Day 9: Radial Velocity to Mass
by Allison Maker
Today we discussed a series of eleven questions in small groups of 2 or 3 throughout the class, relating to the density and composition of the planets and how those relate to their formation. I thought I’d talk about a couple of the questions that I thought had the most interesting (or perhaps unexpected) answers.
2. List all of the reasons you can think of that can lead us to think that Jupiter has a core.
My group came up with a couple of reasons. First, we thought about the inconsistency in density between Jupiter and a body of pure hydrogen and helium—Jupiter is slightly denser, and thus must have some other materials in it. Obviously, these materials could be scattered throughout the planet and are not necessarily condensed into a core, but the presence of a core of some other material is the most straightforward explanation. Next, we discussed that the moment of inertia of Jupiter indicates that there is more material in the center than on the outer part of the planet. Mike Brown explained that moment of inertia could be easily measured just by looking at the shape of the planet. Last, we considered the fact that Jupiter has a pronounced magnetic field, but it turns out that the presence of a core is not necessary to produce a field, something we talk about in the next question:
3. List all of the ingredients that you can think of that are required for a magnetic field created by a dynamo.
We don’t need the presence of a “core” to create this dynamo effect. All we need is some liquid inside the body, and this liquid must be convecting (to create movement for the dynamo), spinning along an axis (so that the convection causes what Mike Brown calls “twirlys” that spin regularly and so keep the field aligned), and be made of some conducting material. In other words, the planet doesn’t even need a metal to have a magnetic field; even water with a lot of dissolved ions would function. These are cool facts, but it just means it’s even harder to find more information about the interior of Jupiter. Maybe the Juno spacecraft, set to reach Jupiter in 2016, can tell us more!
This is the first Coursera-based class that I have ever taken. I still go back and forth on if I like the format of the course. Sometimes I miss having the opportunity to ask the lecturer questions right away. Other times, seeing the correlations directly highlighted on graphs is super awesome.
Today was one of these days where I enjoyed having the extra time in class that was not in lecture format. It is midterms week at Caltech but luckily we do not have a midterm scheduled in this class. At the beginning of the class, I was surprised when Professor Brown passed out papers to everyone with a list of questions. Was it a pop quiz or maybe even midterm? If you haven’t noticed, anything is possible with Professor Brown.
And so he asked us to answer as much as we could for all of the 11 questions in the next 15 minutes. For some of the questions I was really excited that I knew the answer to the question on others I know that the one phrase I wrote couldn’t explain all the questions asked. After these 15 minutes of answering and reflection, I was glad we could exchange ideas with our seat partner. And then each group answered one of the questions…or at least the once we got to.
I turned in the official list of questions with a course evaluation form, but here are the answers we decided on:
1. Compared the density of each planet from highest to lowest if all material is at 1 bar pressure
i. Mercury (because it has a lot of iron)
ii. Earth (slightly denser and bigger than Venus)
iv. Mars (lightest of the terrestrial planet)
vi. Uranus (slightly smaller and less dense than Neptune)
viii. Jupiter (bigger and more dense than Saturn)
2. How can you tell if a gaseous planet has a rocky core
i. The strongest evidence is that the moment of inertial is not what we predict for a homogenous density throughout a planet. As more mass is in the center, it has a smaller moment of inertial.
ii. We determined that the following might give us hints of the fact that it is rocky but we cannot use the information as a deciding factor:
i. Below the H+He curve for radius vs mass plot.
ii. The planet has a magnetic field since we need a core on Earth (especially the liquid part of the core). Later the core was defined as the solid material at the middle of the planet. I guess that was not the answer Professor Brown was looking for.
iii. He raining down (like sugar in a coffee mug settling)
3. What are some of the characteristics needed for convection to occur?
i. Spinning of the planet so that the Coriolis Effect occurs.
ii. As a liquid, convection will occur at descent time scales.
iii. Material composition needs be conducting.
iv. Convective stability is required such that the dT/dz curve of the composition is stable.
