Exoplanet Transits: How Our Predictions Stack Up

Compare the system parameters derived on the worksheet from the canonical values you can find online in Exoplanets.org.

Before we start, let's just take a moment to consider the fact that there's a website (actually there are several such websites) where you can look at the data on hundreds upon hundreds of exoplanets, from their mass to their radius to the properties of their orbit and parent star.

If that doesn't just blow your mind, I don't know what will.

With that out of the way, let's look at the quantities we estimated two posts ago, and how they stack up against the values found here.

Size comparison of Jupiter (L) and WASP 10b (R) (source).

We estimated $R_p$ to be 1.3 Jupiter radii; Exoplanets.org has it at 1.08 Jupiter radii (16% error).

We estimated $M_*$ to be 0.790$M_\odot$; Exoplanets.org has it at 0.790$M_\odot$ as well(!) (0% error).

We estimated $M_p$ to be 5.27 Jupiter masses; Exoplanets.org has it as 3.19 Jupiter masses (39% error).

We estimated $a/R_*$ to be 12; Exoplanets.org has it as 11.87 (0.3% error).

We estimated $T$ to be 2 hours ($7.2 \times 10^3$ seconds); Exoplanets.org has it at $8.0 \times 10^3$ seconds) (10% error).

We estimated $\rho_*$ to be 3.4 g cm$^{-3}$; Exoplanets.org has it at 1.5 g cm$^{-3}$ (130% error).

We estimated $\rho_p$ to be 2.8 g cm$^{-3}$; Exoplanets.org has it at 3.1 g cm$^{-3}$ (9% error).

These are all pretty impressively close, with a couple of exceptions (most notably the stellar density). However, even those discrepancies don't look too bad when you consider that we were working with a given stellar radius of $0.8M_\odot$--the canonical value is actually just short of $0.7 M_\odot$, which would notably improve a our value for stellar density.

All in all, a pretty impressive show for just a couple of graphs and some algebra!

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.

Exoplanet Transits: The Bread and Butter of Detecting Other Worlds

What properties can be obtained from a planetary transit lightcurve and radial velocity curve?

This is a pretty general question, so it's important to start by examining what data we actually collect when we observe the radial velocity of a star and the light curve of a planetary transit. The first of these, the radial velocity curve, we have actually already discussed in great depth here, but we'll do a quick refresher course on that when we get around to using it. Right off the bat, though, we'll be working with the planet's lightcurve.

Lightcurve of WASP-10 b transit.

As you can see, there's a very obviously visible dip in the amount of light we're receiving from the host star, WASP-10, as the planet passes in front of it. As a matter of fact, we can derive a pretty substantial amount of information from the length, depth, and shape of this dip. 

The sharply slanted edges are known as "ingress" and "egress," and are the time it takes for the planet to go from just barely overlapping with the outer edge of the star to sitting entirely in front of it. However, in between ingress and egress the brightness of the star isn't constant--it actually keeps getting dimmer until the very center of the transit. This is due to a process called "limb darkening," where the outer edge of a star's disc actually appears less bright than the center (it has to do with looking through more of the star's atmosphere, and is comparable to why stars are dimmer and harder to observe when they're close to the horizon). This phenomenon is easily visible on any high-resolution image of the Sun:

Example of limb darkening on the Sun (source).

We can also figure out the relative radii of the planet and star from the depth of the transit. Since the planet is emitting essentially no light, the flux received from the star should decrease by a factor equivalent to the area of the planet's disc ($\pi R_p ^2$) over the area of the star's disc ($\pi R_* ^2$), since the planet is blocking that fraction of the star's area, and thus light. If we define the depth of the transit as $\delta$, we can express this mathematically as\[\delta = \frac{R_p ^2 }{R_* ^2}\]On our lightcurve of WASP-10 b it looks like $\delta$ is around 0.3, so we know that the radius of the planet is around $\sqrt{0.03} \approx 1.7 \times 10^{-1}$ times  the star's radius.

The next thing we can figure out from the lightcurve is the ratio of the star's radius to the planet's orbital semimajor axis.

The star on the left, with the planet shown at the start of its transit (top) and end of its transit (bottom). The orbital semimajor axis, a, and the radius of the star, $R_*$, are labeled.

As you can immediately see from the figure above, $tan(\theta) \approx \theta = R_*/a$. The ratio of the transit time to double this angle will be the same as the total period over the angle traversed over one period ($2\pi$, a full circle). Or, in equation form: \[\frac{t_t}{2\theta}=\frac{t_t}{\frac{R_*}{a}}=\frac{P}{2\pi}\]Rearranging this, we can see that the ratio of $a$ to $R_*$ is \[\frac{a}{R_*} = \frac{P}{2\pi t_t}\]If we had a longer lightcurve showing multiple transits we would be able to pull the period off of that, but we can also grab it off of the radial velocity curve (which will make an appearance in a few moments). As it happens, WASP-10 b has a 3.1-day period (meaning the planet's year lasts only 3.1 days!), and from the lightcurve we can tell that the transit time is right around 2 hours, so we can tell that the semimajor axis is 3.1 days/2 hours $\approx$ 12 times the radius of the star.

