Survivable, More or Less: Stellar Habitable Zones

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

The habitable zone is a pretty simple, and neat, idea--essentially, for a given star of a given size (and temperature, and luminosity--but, as we know, these parameters are all connected), there is a range of radii out from the star called the habitable zone which is just the right temperature for liquid water to exist on a planet's surface (for this reason it's also known as the "Goldilocks zone"--not too hot, not too cold).

Habitable zones as a function of $M_*$ (source).

Unfortunately, this is one of those ideas that sounds a lot better than it works. In reality, there are a ton of different factors that could influence where planets (or moons!) could be habitable, from atmospheric composition (imagine a planet as far from the Sun as Mars but with Venus' wicked greenhouse atmosphere), internal heating due to tidal forces (looking at you, Europa), and reflected sunlight from a large parent planet in the case of moons. This is all assuming that liquid water is a must for life, which is a big if. However, despite the oversimplifications inherent, it's still an interesting concept to explore.

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

Doppler amplitude is a scary sounding term, but it's actually something we've gotten pretty familiar with at this point--the maximum radial velocity of the star, $v_*$. As we know, $M_p v_p = M_* v_*$. We can treat $M_p$ as a constant, so if we divide $M_*$ over to the other side, we can turn this into a scaling relation based on mass and velocity:\[v_* - K_{HZ} \sim \frac{v_p}{M_*}\]So our next step is to figure out how $v_p$ scales with stellar mass. Assuming a circular orbit, we know that $v_p = \frac{2\pi a}{P} \sim \frac{a}{P}$, so we can substitute into our scaling relation to get \[K_{HZ} \sim \frac{\frac{a}{P}}{M_*}\]According to Kepler's Third Law, $P \sim (\frac{M_*}{a^3})^{1/3}$, so we can substitute this in as well. \[K_{HZ} \sim a^{-1/2}M_*^{-1/2}\] Bear in mind that this isn't an arbitrary $a$, this is actually $a_{HZ}$--the distance of the habitable zone from the parent star, which scales as $T^{-2} L^{1/2}$. If we hold temperature constant and use the (fuzzy, but reliable) intermediate-mass mass-luminosity relation for main sequence stars $L \sim M$, we can say that $a_{HZ} \sim L^{1/2} \sim M_* ^2$. Plugging this back into our relation for $K_{HZ}$ gives us a final scaling relation:\[K_{HZ} \sim M_* ^ {-\frac{3}{2}}\]
b) How does the transit depth, $\delta$, scale with $M_*$?

In our recent examination of exoplanets we found that $\delta \sim \frac{R_p ^2}{R_* ^2}$. We're holding $R_p$ constant, so all we're interested in is how $M_*$ scales with $R_*$. This is actually a hotly investigated question for parts of the H-R diagram (particularly low-mass stars), but in the region close to one solar mass (where most stars are located), it's pretty reliably $M \sim R$. Thus, we can simply substitute radius for mass into our scaling relation to find that \[\delta \sim M_* ^{-2}\]
c) How does the transit probability, Prob$_{tr, HZ}$, depend on $M_*$?

The transit probability is exactly what it sounds like--the likelihood that an observable transit of an exoplanet (in this case in the habitable zone) will occur. Transit probability is equal to $\frac{R_*}{a_{HZ}}$, so this is again a simple matter of substituting according to other scaling relations--$R\sim M$ for $R_*$ and $a_{HZ} \sim M^2$ for $a_{HZ}$. The result:\[\textrm{Prob}_{tr,HZ} \sim M_* ^{-1}\]
d) How does the number of transits (orbits) per year depend on stellar mass?

The longer a planet's orbital period is, the fewer transits we'll see in a given year--so, intuitively, the number of transits per year scales as $P^{-1}$. We found in part a) that $P \sim \frac{a_{HZ}^{3/2}}{M_* ^{1/2}} \sim M^{5/2}$, so we then know that\[N_{transits} \sim M^{-5/2}\]
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?

All the results we found scaled inversely with mass--in other words, increasing the mass of a target star decreases the amplitude of Doppler shifts, the transit depth, the likelihood of a transit, and the number of observable transits per year. The obvious conclusion is that we should target the smallest stars we can possibly find. However, our simplifying assumptions eliminated a few factors that can be pretty crucial when searching for exoplanets. We ignored temperature entirely, despite the fact that some stars, such as brown dwarfs (semantic discussion of whether brown dwarfs count as stars notwithstanding) are so cold that it would be absurd to search for habitable planets around them--one, discovered recently, is literally as cold as water ice. Perhaps most importantly, though, we disregarded the brightness of our target stars. Dimmer stars around one solar mass may be the most common stars in the galaxy, but brighter A-types--despite being 0.5% as common--constitute the bulk of easily observable stars, since they can be seen from orders of magnitude farther away. Thus, in conducting a wide-ranging search for exoplanets, it is important to balance the competing factors of mass and visibility.

Credit and appreciation go to Anne Madoff, Scott Zhuge, Jennifer Shi, and Louise Decoppet for collaboration on these problems.