4. Draw the H-He curve on a radius vs mass plot. Place and explain Saturn, Jupiter, Uranus, Neptune and hot Jupiter like planets.
(Figure 1: From Guillot AREPS 2005)
i. Jupiter — The difference in mass for its radius means that 10 earth masses are unaccounted for that are heavier than the expected composition of He and H. We now think it has an ice and rock core.
ii. Saturn — Less massive than Jupiter and slightly less He and H for its mass and radius from what the sun and Jupiter is made off of. This is because we think it has an ice and rock core like Jupiter.
iii. Hot Jupiters — red circle
iv. Neptune — lower radius blue dot because Neptune and Uranus have more rock and ice but Neptune is smaller and more massive.
v. Uranus — bigger radius blue dot
5. Draw an adiabatic line on a temperature vs height plot and a curve that is stable vs unstable.
(Figure 2: Unstable dT/dz (red), stable dT/dz (green), adiabatic (purple). dT/dz is unstable if a change in height would cause the parcel to not return to its original position.)
Unfortunately we didn’t get to the other questions before the class ended. Maybe you will hear about the next time! If you were wondering why I liked this particular class, I got to teach someone else about something and learn from what others knew. I learn really well in these kinds of settings.
by Valerie Pietrasz
Today we did a brief review of topics covered both in class and in our video lectures from the past couple of weeks. Were we in high school, this would have been a quiz, but since we are mature college students, we answered the questions individually and discussed our answers with the class.
The first few questions focused on the density of planets in our solar system. First, we ordered the planets from highest to lowest uncompressed density. After (not very) much debate, we agreed on the following order:
Mercury > Earth > Venus > Mars ( > Pluto? Shh, don't tell Mike Brown I put this here.) > Neptune > Uranus > Saturn > Jupiter
(Upon further investigation, Google has informed me that Earth has a higher density than Mercury; however, since we are considering the density of the planet if the materials were not compressed due to gravity, this does not disagree with our conclusions.)
Then we were asked what density told us about composition. For instance, we recreated the mass vs. radius plot of a H + He planet published by Guillot in 2005 and placed our outer planets (Jupiter, Saturn, Uranus, and Neptune) on it as well as hot Jupiters. Jupiter and Saturn are slightly below the H + He curve because they contain predominantly H + He with small amounts of denser material (probably in the form of a solid core); similarly, Neptune and Uranus sit even further below the H + He curve because they contain even more dense material in the form of ice. However, hot Jupiters sit inexplicably above the H + He curve: this implies they are less dense than a planet composed of hydrogen and helium. This is confusing because hydrogen and helium are the two lightest elements in the universe; so how can a planet be less dense than helium and hydrogen? These planets must be inflated somehow. Unfortunately, we don't exactly know how.
Similarly, we listed some of the reasons we have that imply Jupiter has a core. The first, already discussed, is that Jupiter is too dense to only contain the H + He we measure in the atmosphere. The other primary reason is Jupiter's moment of inertia: the measured moment (which we can find by looking at its rotation) does not fit that of a homogenous planet with a mean density of Jupiter's; instead, it is smaller, implying that the denser materials are contained more toward the center of the planet.
We also considered other questions, but kind of ran out of time to talk about them in depth, so I won't worry about 'em here
Day 10: Eleven Questions
by Xinyi Nan
The study of planets around other stars is one of the most exciting fields in planetary science today. But how do we know what we know about these exosolar planetary systems? In today’s class, we took a step back from the cool stuff everyone’s discovering to think in detail about how they did it in the first place.
It’s not immediately obvious, but all the information we have is derived from a few pieces of information we know about the planets’ stars. Just by looking at the light we receive from these distant bodies, we can figure out the stars’ temperatures, masses, composition, and relative velocities. But what do these measurements tell us about planets? A lot more than you might think. In the best-case scenario, we can estimate their radial velocities, periods, orbit radius, masses, and rough composition.