Now it gets both very messy and very cool. Density is defined as mass over volume, so we know that\[\rho_* = \frac{M_*}{4/3 \pi R_*^3}\]. We can also solve Kepler's third law for $a$ to give us \[a = \left(\frac{GM_*P^2}{4\pi^2}\right)^{1/3}\] The algebra from here gets sloppy and hard-to-follow, so we'll skip to the interesting part (if we were writing a textbook, this is where we would say that "the derivation is left as an exercise"). We can put these equations togetehr and solve for the density of the star in terms of known quantities, including the ratio of the semimajor axis to the star's radius:\[\rho_* = \left(\frac{a}{R_*}\right)^3\left(\frac{3\pi}{GP^2}\right)\]In the case of WASP-10, this gives us a density of 3.4 g cm$^{-3}$, or about 2.4 times the density of the Sun.

If we know the radius of the star, we can now figure out a few more definite quantities. Let's say $R_* \approx 0.8R_\odot$. From our semimajor axis-radius ratio, we can immediately say that WASP-10 b is orbiting at $12 \times 0.8 R_\odot = 6.7 \times 10^11$ cm. Since we also have an equation for the ratio of the planetary and stellar radius, we can also say that $R_p = 0.17 \times 0.8 R_\odot = 9.5 \times 10^9$ cm. Since we know the density of WASP-10, we can also compute its mass as $M_* = 3.4 \times 4/3 \pi (0.8 M_\odot)^3 = 2.5 \times 10^{33}$ g, or 1.2 $M_\odot$. The last value we can compute is the radial velocity of the planet, $K_p$. If we assume a circular orbit (and thus constant velocity), this means that $K_p$ can be related to $P$ and $a$ in a distance-rate-time equation, $K_p \times P = 2\pi a$. Solving for $K_p$ gives us a brisk $1.5 \times 10^7$ cm s$^{-1}$.

To continue, we'll finally break out that radial velocity chart.

As you can see, the star's radial velocity fluctuates from around -500 m/s to 500 m/s. This means that it's actually steadily moving around its orbit at this speed--we just only see it moving at that speed when it's going directly toward or away from us (remember, it's a radial velocity plot). Back when we were looking at these earlier, we found that $M_* v_* = M_p v_p$. Luckily for us, we've figured out three out of these four variables, and can now solve for the mass of the planet using the mass of the star and the radial velocities of each--it comes out to $M_p = 10^{31}$ g.

Lastly, but still important, we can figure out a quantity called the "impact parameter," which we'll denote as $b$. This is a value between 0 and 1 which quantifies how close the planet is to transiting directly across the star's equator ($b = 0$) and just skirting its edge ($b=1$).

Visualization of the impact parameter $b$. The planet moves along the horizontal line at the top (source).

$b$ is related to the depth of the transit $\delta$, along with the total transit time $T$ and the ingress time $\tau$ according to $b = 1 - \delta^{1/2}\frac{T}{\tau}$. We can estimate $\tau$ from the light curve to be about 0.3 hours, so finding $b$ is a simple matter of plugging and chugging--it comes out to be about 0.13.

So, summing everything up--from two plots and a given radius, we were able to find:

-The length of an exoplanet's year
-The density of its host star
-The mass of its host star
-The mass of the exoplanet itself
-The radius of the exoplanet
-The radial velocity of the exoplanet
-The radial velocity of the star
-The impact parameter of the planet.

We didn't do this, but we can also find the density of the exoplanet--it's an easy matter of dividing mass by volume. As it turns out, in this case $\rho_p = 2.8$ g s$^{-1}$.

Lots of thanks and credit goes to Anne Madoff, Scott Zhuge, Jennifer Shi, and Louise Decoppet for helping out with this one.

HR 8799 and the Vortex Coronagraph

(source)

This is a pretty cool picture--one that I think really is worth its own post. It's a direct image of the three planets known to orbit the star HR 8799. HR 8799 is an exceptionally young star--just 30 million years old or so--and it is still surrounded by an enormous dust halo over 2000 AU in diameter. This cloud of dust is apparently so thick that it threatens the stability of this young solar system. and could be the subject of a post all its own. However, it is not the subject of this post. The three planets known to orbit inside of it are (actually, there are four--but one is too close in to directly image yet) .

