Case Studies in Exoplanets

a) What are the periods, velocity amplitudes and planet masses corresponding to the two radial velocity time series below? The star 18 Del has $M_* = 2.3 M_\odot$, and HD 167042 has $M_* = 1.5 M_\odot$. Notes: Each data point is a radial velocity measured from an observation of the star's spectrum, and the dashed line is the best- tting orbit model. Prof. Johnson found the planet around HD 167042 when he was a grad student, and each data point represents a trip from Berkeley, CA to Mt. Hamilton and a long night at the telescope. "Trend removed" just means that in addition to the sinusoidal variations, there was also a constant acceleration. What would cause such a "trend?"

The first plot we'll look at is 18 Del's, shown below:

Right off the bat, we can gauge the period as being approximately three years--it looks like there's a trough right over the "2003" tick mark, and the next is over the "2006" one. In actuality, the period looks like it's a hair less, but three years should be a good enough approximation to give us an order-of-magnitude sense of the quantities we're looking for. We can also pull the velocity amplitude right off of the plot--it looks to be right around 100 m/s--or $10^4$ cm/s (again, probably just a bit more, but this should be good enough for our purposes). The math comes in when we figure out the planet mass. For this, we'll start off using the equation we worked out in the last post for $K$, solved for $M_p$:\[M_p = \frac{KPM_*}{2\pi a_p}\]We don't, however, know $a_p$--for that, we'll need to turn to Kepler's third law, $P^2 = \frac{4\pi^2a^3}{GM_*}$. We can solve this for $a$--which is really $a_p$--in order to plug into our previous equation. Solved for $a$, we get\[a_p = \left(\frac{P^2GM_*}{4\pi^2}\right)^{1/3}\] Plugging this in gives us the messy but workable\[M_p = \frac{KPM_*}{2\pi \left(\frac{P^2GM_*}{4\pi^2}\right)^{1/3}}\]This comes out to $1.68 \times 10^{31}$ grams. This holds up well under scrutiny: according to Wikipedia (can we pause for a moment just to consider how incredible it is that there are Wikipedia pages about planets beyond our solar system?), 18 Del B has a mass of at least 10.3 Jupiter masses--or $1.9 \times 10^{31}$ grams.

Next we'll turn to the plot of HD 167042, shown here:

Once again, we can pull the period and velocity right off of the plot. The period looks to be just about one year, and the velocity amplitude around 35 m/s. We can then plug these into the same equation we put together to find the mass of the planet, which gives us a result of $1.47 \times 10^{30}$ grams. This, too, can be checked against the planet's Wikipedia page, which tells us that HD 167042 masses in at a minimum of 1.7 Jupiter masses, or $3.23 \times 10^{30}$ grams.

The constant acceleration in HD 167042 could be due to a number of sources. The most obvious would be the radial velocity of the entire solar system away or toward us due to the proper motion of the star, but there are other possibilities as well, such as another planet (or companion star) tugging on HD 167042 in other ways.

b) What is up with the radial velocity time series below? Sketch the orbit of the planet that caused these variations. (HINT: There's only one planet orbiting a single star).

Here's the plot in question:

As you can see, the star very rapidly accelerates away from us, then more gently accelerates back toward us, reaching a much higher velocity away than toward us. The most likely explanation for this is a highly elliptical orbit, where the planet most closely approaches the star (and is thus moving the fastest) while moving away from us (meaning that the star would move fastest, and toward us, at that point), and then moves toward us at the most distant point in its orbit (meaning that the star would move slowest, and toward us at that point). Such an orbit would look something like this:

Indeed, HD222582 b has one of the most highly elliptical orbits of any planet known, at an impressive .76.

Credit goes to Louise Decoppet and Jennifer Shi for help figuring out these three fascinating exoplanets!