How to calculate the argument of perihelion for exoplanets with incomplete orbital elements?

Question

I'm trying to simulate the Kepler-444 exoplanetary system using data from this website. The problem is that there is no argument of periapsis parameter for the Kepler-444 system, so while I think I'm getting the correct shape of the orbits, their periapsises and apoapsis are all aligned in the same direction, as you can see in the screen dump below.

So my question is how, or if, I can get the periapsis to be correctly aligned with the data that is available on this website for these planets (unfortunately I can't link to a query, but you can search Kepler-444 and the planets should pop up in the table)? The way I'm doing it now is that I take the semimajor axis and the eccentricity data from the website I linked to, and then I calculate the periapsis for each planet and then I set the x axis to be that value while I let y and z be zero. Once there, I use the vis-viva equation to calculate the velocity vectors for the planet, and then I iterate the simulation forwards in time and get the shapes of the orbits. With the solar system planets, I take the argument of periapsis and the inclination of the orbit and rotate the position and velocity vectors that I derived accordingly, and this way I end up with orbits that have the right shape and their periapsises are correctly aligned, too. Naturally I don't get the right position of a planet in its orbit at a given time using this approach, but for now that is not important; I just want the orbits to have the correct shape and to have their periapsises correctly aligned.

I know very little about orbital mechanics, so any help would be much appreciated so that I can learn something and improve the quality of my simulation.

Answer 1

Ford et al. (2008) give the following relationship for eccentric transiting planets:

$$\frac{t_\mathrm{D}}{P} \simeq \frac{R_\ast}{\pi a \sqrt{1-e^2}}\sqrt{(1+r)^2 - b^2} \left(\frac{d_\mathrm{t}}{a}\right)$$

Where $t_\mathrm{D}$ is the total transit duration, $P$ is the orbital period, $R_\ast$ is the stellar radius, $a$ is the semimajor axis, $e$ is the eccentricity, $r \equiv R_\mathrm{p}/R_\ast$ is the ratio of planetary radius $R_\mathrm{p}$ to the stellar radius, $b \equiv d_\mathrm{t} \cos i / R_\ast$ is the impact parameter, $i$ is the inclination and $d_\mathrm{t} = a(1-e^2)/(1+e \cos \omega)$ is the distance between the star and the planet during the transit. By inverting this you would be able to estimate $\cos \omega$ from the parameters listed on the site, which unfortunately won't give you a unique value (and if you're unlucky, the various uncertainties may conspire to give you a value $\left|\cos \omega \right|>1$).

In the case of Kepler-444, the website gives a link to the paper where the orbits were published, Campante et al. (2015). Their table 4 tabulates both $e \cos \omega$ and $e \sin \omega$ for the planets. Unfortunately it is not possible to simply determine the value from these estimates without having access to the full distribution of samples, so this isn't entirely helpful.

The good news is that the statistical significance of the non-zero eccentricities doesn't appear to be too high in this case, so you can probably treat the orbits as circular and not worry too much about it: tightly-packed systems like this usually have very low eccentricities otherwise they would become unstable.

Answer 2

Missing Orbital Parameters

Six orbital parameters (seven if you include an epoch) are the minimum required to uniquely describe a simple two-body orbit like the type you are trying to simulate. The database you linked to contains entries for:

1. Semi-major Axis
2. Eccentricity
3. Argument of Periastron (aka argument of perigee, preiapsis)
4. Inclination (missing some entries)
5. Time of Periastron (also missing some entries)
6. Right Ascension of the Ascending Node (aka big$\Omega$ or just $\Omega$ -- no entries)

It is the lack of values for item #6 that under-specifies the Kepler-144 orbits, as well as the missing values for the inclination and time of periastron. Without more information about the planet's orbits, there is no way to faithfully recreate their orbits without making up some values for the missing parameters.