Those three imageable planets were among the first directly imaged, using the Keck telescopes in 2008, observed using adaptive optics in infrared. But the image we're concerned with here is an entirely different image, taken just last year. What is remarkable about it is that it was taken using an instrument called a vortex coronograph (a coronograph is a device used to block out the light from a star in order to observe things near to it), and that that instrument was attached to portion of the Hale telescope just 1.5 meters in diameter. While by no means a small telescope, this is an astonishingly small aperture to be producing direct images of exoplanets. Coronographs have been used since before exoplanets were even known to exist, when observing our own Sun. However, a significant amount of light has always been able to creep around the dot that blocks it in traditional coronographs, making direct imaging of planets around distant stars extremely difficult. The vortex coronograph evidently solves this problem by using a spiral pattern to obscure the star, blocking its light almost entirely while allowing all of the light from any planets it may have to pass through. And if it is as effective as it seems, it may mean that direct imaging of exoplanets, still responsible for just a handful of exoplanet detections, could become an increasingly viable way of discovering planets--one which might be able to tell us more about the nature of those planets than we have yet been able to discern.

The original direct image of the HR 8799 system, taken using the Keck Telescopes--which are ten times the diameter of the mirror used to produce the image at the start of this post (source).

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!

Exoplanets Tugging on Stars: Stellar Velocities

a) Astronomers can detect planets orbiting other stars by detecting the motion of the star--its wobble--due to the planet's gravitational tug. Start with the relationship between $a_p$ and $a_*$ to find the relationship between speed of the planet and the star, $v_p$ and $v_*$.

Obviously our first step here is to define $a_p$ and $a_*$. Key to understanding this is realizing that planets don't orbit stars per se--in reality, both objects orbit the center of mass of the system. As it happens, though, stars are so enormous compared to planets that this point tends to be well inside the body of the star itself. However, this does mean that the star is "wobbling" as it orbits a point that isn't quite its center. There are some great animations of this happening in different setups here.

The barycenter (center of mass) of Sun and Jupiter is just outside the Sun's surface. Note that this isn't quite the barycenter of the Solar System as a whole--considering all planets and other objects complicates the problem enormously. However, Jupiter is so massive compared to the other planets that our Solar System can be thought of in this regard as being the Sun-Jupiter system (source).

We found in an earlier problem that the offsets of each object from the system's barycenter proportionate to mass--in other words, $a_p \times M_p = a_* \times M_*$. To figure out the velocity of each object, we'll consider the period of each, which we can use in a distance-rate-time equation: the circumference of each's orbit must be equal to its orbital velocity multiplied by its period (note that the period for both objects must be the same). In mathematical terms, we can solve for period and set up the following inequality:\[\frac{2\pi a_*}{v_*} = \frac{2\pi a_p}{v_p}\] Dividing out the $2\pi$ from both sides, we have\[\frac{a_*}{v_*} = \frac{a_p}{v_p}\]

b) Express the speed of the star, $K$, in terms of the orbital period $P$, the mass of the star $M_*$, and the mass of the planet $M_p << M_*$.

We know right off the bat that $KP = 2\pi a_*$--again, this is just distance equals rate times time. From there, all we need to do is substitute in for $a_*$. Since we know that $a_p \times M_p = a_* \times M_*$, we can solve for $a_*$ to find that $a_* = a_p \times M_p / M_*$. We can then substitute this back into our original equation and solve for $K$ to find that\[K = \frac{2\pi M_p a_p}{PM_*}\]

c/d) We can measure the velocity of a star along the line of sight using a technique similar to the way in which you measured the speed of the Sun's limb due to rotation. Specifically, we can measure the Doppler shift of stellar absorption lines to measure the velocity of the star in the radial direction, towards or away from the Earth, also known as the "radial velocity." What is the time variation of the line-of-sight velocity of the star as a planet orbits? Sketch the velocity of a star orbited by a planet as a function of time.

This problem is worded somewhat confusingly--it's important to understand for it that we're discussing the radial velocity of a distant star, being tugged about by a large companion planet. This is also assuming that we're sitting in or close to the plane of that planet's orbit, so it won't be getting tugged up and down, just around in a circle. Knowing all of this, it should be fairly intuitive to realize that the star's radial velocity would follow a sine wave--when the planet is approaching us, the star will be moving away, and when the planet is moving away from us, the star will be moving closer. When the planet is between us and the star, and vise-versa, the star would appear to be sitting still.

Velocity of the star, plotted against time. The amplitude of the star's velocity is $K$ (when it moves directly toward or away from us, we see it moving at its true orbital velocity of $K$ or $-K$), and the length of one period is (surprise!) $P$.

e) What is the velocity amplitude, $K$, of an Earth-mass planet in a 1-year orbit around a Sun-like star?

