stefan boltzmann law

2

The first person to determine the surface temperature of the Sun (and to come up with a sensible result) was a Slovene physicist (also mathematician and poet) Jožef Stefan [ˈjoːʒɛf ˈʃteːfan]. His result of 5700K differs by only 80K from the modern estimate (5780K).

He calculated it using the formula j*=σT4 known as the Stefan–Boltzmann law, (which he first formulated in 1879, his student Ludwig Boltzmann later expanded its application in 1884), σ being the Stefan–Boltzmann constant (also known as only Stefan’s law and Stefan’s constant). He’s the only Slovene scientist to have a law of nature named after him (but other things have been named after him as well, including a crater on the Moon).

Requested by punnybones

When a volcano erupts, it turns the sky gray with its ash, blocking out the sun and transforming the atmosphere into a cold, dark cloud of gas.

Luckily for any pokémon, when this happens, Volcarona appears, and acts as a perfect replacement for the sun.

As you can imagine, the sun’s not an easy thing to replace. If Volcarona wants to nab a job interview for the position of “Sun”, it better put a few good qualifications on its resumé.

First, let’s talk about distance. If Volcarona is a perfect replacement for the sun, we expect it to be identical in relative size, brightness, and temperature. When you look into our sky, the sun (because its so far away), has about ½ a degree of diameter across the sky. The sun’s actual diameter is about 1,391,684 km. Volcarona, who is only 1.6 meters tall, must be a lot closer for that effect. Volcarona is 869802500x smaller than the sun, so must be 869802500x closer to appear the same.

Volcarona flies at about 171 meters (560 ft) above the planet’s surface to look the same size as the sun.

Now for brightness. It’s important to understand what Astronomers mean by luminosity. Brightness is quantified over all wavelengths, not just the ones we humans can see. So even though the sun emits lots of visible and ultraviolet waves, Volcarona probably emits more photons at infrared levels.

Furthermore, there’s a difference between apparent and absolute brightness. Apparent Brightness is how bright something looks to us, from Earth. For the sun (and for Volcarona), its about -26.7. Absolute Brightness is at a fixed distance (1 parsec). For the sun, it’s 4.83. You can imagine, because Volcarona is so much closer, it’s not nearly as bright, even if it looks the same.

Volcarona has a absolute brightness of M=49.6, over 500,000,000,000,000,000x less bright than the sun. This corresponds to a Luminosity of 480 million Watts.

Of course, the more important thing about our sun is the heat we gain from it. Our sun is a lot hotter near its atomic-fusion core than it is at its surface, but overall the sun can be treated as one uniform body with an effective temperature or 5780 K (5507 C / 9944 F). Since we have the luminosity, we can find the temperature using a variation of the Stefan-Boltzmann Law,

Volcarona’s temperature is 5700 K / 9800 F / 5400 C.

This is interesting, because it’s almost the exact same as the effective temperature of the sun. Volcarona is a lot closer, smaller, and a lot less bright, but has the same temperature. Of course, there’s no way Volcarona could survive with this, but if it wants to replace the sun, that’s what it has to do.

Someone in my class just asked: "How do we know the temperature of the sun? A probe wouldn't be able to get that close." Now I'm curious.

Actually it’s pretty easy. If you know how large the Sun is and how bright it is then we can get it’s surface temperature.

To get it’s luminosity, you measure it’s brightness from Earth and since the Sun emits light equally in all directions (i.e. like a sphere of light expanding from the Sun), all you do is use the equation for the surface area of a sphere which is 4*π*(distance to the Sun)^2 and multiply that by the brightness you detect here on Earth. This is the Sun’s luminosity.

Once you get luminosity you can divide it by the size of the Sun and this should give you the temperature.

Due to something called the Stefan-Boltzmann Law you need to do a few minor adjustments to the equation namely, put the Stefan-Boltzmann-constant (represented by a “σ“ with a value of 5.67*10^-8 W/m^2K^4) in the equation (in the denominator) and raise the entire thing to the power of ¼.

It will look like this: (Luminosity/σ*Size of the Sun)^(¼)

All this is is an adjustment of the Steffan-Boltzmann Equation which relates luminosity to size and temperature.