# Surface temperature of a planet revolving a sun

In summary, the surface temperature of a small planet orbiting around the sun with a circular orbit is ##\theta_0=273.15## degrees Kelvin.

## Homework Statement

Find the surface temperature of a small planet having circular orbit around the sun with time period T,assuming sun and planet to be black bodies. Take radius of sun = R, its mass = M and its surface temperature as ##\theta_0##.

## Homework Equations

##P=eA\sigma T^4##
Total energy of planet is ##\frac{GMm}{2R}##
##I=\frac{P}{4\pi d^2}##

## The Attempt at a Solution

Power emitted by sun is ##P=eA\sigma\theta^4_0##.
The energy of planet is the sum of its kinetic and potential enegies (E) + the energy received from sun.
From these relations, I am not able to find the planet's surface temperature.
Since the planet is also a black body, the energy absorbed by planet = the energy imcident on the planet. I can find the energy it received during one complete revolution.
But I am not able to get any usefull relation.

You do not need to know the energy during one revolution. The energy loss rate needs to equal the energy gain from absorbing sunlight.

## Homework Statement

Find the surface temperature of a small planet having circular orbit around the sun with time period T,assuming sun and planet to be black bodies. Take radius of sun = R, its mass = M and its surface temperature as ##\theta_0##.

## Homework Equations

##P=eA\sigma T^4##
Total energy of planet is ##\frac{GMm}{2R}##
##I=\frac{P}{4\pi d^2}##

## The Attempt at a Solution

Power emitted by sun is ##P=eA\sigma\theta^4_0##.
The energy of planet is the sum of its kinetic and potential enegies (E) + the energy received from sun.
From these relations, I am not able to find the planet's surface temperature.
Since the planet is also a black body, the energy absorbed by planet = the energy imcident on the planet. I can find the energy it received during one complete revolution.
But I am not able to get any usefull relation.

You need to calculate the power incident on the planet from the sun. This can be approximated by the ratio of the cross sectional area presented to the sun (pi*r_planet^2) divided by the total area (solid angle) 4*pi*r_orbit^2 times the power output by the sun (assumes isotropic radiation). At equilibrium, the power deposited into the planet from the sunlight must be equal to the power radiated by the planet to space. This can be used to calculate the black body temperature for the planet. So, to figure all of this out, you need to find out what the orbital radius is.

Quantum Defect said:
You need to calculate the power incident on the planet from the sun. This can be approximated by the ratio of the cross sectional area presented to the sun (pi*r_planet^2) divided by the total area (solid angle) 4*pi*r_orbit^2 times the power output by the sun (assumes isotropic radiation). At equilibrium, the power deposited into the planet from the sunlight must be equal to the power radiated by the planet to space. This can be used to calculate the black body temperature for the planet. So, to figure all of this out, you need to find out what the orbital radius is.
##T=\frac{2\pi R_0}{v}##
And ##v=\sqrt{\frac{GM}{R}}##
So ##T=2\pi R_0\sqrt{\frac{R}{GM}}##
So ##R_0=\frac{T}{2\pi}\sqrt{\frac{GM}{R}}##
But radius of planet is not given.

Assume a radius r and see what comes out. Do not enter a value for it.

Edit: Also, v is irrelevant.

sorry, the value of Radius of orbit is wrong in post 4.
Its ##R_0^{3/2}=\frac{T}{2\pi}(GM)^{1/2}##
The power received by planet is ##P=\pi r^2\frac{P}{4\pi R_0^2}##
Now P emitted=##4\pi r^2\sigma\theta^4##
So the r^^2 terms goes out and i got the correct answer.

## 1. What is the surface temperature of a planet?

The surface temperature of a planet refers to the average temperature of the planet's surface. It is determined by factors such as the distance from the sun, the composition of the planet's atmosphere, and its rotation and tilt.

## 2. How does the distance from the sun affect a planet's surface temperature?

The distance from the sun is a major factor in determining a planet's surface temperature. The closer a planet is to the sun, the more energy it receives, resulting in a higher surface temperature. Conversely, the further away a planet is, the less energy it receives, resulting in a lower surface temperature.

## 3. Why is the composition of a planet's atmosphere important in determining its surface temperature?

The composition of a planet's atmosphere plays a crucial role in regulating its surface temperature. Certain gases, such as carbon dioxide and water vapor, act as greenhouse gases, trapping heat and causing the planet's surface temperature to increase. Other gases, such as nitrogen and oxygen, do not have this effect and help maintain a stable surface temperature.

## 4. How does a planet's rotation and tilt affect its surface temperature?

A planet's rotation and tilt also play a role in its surface temperature. A faster rotation can distribute heat more evenly, resulting in a lower temperature difference between day and night. A greater tilt can also lead to more extreme seasons and temperature variations on the planet's surface.

## 5. Is the surface temperature of a planet constant?

No, the surface temperature of a planet is not constant. It can vary greatly depending on the factors mentioned above, as well as other factors such as the planet's albedo (reflectivity) and the presence of geological features such as oceans and mountains. Climate change can also impact a planet's surface temperature in the long term.

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