Starstruck: Habitable Zone

The so-called “habitable zone” around a star is the range of semimajor axes corresponding to
planetary surface temperatures warm enough for liquid water. How does the location of the
center of the habitable zone scale with stellar mass? (Recall the scaling relationships from our activities on stellar structure.
We will have liquid water from 0 degrees Celsius to 100 degrees Celsius. This is equivalent to 273K to 373K.
We know from our activities on stellar structure that mass scales with radius. We also know relationships for luminosity, flux, temperature and radius:
$\bg_black F\sim T^{4}$
$\bg_black L\sim R^{2}F$
$\bg_black L\sim R^{2}T^{4}$
$\bg_black a{_{}}^{2}T_{a}^{4}\sim R{_{*}}^{2}T_{*}^{4}$
Note that Ta is NOT the value of the temperature on the planet. It is the effective temperature, due to the sun, at a distance (a) from the sun. Luminosity remains constant, thus the further away from the star you get, the smaller the flux is and the smaller temperature is.
$\bg_black {a} \sim \sqrt{\frac{R{_{*}}^{2}T_{*}^{4}}{T_{p}^{4}}}$

MORE!!! After Professor Johnson's comment... I didn't go quite far enough!!

I want an expression for a in terms of M. This means that if I use the current expression but find scaling relations for R, T* and Tp in terms of a and M, I should have a better answer.

M scales with R.

$\bg_black \frac{dP}{dr}=\frac{GMp}{{r}^{2}}$

$\bg_black P=-\frac{GMp}{{r}}$

This is the energy due to pressure. Balance this with thermal energy:

$\bg_black P=-\frac{GMp}{{r}}$

$\bg_black \frac{3}{2}kT=\frac{GM(\frac{M}{V})}{{r}}$

$\bg_black \frac{3}{2}kT=\frac{GM(\frac{M}{\frac{4}{3}\pi r^{3}})}{{r}}$

$\bg_black \frac{3}{2}kT=\frac{3GM^2}{{4\pi r^{4}}}$

For temperature, we can just use flux, since we have a direct relationship between flux and temperature. flux is dependent on the distance from the star.

(more coming!!! sorry!!)

2 comments:

John JohnsonDecember 4, 2011 at 4:51 PM
I think the question asked how a scales with the mass of the star, not the radius and effective temperature. You're almost there! Please add a few lines to find b in the expression a ~ M^b
JackieDecember 6, 2011 at 12:51 PM
How do you go from T_a to T_p?

Also, check the units on your last expression. Do they work? Where did things go wrong?

If you have time today, try to work this problem with one of your classmates. Or you can post your full solution on here and send me an email to let me know, and I'll try and send you feedback. It's a great exam review question! If you don't have time, though, don't worry - you're doing great in the class and that's definitely going to extend to your exam too :)

Jackie

Sunday, December 4, 2011

Habitable Zone

2 comments: