# HELP: Weeks 6-7 (Applying physical laws)

This post is a short summary of the last few weeks’ work.

Habitable Zone Research

Owen did some research into calculating the habitable zone for exoplanets. He found a pair of very simple equations corresponding to the inner and outer edge of the habitable zone:

The inner Edge of the HZ (r_i) is the distance where runaway greenhouse conditions vaporize the whole water reservoir and, as a second effect, induce the photodissociation of water vapor and the loss of hydrogen to space.  The outer edge of the HZ (r_o) is the distance from the star where a maximum greenhouse effect fails to keep the surface of the planet above the freezing point, or the distance from the star where CO2 starts condensing.

The article Owen found these equations on backs using the Stellar Flux over equilibrium temperature which cancels any dependence on albedo. This makes determining values for Stellar Flux’s easier as albedo differs only slightly in every spectral type of star. This article came to the conclusion of using 0.53W/m^2 and 1.1W/m^2 for the outer and inner stellar flux’s, respectively, by using the bolometric correction. These values were clarified in [Kasting et al., 1993, cited below; Whitmire et al., 1996].

We wanted to create our own formula, with different assumptions, to compare with equations Owen had found. Amaia did some research into this and found an equation that we could manipulate:

We rearranged this equation to make the orbital radius D the subject and then subbed in temperature limits to find the inner and outer limits of the habitable zone radius. For our temperature limits we used:

• 647K – critical temperature of water
• 273K – freezing point of water

When creating our own formula for the habitable zone, we made a few assumptions on the way. We started off by assuming the planets are black bodies, meaning the albedo a is 0 and emissivity ε is 1. We set the ratio of the area of the planet that absorbs power Aabs and the area of the planet that radiates power Arad to ½, which assumes a slow-rotating planet and makes sense as only about half of the planet will be facing the star at a time. We have assumed a circular orbit as we use a sphere radius of D. We have also assumed no greenhouse effect and an even temperature around the planet, which is not the case but is taken as an average.

Determining Parameters to Define our ‘Habitable Planet’

Amaia and Harry selected which parameters we would use as our definition of a habitable planet and have determined value ranges for each parameter.

• Max Gravity: (3-4)g
• Minimum mass: 0.3 Earth Masses
• Stellar star classification: F, G, K
• Temperatures associated with these spectral types: F(6000K-7600K), G(5000K-6000K), K(3500K-5000K)
• Habitable Zone: We will use the equation Owen found and the one we create ourselves
• Planet Density: >2000kg/m^3 (anything less is probably a gaseous planet)

Kepler’s Law

Harry found that Kepler’s law does hold for our dataset (with very circular orbits of single-star systems). Nevertheless, we couldn’t just decide to dismiss all eccentric orbits as this would eliminate a huge amount of our data.

Amaia suggested to use the circular, single star orbits to prove that Kepler’s law fits the dataset, and then use the specific relationship for the whole dataset to fill in the values. We spoke to David about this issue, and he suggested that orbital periods longer than the time we’ve been observing exoplanets for can skew the data. This might be contributing to the issue too. The periods longer than we’ve been studying exoplanets will have large errors and therefore give large errors. David also pointed out that Kepler’s Law should use the average distance not just “a”. The distance measured as “a” for some planets may simply be the detected distance, not an actual calculation of “a”. This is more to do with the way the database is written. If there is an accurate period given, it should always give you the average distance “a”. So, we just used Kepler’s law to fill in the gaps and propagated the errors.

Applying Kepler’s law to our dataset proved to be difficult and so Davids feedback was to not bother checking specifics with weird periods and eccentricities. We Just assumed Kepler’s law works for all because they are all within the same order of magnitude, so this is a reasonable assumption for astrophysics. Plus, the number of stars in the system doesn’t have much of an effect. In the end we concluded that it’s hard to prove that Kepler’s law holds for the data set we have, but we can assume it does for every star because it’s a geometric law. Also applies to semi major axis of course.

Calculating errors for our data

Amaia has calculated all the errors with a lengthy code that contains a lot of separate functions. Amaia found the difference between actual radius and calculated radius (and for mass) for the ones we have both for. Using bins instead of plotting all of them, she found the average deviation for the values in that bin. Too many small bins lead to empty bins, so she widened them enough that it worked. She added a linear relationship on either end of the mass-radius ranges rather than extrapolating outside the range to fit higher and lower values, because extrapolating didn’t return sensible values. Extrapolating for the other values were however successful.

Multiple Star Systems

In our dataset, we realise that a portion of these exoplanets orbit star systems of more than one star. Owen did some research into multiple star systems to see whether this would affect our results.

When a planet orbits a multiple star system, it can orbit the stars in several ways. For example, in a binary star system, the planet can either orbit one star (S-type orbit) or orbit both stars (P-type orbit). This qualitive information is not available in our dataset. So before proceeding into further research, we knew that this research wouldn’t be changing our results but would be good to include in our discussion section for the report.

Multiple-star systems can perturb a planet’s orbit, precluding any chance for life as we know it to survive. But even for planets in stable orbits, these stars can produce habitable zones that change dramatically as the stars move around each other. Habitable planets can dip out of the HZ for a small amount of time, and the resilience of the planets habitability strongly depends on its climate inertia. Combining orbital dynamics with simple climate models we demonstrate that the size of circumstellar habitable zones depends on a planet’s climate inertia. The higher a climate’s resilience to variations in the incident light, the higher the chances for planets to remain in a habitable state. In systems like α Centauri, a low climate inertia shrinks the habitable zone by 50%