# 6.3 How hot?!

How do you know how incredibly hot these hot Jupiter planets are? You can work out an estimate based on three things: how big their parent star is, how hot their parent star is, and how far away the planet is from their parent star.

We can think about how we’d expect the temperature at the surface of the planet to depend on these three things. A bigger, hotter star releases more energy which the planet absorbs, so the planet will be hotter. A planet that is further away will absorb less of this energy from its star and so it will be cooler.

The precise relationship between these properties and the surface temperature of the planet is complicated, but the following facts will help you do a bit of detective work.

- If the temperature of the star is four times higher, then the temperature of the planet will be four times higher – the planet temperature is
*proportional*to the star temperature. - If the radius of the star is four times greater, then the temperature of the planet will be twice as high. (This is saying that the planet temperature is proportional to the square root of the star’s radius.)
- If the distance between the planet and the star is four times greater, then the temperature of the planet will be halved. (This time the planet temperature is
*inversely*proportional to the square root of distance from the star. Inversely proportional just means that if the first thing increases then the other decreases.)

## Activity 6 Hotter or colder?

The interactive application allows you to calculate the temperature a planet is expected to have based on the mass of its parent star and the planet’s orbital distance from the star. This temperature is called the equilibrium temperature because it is calculated assuming that the planet is in a heat equilibrium, radiating exactly as much heat as it absorbs – it depends on the factors discussed above. The interactive application assumes the star is a main sequence star, and for main sequence stars the radius and temperature on which the calculation depends are specified by the star’s mass.

HD 209458 is a 1.15 M_{Sun} main sequence star. HD 209458 b has a circular orbit of radius 0.05 AU. Adjust the sliders to these parameters.

What is the planet equilibrium temperature of HD 209458 b?

### Answer

Express the equilibrium temperature of HD 209458 b in °C.

### Answer

1037 °C. Remember, to do the conversion, simply subtract 273. 1310 − 273 = 1037.

What happens if you increase the mass of the star?

### Answer

Both the star and the planet get hotter.

What happens if you increase the planet’s orbital distance from its star?

### Answer

The planet gets cooler.

Reset the sliders to the values for HD 209458 b (1.15 M_{Sun} and 0.05 AU).

Can you work out what temperature HD 209458 b should be if it was four times as far from its star? Express your answer in kelvin. Use the interactive application to find out or check your answer.

### Answer

With *a*_{total} = 0.2 AU, the interactive application gives 657 K. This is half the temperature of HD 209458 b (give or take a few – the precision of the sliders is not perfect!), in accord with fact 3 above.

Can you find different combinations of stellar mass and planet distance that produce planets with the same temperature as HD 209458 b?

### Answer

Yes, you should be able to. If you increase the stellar mass, the planet becomes hotter. Then if you move the planet further away from its star until it cools to an equilibrium temperature of 1310 K, you can get a combination that fits. By playing in this way, you should be able to find an appropriate combination for every value of the stellar mass covered by the slider in the interactive application.

## Maths help

The following OpenLearn resources may help with the maths in this section. Open the link in a new tab or window by holding down Ctrl (or Cmd on a Mac) when you click on the link. Remember to return here when you have finished.

- direct proportion (Section 2.2) [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]
- inverse proportion (Section 2.3)