# Why is the Sun's brightness and radius increasing, but not its temperature?

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On the Sun's article on Wikipedia, there is an image showing how the Sun's brightness, radius and temperature have changed over time:

For the past (and next) few billion years, I see the luminosity continuously rising, the temperature remaining about the same, and the radius also rising.

This seems bizarre to me. I would imagine that, for luminosity to increase, the temperature would have to increase. And for temperature to increase, the radius would have to decrease (as the core becomes denser, temperatures rise).

So why does it work this way?

PS: I would also appreciate a concrete explanation for why all three decreased at the beginning of the Sun's life, but suddenly began increasing.

The effective temperature $T_mathrm{eff}$ of a star, which is presumably what's been plotted, is defined through its relationship with the star's radius $R$ and luminosity $L$ by

$$L=4pi R^2sigma T_mathrm{eff}^4$$

This comes from the assumption that the star radiates like a black body at the photosphere. While this isn't strictly true, it's quite accurate, and regardless, that's how we define the effective temperature. The actual surface temperature will be slightly different but also behave roughly as plotted.

So, even if $T_mathrm{eff}$ is constant, the star expands if it grows brighter. Also, you can see that the sensitivity to temperature is steeper than radius, so a moderate change in luminosity can be absorbed by a relatively small change in effective temperature.

While the luminosity is basically determined by the simple behaviour of the nuclear reactions in the core (in terms of temperature and density), the surface properties depend on how energy is being transported near the surface. For radiation, you have to consider what the opacity of the material is, which itself depends on ionization states and whatnot. It's easy enough to see why the luminosity grows (the core gets denser and also hotter, producing energy faster) but the determination of the surface properties is more complicated. For the Sun, it turns out the way shown in the plot after you solve all the equations with the relevant opacities.

Also, as an extreme counterexample to "brighter means hotter and smaller", remember that red giants are much brighter but also much cooler!

PS: I'm not sure the source of the data but I would guess the wiggle at the start is because of the star finishing its contraction onto the main sequence. That is, before the first minimum, energy is being released by gravitational contraction. After it, the energy from nuclear reactions starts to dominate.

## What is the relationship between stellar temperature, radius, and luminosity?

For stars in their main sequence, as stellar mass increases, so do diameter, temperature and luminosity. The relationship is represented in the Hertzsprung-Russel diagram.

#### Explanation:

In the H-R diagram shown below, the brightness (luminosity) is presented on the y axis, and temperature on the x axis (from right to left). The main sequence is the population of stars shown diagonally from top left to bottom right.

Brightness clearly increases with temperature, and with any incandescent (glowing from heat) object, the hotter the object the bluer its light.

What makes a star hotter is a more rapid rate of fusion in the core, which is driven by higher pressure from higher mass.
So the bigger the star (mass and diameter), the brighter it is, the hotter it is, and the bluer it is. Smaller stars are cooler and redder.

Stars off the main sequence - red giants and white dwarfs - don't follow the same pattern. Red giants produce tremendous energy, but they are puffed up, so the surface area is massively increased. As a result, their surface temperature is low, so they are bright but red.

White dwarfs are dying naked stellar cores and very small. They produce less energy, but have a very high surface temperature, so white but dim.

## How does distance affect brightness of stars?

The intensity or brightness of light as a function of the distance from the light source follows an inverse square relationship. Notice that as the distance increases, the light must spread out over a larger surface and the surface brightness decreases in accordance with a "one over r squared" relationship.

Also, what are the factors that determine the brightness of a star? In conclusion, many factors affect the brightness of a star, and these include (but are not limited to) surface area, mass, evolutionary stage, temperature, and distance (if you are talking about apparent magnitude).

Likewise, people ask, how do you measure the brightness of a star?

We measure the brightness of these stars using the magnitude scale. The magnitude scale seems a little backwards. The lower the number, the brighter the object is and the higher the number, the dimmer it is. This scale is logarithmic and set so that every 5 steps up equals a 100 times decrease in brightness.

How is brightness calculated?

The brightness of a light emitting diode (or anything else) is measured by luminous flux, which is the impact of the light on an eye, adjusted for different wavelengths. Luminous flux is measured by lumens, which corresponds to candelas of light, or luminous intensity, emitted over a solid angle.

## The Proto-Sun

The proto-Sun slowly contracted while embedded deep within its birth cloud.

At this stage it was only visible as a bright infrared source, since only infrared light can penetrate the surrounding gas and dust clouds.

