# Simplified formula for temperature between the star and background temperature

I am looking for a simplified formula to calculate the temperature of an object in space and how distance from the star affects said temperature.

Something that closely represents reality will suffice. It does not have to be accurate. It is just to model temperature damage in a semi-realistic sci-fi space game.

Basically if I go out into space and I grab a thermometer and hold it close to the sun and take a reading and then take a few steps back and take a reading again and repeat this what would my curve look like?

So I basically have the surface temperature of a star and the distance between the star and my "thermometer" and I have to find out roughly what temperature it would read.

PS: Is this even how temperature in space works?

• Are you asking about a planet, (with atmosphere) or a spacecraft? – user184116 Feb 8 '18 at 20:33
• @sammygerbil The OP is not asking about measuring the temperature of a star. The surface temperature is given. He/she is asking about the temperature of an absorbing body at equilibrium a certain distance from the star. Since a thermometer is indeed an absorbing body, it makes sense to use it in this context. – probably_someone Feb 8 '18 at 21:50
• check this out physics.stackexchange.com/questions/114972/… – pentane Feb 11 '18 at 1:35
• or, more directly, here at wikipedia en.wikipedia.org/wiki/… – pentane Feb 11 '18 at 1:43

Since you're only interested in a rough approximation, we can assume that both the star and the body you're interested in are blackbodies. In that case, we can simply set the total power incident on the body equal to the total power radiated by that body, and from that, solve for the body's temperature as a function of distance.

The total power $L$ radiated by a star with radius $R_s$ and surface temperature $T_s$ is given by the Stefan-Boltzmann law:

$$L=4\pi R_s^2 \sigma T^4$$

where $\sigma$ is the Stefan-Boltzmann constant. Suppose the body is sitting at a distance $r$ from the star. At that distance, the power per unit area incident on the body is

$$\frac{P}{A}=\frac{L}{4\pi r^2}$$

since the total area over which power is being emitted is a sphere of radius $r$, which has a surface area of $4\pi r^2$. If your body has a cross-sectional area of $A_{cs}$, then the power incident on that body is

$$P_{in}=\frac{LA_{cs}}{4\pi r^2}$$

If the body's temperature is stable, then it must be emitting as much power as it receives. If the body has total surface area $S$, then it's radiating an amount of power also given by the Stefan-Boltzmann law:

$$P_{out}=S\sigma T_b^4$$

for a given temperature $T_b$. Setting $P_{in}=P_{out}$ and solving for $T_b$, we get

$$T_b=T_s\sqrt{\frac{R_s}{r}}\left(\frac{A_{cs}}{S}\right)^{1/4}$$

So you can see that the temperature of your body decreases as $\sqrt{\frac{1}{r}}$ with distance. If the body you're talking about is a sphere of radius $a$, then we can make this even simpler: $A_{cs}=\pi a^2$ and $S=4\pi a^2$, so $A_{cs}/S=1/4$, and

$$T_b=T_s\sqrt{\frac{R_s}{2r}}$$

If your body isn't a sphere, then the ratio $A_{cs}/S$ will change with its orientation, as it presents more or less exposed surface to the star. But the spherical approximation should probably work for your purposes.