptolemy's main claims to fame.

It follows, from the above discussion, that the geocentric model of Ptolemy is equivalent to a heliocentric model in which the various planetary orbits are represented as eccentric circles , and in which the radius vector connecting a given planet to its corresponding equant revolves at a uniform rate. In fact, Ptolemy's model of planetary motion can be thought of as a version of Kepler's model which is accurate to first-order in the planetary eccentricities--see Cha. 4. According to the Ptolemaic scheme, from the point of view of the earth, the orbit of the sun is described by a single circular motion, whereas that of a planet is described by a combination of two circular motions. In reality, the single circular motion of the sun represents the (approximately) circular motion of the earth around the sun, whereas the two circular motions of a typical planet represent a combination of the planet's (approximately) circular motion around the sun, and the earth's motion around the sun. Incidentally, the popular myth that Ptolemy's scheme requires an absurdly large number of circles in order to fit the observational data to any degree of accuracy has no basis in fact. Actually, Ptolemy's model of the sun and the planets, which fits the data very well, only contains 12 circles ( i.e. , 6 deferents and 6 epicycles).

Ptolemy is often accused of slavish adherence to the tenants of Aristotelian philosophy, to the overall detriment of his model. However, despite Ptolemy's conventional geocentrism, his model of the solar system deviates from orthodox Aristotelism in a number of crucially important respects. First of all, Aristotle argued--from a purely philosophical standpoint--that heavenly bodies should move in single uniform circles . However, in the Ptolemaic system, the motion of the planets is a combination of two circular motions. Moreover, at least one of these motions is non-uniform . Secondly, Aristotle also argued--again from purely philosophical grounds--that the earth is located at the exact center of the universe, about which all heavenly bodies orbit in concentric circles. However, in the Ptolemaic system, the earth is slightly displaced from the center of the universe. Indeed, there is no unique center of the universe, since the circular orbit of the sun and the circular planetary deferents all have slightly different geometric centers, none of which coincide with the earth. As described in the Almagest, the non-orthodox (from the point of view of Aristolelian philosophy) aspects of Ptolemy's model were ultimately dictated by observations . This suggests that, although Ptolemy's world-view was based on Aristolelian philosophy, he did not hesitate to deviate from this standpoint when required to by observational data.

From our heliocentric point of view, it is easily appreciated that the epicycles of the superior planets ( i.e. , the planets further from the sun than the earth) in Ptolemy's model actually represent the earth's orbit around the sun, whereas the deferents represent the planets' orbits around the sun--see Fig. 29. It follows that the epicycles of the superior planets should all be the same size ( i.e. , the size of the earth's orbit), and that the radius vectors connecting the centers of the epicycles to the planets should always all point in the same direction as the vector connecting the earth to the sun.

We can also appreciate that the deferents of the inferior planets ( i.e. , the planets closer to the sun than the earth) in Ptolemy's model actually represent the earth's orbit around the sun, whereas the epicycles represent the planets' orbits around the sun--see Fig. 33. It follows that the deferents of the inferior planets should all be the same size ( i.e. , the size of the earth's orbit), and that the centers of the epicycles (relative to the earth) should all correspond to the position of the sun (relative to the earth).

Figure 1: Hipparchus' (and Ptolemy's) model of the sun's apparent orbit about the earth (right) compared to the optimal model (left). The radius vectors in both models rotate uniformly. Here, is the sun, the earth, the geometric center of the orbit, the equant, the perigee, and the apogee. The radius of the orbit is normalized to unity.

The geocentric model of the solar system outlined above represents a perfected version of Ptolemy's model, constructed with a knowledge of the true motions of the planets around the sun. Not surprisingly, the model actually described in the Almagest deviates somewhat from this ideal form. In the following, we shall refer to these deviations as errors'', but this should not be understood in a perjorative sense.