In class, we spent some time looking at the orbital mechanics behind the radial velocity method and transit method, which can be used to get information about the planet in special circumstances. One thing that struck me was how special these circumstances were. Both of these methods involve looking at tiny variations in spectral data from potentially planet-hosting stars. This means that the stars need to be close enough and our equipment good enough for us to tell when variations are statistically significant. In addition, the plane of orbit has to be at a good angle relative to us, because we can only measure changes in velocities if the star is moving toward or away from us and we can only see transits if the planet crosses the star in our line of sight. Identifying changes in measurements also becomes more complicated when there is more than one planet in a system, especially if the planets exert opposing tugs on the star.
All of these factors make the fact that we have found so many other planets all the more astounding. Right now most of the new data on exoplanets is just coming from the Kepler space telescope, which is only looking at a tiny region of the sky. Imagine how many other planets there actually are if we already know about so many just by using our limited techniques. Just imagine how much more there is to find.
Our trusty TAs informed our class today that Mike Brown was missing, and we had a mystery to solve! The mystery of Mike's absence was quickly revealed. Our jovial teacher of Jovian processes was busy being recognized by the National Academy of Sciences. That wasn't the mystery. The case was this:
A faraway planet had been detected by the Chandler wobble of a distant star. When a planet is gravitationally bound to a star, both the planet and the star orbit around their barycenter (their "very center" of mass). As the star orbits, the light emitted from the star is Doppler shifted, and back on Earth we measure alternating red and blue shifted light. One such star had just been detected. Our task was to learn as much about its enigmatic planet as we could.
The suspect: Mr. Enigmatic Planet.
Our only evidence was the following:
1. From the brightness of the star, our TA crime science lab had determined the star's temperature. From the star's temperature, they deduced the star's mass, Ms.
2. From the maximum Doppler shift, we could determine the radial velocity of the star, Ks.
3. The Doppler shift periodically changed from red shifted light to blue shifted light. This rate was equal to the orbital period of the star, which was equal to the orbital period of the planet, P.
That wasn't much to go on. Our TAs insisted we had to find the mass of the planet to close the case. We weren't sure it was possible.
We started out by deducing the radius of the planet, a. (For simplicity's sake, we assumed the planet was orbiting in a circle.) We knew the gravitational force on the planet. We knew this was equal to the centripetal acceleration of the planet. We also knew the velocity of the planet was the circumference of its orbit divided by its period. With some tricky crime science plug-and-chug magic, we found the radius of the planet as a function of its orbital period, a3=P2GM/(4π2).
We had hit a dead end. But wait! Our TAs reminded us that the total momentum of the star-planet system was zero! The momentum of the planet had to equal the momentum of the star, which we knew! At last, we found the mass of the planet was
Our case was closed. Sort of. If you actually read that last equation instead of skimming over it (like I normally do), you might have noticed the sin(i) term in there. The angle i refers to the tilt of the far away solar system with respect to the observer (us). Unfortunately, we didn't know the tilt of the solar system, so we only found the planet's minimum mass. Maybe next time we'll be lucky enough to see a transit of the planet, which would tell us that sin(i)≈1.
by Lori Dajose
Today Mike Brown wasn’t in class (something about being inducted to the National Academy of Sciences or whatever) so Michael led a discussion on what appears to be the most up-and-coming part of planetary science: finding exoplanets.
We’ve only been finding exoplanets for about 20 years, but the numbers are increasing exponentially. So how do we detect an exoplanet? They’re so tiny and far away, it’s really difficult to just see one, the way we can see Mars in the night sky. Even with really powerful telescopes, only a handful of planets have been directly imaged. But astronomers are clever people, and have devised several ways to indirectly search for other worlds.
The first method we talked about was radial velocity (RV). The underlying concept behind this detection method is that a star and planet will orbit—or dance, to put it more poetically—around their shared center of mass, their “barycenter.” Even though we usually can’t directly see a planet, we can detect its gravitational influence on a star, by detecting the star’s motion about the barycenter.
