# Help with planetary temp. equation

1. Feb 11, 2005

### Mean-Hippy

I an trying to calculate the temperature of a hypotethical planet orbiting the CHARA star at a distance of 1 AU. Now I know that the luminosity of Sol is 4x10^26 watts and that Chara's is about 20% more so 4.8x10^26 watts.

I found the following equation to calculate it:

( L )
Temp ( in K ) = (--------------) 1/4
(16*pi*S*R^2 )

L = Luminosity of star
S = Stefan-Boltzmann constant wich should be 5.67x10^-8
R = radius of planetary orbit in meters where 1 AU is .94x10^12, squared....

Heres where I foud this equation:
http://spiff.rit.edu/classes/phys230/lectures/life/life.html

I end up with stupid 476510.66 K ... this is supposed to be a planet, not a quasar ! Where am I wrong ?

Thanks for help !

2. Feb 12, 2005

### saltydog

I get 117.5 degrees K. Perhaps you're not squaring the radius.

Last edited: Feb 12, 2005
3. Feb 12, 2005

### marcus

you are less apt to make errors if you use natural units

the solar const is 5.7E-117 so for that star (20% briter) the power per unit area at 1 AU is 6.84E-117

you are calculating equilibrium temp for a dark ball, so you have to divide by 4 giving
1.7E-117

the ball will heat up until it radiates away at that rate

by stef-boltz law you must multiply by 60/pi2, which is same as multiply by 6, and then take fourth root.
multiply by 6
E-116

take fourth root
E-29

Beautiful example, that temperature is about 49 degrees Fahrenheit!

If you would like it in Kelvin, it is about 283 Kelvin.

this is close to the global avg. surface temp on earth. the black ball temp at earth's distance from sun is somewhat lower but the greenhouse effect makes up the difference

saltydog I do not see how you got 117 Kelvin
I am pretty sure the rite answer is around 280 Kelvin
I could do it more precisely but it would still come out around 283 K like what I got

[edit, to clarify in response to comment by Chronos: Earth atmosphere has two effects, one cooling and one warming. they nearly cancel but the greenhouse warming effect slightly wins out over the cooling albedo effect of reflection off the tops of clouds. so Earth is a little warmer than the dark sphere equilibrium temp. BUT if there were a lot more clouds to reflect light away, ceteris paribus, the cooling effect could win out.]

Last edited: Feb 13, 2005
4. Feb 12, 2005

### marcus

5. Feb 12, 2005

### saltydog

I just plugged-in the values that mean hippy reported, into the equation. I know, kind of cold (liquid oxygen?).

$${\root{4}\of{\frac{4.8*{{10}^{26}}}{16\multsp \pi \multsp 5.67*\multsp {{10}^{-8}}\multsp {{(0.94 * \multsp {{10}^{12}})}^2}}}} }$$

6. Feb 12, 2005

### Mean-Hippy

Well thankyou very much gentlemen !

7. Feb 13, 2005

### Chronos

I refuse to vacation on a planet without an atmosphere.

8. Feb 13, 2005

### marcus

I talked this over with the manager and he was very concerned by your reluctance. He has ordered an an atmosphere from Comets-R-Us which
will be fine-tuned to have the nice feature that its albedo effect (reflection from the tops of clouds, and snowfields) will exactly cancel its greenhouse effect.

therefore the planet will have exactly the same avg temperature of E-29
(283 kelvin)------equilibrium temp for a dark sphere at that distance from star----as it would with no atmosphere at all

they are offering some great rates on second homes and timeshares, so I would definitely consider it if you are looking for a place to spend the hols.