Ptolemy's first error lies in his model of the sun's apparent motion around the earth, which he inherited from Hipparchus. Figure 1 compares what Ptolemy actually did, in this respect, compared to what he should have done in order to be completely consistent with the rest of his model. Let us normalize the mean radius of the sun's apparent orbit to unity, for the sake of clarity. Ptolemy should have adopted the model shown on the left in Fig. 1, in which the earth is displaced from the center of the sun's orbit a distance (the eccentricity of the earth's orbit around the sun) towards the perigee (the point of the sun's closest approach to the earth), and the equant is displaced the same distance in the opposite direction. The instantaneous angular position of the sun is then obtained by allowing the radius vector connecting the equant to the sun to rotate uniformly at the sun's mean orbital angular velocity. Of course, this implies that the sun rotates non-uniformly about the earth. Ptolemy actually adopted the Hipparchian model shown on the right in Fig. 1. In this model, the earth is displaced a distance from the center of the sun's orbit in the direction of the perigee, and the sun rotates at a uniform rate ( i.e. , the radius vector rotates uniformly). It turns out that, to first-order in , these two models are equivalent in terms of their ability to predict the angular position of the sun relative to the earth--see Cha. 4. Nevertheless, the Hippachian model is incorrect, since it predicts too large (by a factor of ) a variation in the radial distance of the sun from the earth (and, hence, the angular size of the sun) during the course of a year (see Cha. 4). Ptolemy probabaly adopted the Hipparchian model because his Aristotelian leanings prejudiced him in favor of uniform circular motion whenever this was consistent with observations. (It should be noted that Ptolemy was not interested in explaining the relatively small variations in the angular size of the sun during the year--presumably, because this effect was difficult for him to accurately measure.)

Ptolemy's next error was to neglect the non-uniform rotation of the superior planets on their epicycles. This is equivalent to neglecting the orbital eccentricity of the earth (recall that the epicycles of the superior planets actually represent the earth's orbit) compared to those of the superior planets. It turns out that this is a fairly good approximation, since the superior planets all have significantly greater orbital eccentricities than the earth. Nevertheless, neglecting the non-uniform rotation of the superior planets on their epicycles has the unfortunate effect of obscuring the tight coupling between the apparent motions of these planets, and that of the sun. The radius vectors connecting the epicycle centers of the superior planets to the planets themselves should always all point exactly in the same direction as that of the sun relative to the earth. When the aforementioned non-uniform rotation is neglected, the radius vectors instead point in the direction of the mean sun relative to the earth. The mean sun is a fictitious body which has the same apparent orbit around the earth as the real sun, but which circles the earth at a uniform rate. The mean sun only coincides with the real sun twice a year.

Ptolemy's third error is associated with his treatment of the inferior planets. As we have seen, in going from the superior to the inferior planets, deferents and epicycles effectively swap roles. For instance, it is the deferents of the inferior planets, rather than the epicycles, which represent the earth's orbit. Hence, for the sake of consistency with his treatment of the superior planets, Ptolemy should have neglected the non-uniform rotation of the epicycle centers around the deferents of the inferior planets, and retained the non-uniform rotation of the planets themselves around the epicycle centers. Instead, he did exactly the opposite. This is equivalent to neglecting the inferior planets' orbital eccentricities relative to that of the earth. It follows that this approximation only works when an inferior planet has a significantly smaller orbital eccentricity than that of the earth. It turns out that this is indeed the case for Venus, which has the smallest eccentricity of any planet in the solar system. Thus, Ptolemy was able to successfully account for the apparent motion of Venus. Mercury, on the other hand, has a much larger orbital eccentricity than the earth. Moreover, it is particularly difficult to obtain good naked-eye positional data for Mercury, since this planet always appears very close to the sun in the sky. Consequently, Ptolemy's Mercury data was highly inaccurate. Not surprisingly, then, Ptolemy was not able to account for the apparent motion of Mercury using his standard deferent-epicycle approach. Instead, in order to fit the data, he was forced to introduce an additional, and quite spurious, epicycle into his model of Mercury's orbit.

Ptolemy's fourth, and possibly largest, error is associated with his treatment of the moon. It should be noted that the moon's motion around the earth is extremely complicated in nature, and was not fully understood until the early 20th century CE. Ptolemy constructed an ingenious geometric model of the moon's orbit which was capable of predicting the lunar ecliptic longitude to reasonable accuracy. Unfortunately, this model necessitates a monthly variation in the earth-moon distance by a factor of about two, which implies a similarly large variation in the moon's angular diameter. However, the observed variation in the moon's diameter is much smaller than this. Hence, Ptolemy's model is not even approximately correct.

## The Equation of Time

Universal time (clock time) is based on the motion of an imaginary “mean sun”. This can differ from local solar time, even on the Greenwich meridian, by about 15 minutes over the course of the year. (Note- the difference between a solar day and 24 hours may only be up to 30 seconds but the difference can add up day after day reaching a peak of +14 minutes or -16 minutes).

The Equation of Time is a means of calculating the difference for any particular date. The easiest way to do it is to enter your data into a computer program and let it calculate it all for you. There is one at http://www.go.ednet.ns.ca/

One must also take into account your longitude as the Sun will be at its highest for people at different longitudes. For every degree east of the time zone meridian ( 0 if you live in the UK) you must subtract 4 minutes from your local time.