# Quick help with equation P = a * T^4

• leonmate

#### leonmate

Hi all,

I have an exam tomorrow on the evolution of stars. Revision is going OK, but I've come across this equation. Some quick help would be very appreciated!

P = αT4

I'll give you some context:

I'm solving a question that states a star in Virial equilibrium is contracting quasi-statically (slowly). What effect would this have on the temperature of a very large star. (can assume gas pressure is a negligible fraction of the total pressure and temperature is uniform).
I've started the question with:

Ω = -3 ∫ P/ρ dm

Cancelled density ρ, by subbing dm = 4πr2ρ

Then the solution I have shows a substitution for pressure P as above:

P = αT4

Using this we can solve the problem by showing temperature T ∝ 1/R

I'm sure the equation is a correct one. I was just wondering if anyone knew a little more on it? When can/can't I use this substitution.

Thanks

This just looks like the Stefan–Boltzmann law, as radiated power and pressure are proportional to each other for radiation. If radiation pressure dominates, this is an approximation for the total pressure.

## What is the equation P = a * T^4 used for?

The equation P = a * T^4 is used to calculate the power radiated by a black body, where P is the power in watts, a is the Stefan-Boltzmann constant, and T is the temperature in Kelvin.

## How do I solve for a in the equation P = a * T^4?

To solve for a, you would need to know the power (P) and temperature (T) values, and then divide P by T^4. The resulting value would be the constant a.

## Can the equation P = a * T^4 be used for objects other than black bodies?

Yes, the equation can be used for objects that emit thermal radiation, but it will only give an accurate result for black bodies. For other objects, the value of a may need to be adjusted to account for factors such as reflectivity or emissivity.

## What are the units for the constant a in the equation P = a * T^4?

The units for the constant a in the equation P = a * T^4 are watts per square meter per Kelvin to the fourth power (W/m^2/K^4).

## Is the equation P = a * T^4 valid for all temperatures?

The equation P = a * T^4 is valid for any temperature above absolute zero (0 Kelvin), but it becomes less accurate at extremely high temperatures. This is because it assumes that all energy radiated by the object is emitted as thermal radiation, which may not be the case at very high temperatures.