When we look at a star, we get a spectrum of its light. If the star is relatively stationary, this spectrum will be rather fixed over time. But if the star is wobbling around some center of mass it shares with a companion planet, the spectrum we observe will be red-shifted or blue-shifted by the Doppler effect, depending on if the star is moving towards or away from us. From measuring this shift over time, we can get the star AND the planet’s orbital period, P.
However, we can’t always see the star moving towards or away from us—for example, if the plane of the planet’s orbit is perpendicular to our line of sight, the motion of the star will be such that we see no Doppler shift. The angle a planet’s orbital plane makes with our line of sight is called the inclination angle (and denoted cleverly by an “i”). In this example, i = 90º. So the amount of spectral shift we see is constrained by sin(i). Because we can’t always know the inclination angle, we let k = v*sin(i), where k is the velocity we actually measure.
Later in the class, we used the fact that gravitational force is equal to centripetal force to derive Kepler’s third law, which states that the square of an orbital period is directly proportional to the cube of the orbit’s semi-major axis. I won’t go through all the math, but this was our end result:
By adding the fact that total momentum in the barycentric frame is conserved, we were able to derive an expression for the minimum mass of the planet!
Today we mostly focused on the radial velocity technique, but near the end of the class, Danielle shared some cool things about the other main exoplanet detection technique, the transit method. If a star has a companion planet orbiting somewhere near our line of sight, we can observe the planet’s shadow on the star at some points during its orbit. This is less romantic than it sounds (Shadows on the Stars is a really awesome fantasy novel), but it’s still really neat—essentially we collect the photons from the star over time, and if a planet is passing in between us and the star, it blocks out some of those photons for a short period. This manifests as a dip in an otherwise steady emission of starlight. These dips will come periodically, and thus we can measure the planet’s—you guessed it—orbital period.
There are several more properties of exoplanets we can obtain through these techniques, but this post is getting rather long. In conclusion, it’s really quite amazing how much we can learn about exoplanets—without ever having actually seen them!
by Junjie Yu
Today we talked about how to detect the presence of an exoplanet and to measure its mass using the powerful radial velocity method.
A very large planet, as large as Jupiter, for example, would cause the Sun to wobble slightly as the two objects orbit around their center of mass. The variations in the speed are in the radial velocity of the Sun with respect to the planet. The radial velocity can be deduced from the displacement in the Sun’s spectral lines due to the Doppler effect. When the Sun moves towards us, its spectrum is blueshifted (increase in frequency), while it is redshifted (decrease in frequency) when it moves away from us. Since the radius of the star's orbit around the center of mass is very small, its velocity is much smaller than that of the planet. Still, tiny velocity variations can be detected with modern spectrometers. This theory is commonly used when searching for exoplanets.
Before jumping in to the final answer, let’s have a review of the Kepler's third law of planetary motion. Let M be the mass of the star, rp the distance of a planet from the star, and Tp the period of orbital rotation of the planet around the star. Equating the gravitational and centripetal accelerations gives G*M/rp2 = ωp2rp = (2π/Tp)2*rp. Rearranging this equation we get Kepler’s third law, which states that the square of the orbital period of the planet is proportional to the cube of radius of its orbit, or rp3/Tp2 = G*M/4π2. Similarly, the planet’s velocity is given by vp = (2π*G*M/Tp)1/3.
Next, we want to use this relation to determine the mass of the planet. Based on momentum conservation within the star-planet system, we have mp*vp*sini = M*K, where mp is the mass of the planet, K is the velocity of the star, and i is the tilt of the far away solar system with respect to the observer. By plugging in the relation between velocity and period, we can easily get mp = [M2*Tp/(2π*G)]1/3*K/sini . The star’s velocity and orbital period can be measured, and the mass of the star can be determined in other ways. If we know the value of sin3i, we can determine the mass of the exoplanet; even if we don't, we can still have an estimate of its minimum mass. Isn’t it fascinating?