For example, below are two views of a present-day stellar nursery, the Orion Nebula.

Credit: OSU Astronomy Department

The view on the left is what is seen in visible light. On the right is the same view at infrared wavelengths. Notice how only a handful of faint red stars are seen at visible wavelengths, while in the infrared, we can peer through the dusty Orion molecular cloud and see the rich cluster of young stars recently formed deep within. The quartet of bright stars in the center of the nebula, known collectively as the "Trapezium", will eventually blow away much of the surrounding gas and dust. When that happens in a million years or so, the rich cluster will be visible in the night sky of earth.

A disk of material formed around the proto-Sun, out of which the planets were formed.

## Why is the Sun's brightness and radius increasing, but not its temperature? - Astronomy

The dependence on mass comes about because the sheer weight of the star's mass determines its central pressure, which in turn determines its rate of nuclear burning (higher pressure = more collisions = more energy), and the resulting fusion energy is what drives the star's temperature. In general, the more massive a star is, the brighter and hotter it must be. It is also the case that the gas pressure at any depth in the star (which also depends on the temperature at that depth) must balance the weight of the gas above it. And finally, of course, the total energy generated in the core must equal the total energy radiated at the surface.

This last fact generates yet another constraint, because the energy radiation of a sphere suspended in a vacuum obeys a law known as the Stefan-Boltzmann Equation:

L = C R 2 T 4 (Total luminosity of a hot sphere)

Here L is the luminosity of the star, C is a constant 1 , R is the radius of the star in meters, and T is the surface temperature of the star in K°. Note how swiftly the energy radiated by a star rises with T: doubling the temperature causes its energy output to increase by 16 times.

1 &ndash Very well, if you must know, the constant is equal to 5.67 x 10 -8 W m -2 K -4 .

The main sequence exists precisely because of the inflexible nature of hydrostatic equilibrium. Stars with very low masses (as little as 7.5% that of the Sun) lie at the lower right of the H-R diagram. They must lie at the lower right. This part of the H-R diagram corresponds to extremely low luminosity &ndash as little as a ten thousandth that of the Sun &ndash and low surface temperature, equivalent to the dull orange-yellow glow of molten metal. These stars do not have enough mass to create the pressure necessary to make the nuclear burning in their cores go any faster. High-mass stars (upwards of 40 solar masses) reside at the upper left, as they must. Contrary to the low-mass stars, their immense masses and high central pressures give rise to giants that can be 160,000 times more luminous than the Sun, and so hot that they give off more energy in the ultraviolet than they do as visible light. The Sun lies almost exactly halfway between these extremes, and thus it is neither extremely dim nor extremely bright as stars go. It shines with a bright yellowish-white color.

2 &ndash Astronomers traditionally classify main-sequence stars with letters, like so:
O - 30,000 to 40,000 K°
B - 10,800 to 30,000 K°
A - 7240 to 10,800 K°
F - 6000 to 7240 K°
G - 5150 to 6000 K°
K - 3920 to 5150 K°
M - 2700 to 3920 K°

Since its birth 4.5 billion years ago, the Sun's luminosity has very gently increased by about 30%. 3 This is an inevitable evolution which comes about because, as the billions of years roll by, the Sun is burning up the hydrogen in its core. The helium "ashes" left behind are denser than hydrogen, so the hydrogen/helium mix in the Sun's core is very slowly becoming denser, thus raising the pressure. This causes the nuclear reactions to run a little hotter. The Sun brightens.

## Thought Questions

Someone suggests that astronomers build a special gamma-ray detector to detect gamma rays produced during the proton-proton chain in the core of the Sun, just like they built a neutrino detector. Explain why this would be a fruitless effort.

Earth contains radioactive elements whose decay produces neutrinos. How might we use neutrinos to determine how these elements are distributed in Earth’s interior?

The Sun is much larger and more massive than Earth. Do you think the average density of the Sun is larger or smaller than that of Earth? Write down your answer before you look up the densities. Now find the values of the densities elsewhere in this text. Were you right? Explain clearly the meanings of density and mass.

A friend who has not had the benefit of an astronomy course suggests that the Sun must be full of burning coal to shine as brightly as it does. List as many arguments as you can against this hypothesis.

Which of the following transformations is (are) fusion and which is (are) fission: helium to carbon, carbon to iron, uranium to lead, boron to carbon, oxygen to neon? (See Appendix K for a list of the elements.)