We have the equation for $K$ that we came up with in part b), so this is merely a matter of plugging in $M_\oplus$ for $M_p$, 1 AU for $a_p$, 1 year ($3.16 \times 10^7$ seconds) for $P$, and $M_\odot = 2\times10^{33}$ g for $M_*$. The result: a whopping 9 cm/s, which WolframAlpha helpfully lets us know is, among other things, 10% faster than the top speed of a sloth.

Credit and appreciation goes out to Jennifer Shi, Louise Decoppet, and Scott Zhuge for working through this one with me.

Heading Inside a White Dwarf: Stellar Scaling Relationships, Part III

A white dwarf can be considered a gravitationally bound system of massive particles.

a) What is the relationship between the total kinetic energy of the electrons that are supplying the pressure in a white dwarf, and the total gravitational energy of the WD?

What we have here is an application of our old friend the Virial Theorem, $1/2U=KE$. The total kinetic energy of all of the electrons in the white dwarf is therefore the sum of all of the electrons' individual kinetic energies (bear in mind that these are not all of the particles in the WD, and thus their total kinetic energy is not the total kinetic energy within the WD--this'll be important to remember later). If we assume $N$ electrons, this gives us \[-\frac{3GM^2}{5R} = Nm_{e^-}v_{e^-}^2\]

b) According to the Heisenberg uncertainty Principle, one cannot know both the momentum and position of an election such that $\Delta p \Delta x > \frac{h}{4\pi}$. Use this to express the relationship between the kinetic energy of electrons and their number density $n_{e^-}$.

Our first step is going to be to write the kinetic energy of the electrons in terms of their momentum, similar to what we did in part c) of the last post. This gives us \[KE = \frac{Np_{e^-}^2}{2m_{e^-}^2}\] as the kinetic energy of all of the electrons. We can then substitute in $\Delta p_{e^-}$ for $p_{e^-}$, and now we'll stop worrying too much about constants and equations and move into the realm of scaling relationships. Assuming we are approaching the limit where $\Delta p_{e^-} \Delta x = \frac{h}{4\pi}$, we know that $\Delta p_{e^-} \sim \frac{1}{\Delta x}$ (everything else in there is just a constant). Next, we know that number density $n_{e^-}$ is equal to $\frac{N}{V}$, and can approximate $V$ as $\Delta x ^3$, so $n_{e^-} \sim \Delta x ^{-3}$. We can then relate $n_{e^-}$ and $\Delta p_{e^-}^2$ as they appear in our equation as $\Delta p_{e^-} ^2 \sim n_{e^-}^{2/3}$. Finally, we can substitute $\frac{M}{m_{e^-}}$ in for $N$, giving us a final relation between the kinetic energy of electrons and their number density:\[KE=\frac{Mn_{e^-}^{2/3}}{m_{e^-}^2}\] Since $m_{e^-}$ is a constant, the scaling relationship here is $KE \sim Mn_{e^-}^{2/3}$.

c) What is the relationship between $n_{e^-}$ and the mass $M$ and radius $R$ of a WD?

Once again, we know that $n_{e^-} = \frac{N}{V}$, and that $N = \frac{M}{m_{e^-}}$. Volume, of course, is $4/3\pi R^3$. Dropping the constants $m_{e^-}$ and $4/3\pi$, we have\[n_{e^-} \sim \frac{M}{R^3}\]

d) Substitute back into your Virial energy statement, aggressively yet carefully drop constants, and relate the mass and radius of a WD.

Remembering from part b) that $KE \sim Mn_{e^-}^{2/3}$, and from part c) that n_{e^-} \sim \frac{M}{R^3}, we know that\[KE \sim \frac{M^{5/3}}{R^2}\] We can plug this into the right side of our Virial statement, giving us \[-\frac{3GM^2}{5R} \sim \frac{M^{5/3}}{R^2}\] As for the left side, it is a simple matter to start aggressively dropping constants ($3/5$ and $G$) and reveal that the left side scales as $\frac{M^2}{R}$. If we solve for $M$ in the resulting scaling relation, we find that \[M\sim\frac{1}{R^3}\]This is actually a pretty fascinating result, which tells us that the more massive a WD gets, the smaller it is. While initially extremely unintuitive, this actually makes a fair amount of sense when we consider the process of gravitational contraction that has made the WD so small to begin with, and the electron degeneracy pressure that is preventing it from collapsing into a neutron star. The heavier it is, the harder it will be for degenerate electrons to hold the WD up, and the more it will contract under the inexorable force of gravity.

Credit and thanks to Anne Madoff, Scott Zhuge, Louise Decoppet, and Jennifer Shi to working through to this surprising and cool result with me.