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

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 temperature given is the surface temperature. The core temperature does indeed increase with the radius and luminosity. Apr 17 at 23:42

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

$$L=4\pi R^2\sigma T_\mathrm{eff}^4$$

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.

• Also, it is the luminosity not the surface temperature of the sun that we need to worry about. Some models have all of the oceans on Earth "boiling off" over the next billion years. Apr 19 '16 at 18:35