Why is a higher temperature required to fuse hydrogen to helium by means of the CNO cycle than is required by the process that occurs in the Sun, which involves only isotopes of hydrogen and helium?

Earth’s atmosphere is in hydrostatic equilibrium. What this means is that the pressure at any point in the atmosphere must be high enough to support the weight of air above it. How would you expect the pressure on Mt. Everest to differ from the pressure in your classroom? Explain why.

Explain what it means when we say that Earth’s oceans are in hydrostatic equilibrium. Now suppose you are a scuba diver. Would you expect the pressure to increase or decrease as you dive below the surface to a depth of 200 feet? Why?

What mechanism transfers heat away from the surface of the Moon? If the Moon is losing energy in this way, why does it not simply become colder and colder?

Suppose you are standing a few feet away from a bonfire on a cold fall evening. Your face begins to feel hot. What is the mechanism that transfers heat from the fire to your face? (Hint: Is the air between you and the fire hotter or cooler than your face?)

Give some everyday examples of the transport of heat by convection and by radiation.

Suppose the proton-proton cycle in the Sun were to slow down suddenly and generate energy at only 95% of its current rate. Would an observer on Earth see an immediate decrease in the Sun’s brightness? Would she immediately see a decrease in the number of neutrinos emitted by the Sun?

Do you think that nuclear fusion takes place in the atmospheres of stars? Why or why not?

Why is fission not an important energy source in the Sun?

Why do you suppose so great a fraction of the Sun’s energy comes from its central regions? Within what fraction of the Sun’s radius does practically all of the Sun’s luminosity originate (see Figure 16.16)? Within what radius of the Sun has its original hydrogen been partially used up? Discuss what relationship the answers to these questions bear to one another.

Explain how mathematical computer models allow us to understand what is going on inside of the Sun.

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Why does the luminosity increase?

As core hydrogen burning proceeds, the number of mass units per particle in the core increases. i.e. 4 protons plus 4 electrons become 1 helium nucleus plus 2 electrons. But pressure depends on both temperature and the number density of particles. If the number of mass units per particle is $mu$ , then $P = frac< ho k_B T>, (1)$ where $m_u$ is the atomic mass unit and $ho$ is the mass density.

As hydrogen burning proceeds then $mu$ increases from about 0.6 for the initial H/He mixture, towards 4/3 for a pure He core. Thus the pressure would fall unless $ho T$ increases.

An increase in $ho T$ naturally leads to an increase in the rate of nuclear fusion (which goes as something like $ho^2 T^4$ in the Sun) and hence an increase in luminosity.

This is the crude argument used in most basic texts, but there is a better one.

The luminosity of a core burning star, whose energy output is transferred to the surface mainly via radiation (which is the case for the Sun, in which radiative transport dominates over the bulk of its mass) depends only on its mass and composition. It is easy to show, using the virial theorem for hydrostatic equilibrium and the relevant radiative transport equation (e.g. see p.105 of these lecture notes), that $L = fracM^3, (2)$ where $kappa$ is the average opacity in the star.

Thus the luminosity of a radiative star does not depend on the energy generation mechanism at all. As $mu$ increases (and $kappa$ decreases because of the removal of free electrons) the luminosity must increase.

Explaining this is a bit trickier and ultimately does depend on the details of the nuclear fusion reactions. Hydrostatic equilibrium and the virial theorem tells us that the central temperature depends on mass, radius and composition as $T_c propto frac$ Thus for a fixed mass, as $mu$ increases then the product $T_c R propto mu$ must also increase.

Using equation (2) we can see that if the nuclear generation rate and hence luminosity scales as $ho^2 T_c^$ then if $alpha$ is large, the central temperature can remain almost constant to provide the increased luminosity and hence $R$ must increase significantly. Thus massive main sequence stars, in which CNO cycle burning dominates and $alpha>15$ , experience a large change in radius during main sequence evolution. In stars like the Sun with $alpha sim 4$ , the central temperature increases as $mu$ and $ho$ increases and the radius goes up, but not by very much.

## Why is the Sun's brightness and radius increasing, but not its temperature? - Astronomy

If we think of a beam of light as a stream of particles (like a stream of water, for example) we can speak of a flux , or flow of light. Flux is the amount of light that comes from a certain area (usually one square meter) in a certain amount of time (usually one second). The amount of flux given off by an object depends only on its temperature, according to the Stefan-Boltzmann Law:

where T is the temperature (in K) and the Greek letter Sigma is a constant term called the Stefan-Boltzmann Constant. Its value is unimportant for this class, as we shall see. Flux is not the true measure of an object's energy output. For example, a flashlight and a searchlight have similar temperatures, therefore similar fluxes. But from a distance of 100 yards, the searchlight is the brighter of the two. Why? Because the searchlight is bigger!

