WS 11.2, Problem 3: Goldilocks Zones

A Goldilocks Zone, or a "habitable zone," is a distance from a star where it is neither too hot nor too cold for a planet to have liquid water.  This is the criterion for life that most people have a greed on, but we're not sure that it's the only or most important factor in determining whether or not life can exist on a planet, so it's not the end all be all of life on other planets.

Now let's fi gure out what stars make the best targets for both the radial velocity technique and the transit method, assuming we are primarily interested in fi nding planets in their stars' habitable zones. For a planet in the habitable zone of a star of mass $M_*$...

a)  How does the Doppler amplitude, $K_{HZ}$, scale with stellar mass?

To solve this problem, let's first recall the equation for the Doppler amplitude that we found on a previous worksheet, generously dropping unnecessary constants.

$K\sim\frac{1}{M_*^{4/6}P^{2/6}}$

This isn't as obvious as it might seem, though, because an object's period depends on its mass, so we have to do some more algebra.

$P^{1/3}\sim\left ( \frac{a_P^{3/2}}{M_*^{1/2}} \right )^{1/3}=\frac{a_P^{3/6}}{M_*^{1/6}}$

Of course, the semimajor axis of a planet, $a_P$, is also dependent upon its star's mass.

$a_P^{3/6}\sim M_*^{3/6}$

When you combine all of these scaling factors, you find that

$K\sim\frac{1}{M_*^{4/6}M_*^{2/6}}=\frac{1}{M_*}$

b)  How does the transit depth, $\delta_{HZ}$, scale with stellar mass?

For this, we have to remember that the depth of the transit depends on the ratio of the radii of the star and planet.

$\delta\sim\frac{R_P^2}{R_*^2}$

We can use mass-radius scaling relations to simplify this

$\delta\sim\frac{1}{R_*^2}$

$M\sim R$

$\delta\sim\frac{1}{M_*^2}$

c)  How does the transit probability, Prob$_{HZ}$, depend on stellar mass?

We learned on worksheet 11.1 that the probability that a planet will pass in front of its star is

$\frac{R_*}{a_P}

Now we have to remember both the mass-radius scaling relation and the relationship between a star's mass and $a_P$.

$M\sim R$

$a_P=\frac{M_*a_*}{M_P}$

Then we can rewrite the probability as

$\frac{M_*M_P}{M_*a_*}$

So the probability of a transit happening doesn't really depend on the star's mass at all.


d)  How does the number of transits (orbits) per year depend on stellar mass?

This wording of this question is a little unclear, but it reminds me of part (c) from problem 2 on this same worksheet.

If the Sun were half as massive and the Earth had the same equilibrium temperature, how many days would our year contain?

To solve this, we set up a proportion between the period of a habitable zone planet and the period of our Earth under the conditions specified in the problem.

$\frac{P_{HZ}}{P_\oplus}\sim\frac{a_{HZ}^{3/2}M_\odot^{1/2}}{M_*^{1/2}a_\oplus^{3/2}}$

If we drop the constants, we get

$P_{HZ}\sim\frac{a_{HZ}^{3/2}}{M_*^{1/2}}$

Now we can substitute in the equation for the semimajor axis of the habitable planet and simplify to get

$P_{HZ}\sim M_*$

But the problem asks for the relationship between a star's mass and the number of orbits of a planet, which is inversely proportional to its period, so

Transits$\sim\frac{1}{M_*}$

e)  Based on this analysis, what are the best kinds of target stars for the search for habitable zone planets? What factors did we ignore in this analysis?

From parts (a) through (d), we can see that the Doppler amplitude and number of transits are both inversely proportional to stellar mass and the depth of the transit goes as the inverse square of the stellar mass.   We want all of these quantities to be as large as possible.  A high Doppler amplitude makes a transit more noticeable on an RV graph; a deep transit is more noticeable on a light curve; the more transits there are, the more opportunities we have to gather data.

Based on this, we can see that lower-mass stars make better targets for finding habitable-zone planets.


In doing this, we ignore the fact that lower-mass stars are harder to detect in general because their luminosities are so low.