We call the total energy output per second of an object its luminosity. It depends not only on Flux (temperature) but also on size (or, more accurately, surface area). Stars are for the most part spherical, so we can compute their surface areas easily, using A = 4(pi)R 2 , where R is the radius of the sphere. Therefore

Luminosity = (Flux)(Surface Area) = (SigmaT 4 ) (4(pi)R 2 )

While it is possible to compute the exact values of luminosities, it requires that we know the value of Sigma. We can get around this by comparing the luminosities of two objects, either two different objects, or the same object before or after some great change in temperature, radius, or both:

L 1 / L 2 = (SigmaT 1 4 ) (4(pi)R 1 2 ) / (SigmaT 2 4 ) (4(pi)R 2 2 )

Sigma and 4 and pi all drop out, leaving us with:

L 1 / L 2 = (T 1 4 R 1 2 ) / (T 2 4 R 2 2 )

Or, to arrange it another way:

L 1 / L 2 = (T 1 4 / T 2 4 ) (R 1 2 / R 2 2 )

Often we will use the Sun as object #2, comparing its luminosity to that of another star.

## Sample Calculations

But even luminosity is not the true measure of how bright an object appears. A flashlight at 2 feet is blinding, but a searchlight at 10 miles is not. How can this be if the searchlight has a higher luminosity?

As light leaves a source, it spreads out in a spherical pattern. As the photons get farther away from the light source, they spread out and become less and less concentrated (there are only so many photons to go around, after all). Thus the light source appears dimmer the farther away it is. This is expressed mathematically using the inverse-square relation:

B = (SigmaT 4 ) (4(pi)R 2 ) / (4 (pi) D 2 )

Remember that R is the actual radius of the light source, and D is the distance of the light source. The units for R and D don't really matter, as long as they are the same (both R and D in km, for example). We can get around using Sigma here the same way we did above, by comparing the brightnesses of two objects.

Apparent brightness decreases as the square of the distance. Thus, if I moved a light source from 1 foot to 5 feet away from me, it would appear 25 times dimmer.

## NSO scientists investigate variations in the Sun’s internal temperature and brightness

Global frequencies of surface-gravity or fundamental modes of oscillations are used to estimate the variation in the seismic radius. These modes propagate along the solar surface and their frequencies are believed to be modified by both the magnetic field, and the seismic radius.

The comparison between TSI (green line) and changing seismic radius (blue dots) demonstrate a clear anti-correlation between the two. The solar irradiance increases with decreasing seismic radius or the seismic radius shrinks with increasing magnetic activity. This shrinkage is believed to be caused by an increase in the radial components of small-scale magnetic fields located a few megameters below the surface, or the thermal structure inside the Sun.

The major contribution to TSI variation comes from the changes in the solar magnetic field. However, this paper shows that there is a secondary contribution from changes in the seismic radius, or internal thermal structure of the Sun.

## Relationship between star radius and luminosity

In a PDF presentation on star formation that I'm currently reading, I ran into the following statement:

"If we observe an increase in a star's temperature but without any changes in its luminosity, it means the star is shrinking (its radius is decreasing)"

I'm having trouble understanding this. Here's why:

Luminosity is the total energy output of a star over a given time. If I'm understanding this definition correctly, it means an increase in a star's temperature will increase its luminosity. Assuming this is true, the statement I quoted above seems to imply that a decrease in a star's radius, assuming nothing else changes, should reduce its luminosity. (I think this must be true in order for luminosity to remain constant despite an increase in temperature.)

But why would that be the case? Intuitively I have always thought that reducing a star's radius, without changing its temperature, would keep the luminosity constant. The energy output per unit area would change, but the total for the sun would remain the same, my thinking went. Example:

A star has a surface area of 1,000 units and is emitting 1,000 units of energy per second (luminosity), therefore each single area unit is emitting a single energy unit. If the surface area of the star is reduced to 500 units and nothing else is changed, I thought the luminosity would still equal 1,000 energy units, only now each single area unit is emitting two energy units instead of one.

Is my line of reasoning above incorrect? Is there some intrinsic limit to how much energy a "single unit area" can output? What am I missing?