Comparison to Wasp-10b

This post is going to compare the parameters we get from analyzing these graphs

both graphs provided on Worksheet 11.1

with actual values from exoplanet.org.  

First, let's start with the impact parameter, b.  

$b=1-\delta^{1/2}\frac{T}{\tau}$

In the light curve above, 

$\delta$=0.028  

T = 102 minutes

$\tau\approx$18 minutes

Using these values and the equation above, I found b=0.99, which is a lot higher than the value given by exoplanet.org, b=0.3.  I believe the reason for this difference is a difference in data.  For example, in the light curve above, the length of the transit looks to be a little longer than an hour and a half, but on exoplanets.org, the transit is at least half an hour longer than that.  

Next, let's look at the ratio of the radii of the planet and star.  

$\delta=\frac{R_P^2}{R_*^2}$

$\frac{R_P}{R_*}=\delta^{1/2}$

For this system, we find that the ratio is equal to 0.17, so the planet is 0.17 times as big as the star.  

On worksheet 11.1, it says that the star is 0.8$R_\odot$.  With this information and the ratio we just found, we can tell that the planet is about 1.36 Jupiter radii big.  This is closer to the website's value of about 1.08 Jupiter radii.  The reason for this difference could be that the website lists the sun as being 0.7 solar radii big.  

Now we'll look at the ratio $a/R_*$.

On the worksheet, not taking into account the impact parameter, I used the equation 

$\frac{a}{R_*}=\frac{P}{\pi T}$

This equation gives us a ratio value of about 13.3, which is pretty close to the website's value 11.9.  The difference can be attributed to the fact that the website found a semimajor axis value dependent on the impact parameter.  

Finally we'll look at the densities of the star and planet.

We all know that density equals mass over volume.  But the fun thing about the mass of a star is that it can be related to semimajor axis and period of the system using Kepler's Third Law.

$M_*=\frac{4\pi^2a_P^3}{GP^2}$

$\rho=\frac{3\pi a_P^3}{R_*^3GP^2}=\left ( \frac{a_P}{R_*} \right )^3\frac{3}{GP^2}$

When we substitute provided values into this equation, we find that the density of the star is about 1.45 grams per cubed centimeter, which is pretty darn close to the website's value of 1.51 g/cm^3.  

WS 11.2, Problem 3: Goldilocks Zones

A Goldilocks Zone, or a "habitable zone," is a distance from a star where it is neither too hot nor too cold for a planet to have liquid water.  This is the criterion for life that most people have a greed on, but we're not sure that it's the only or most important factor in determining whether or not life can exist on a planet, so it's not the end all be all of life on other planets.

Now let's fi gure out what stars make the best targets for both the radial velocity technique and the transit method, assuming we are primarily interested in fi nding planets in their stars' habitable zones. For a planet in the habitable zone of a star of mass $M_*$...

a)  How does the Doppler amplitude, $K_{HZ}$, scale with stellar mass?

To solve this problem, let's first recall the equation for the Doppler amplitude that we found on a previous worksheet, generously dropping unnecessary constants.

$K\sim\frac{1}{M_*^{4/6}P^{2/6}}$

This isn't as obvious as it might seem, though, because an object's period depends on its mass, so we have to do some more algebra.

$P^{1/3}\sim\left ( \frac{a_P^{3/2}}{M_*^{1/2}} \right )^{1/3}=\frac{a_P^{3/6}}{M_*^{1/6}}$

Of course, the semimajor axis of a planet, $a_P$, is also dependent upon its star's mass.

$a_P^{3/6}\sim M_*^{3/6}$

When you combine all of these scaling factors, you find that

$K\sim\frac{1}{M_*^{4/6}M_*^{2/6}}=\frac{1}{M_*}$

b)  How does the transit depth, $\delta_{HZ}$, scale with stellar mass?

For this, we have to remember that the depth of the transit depends on the ratio of the radii of the star and planet.

$\delta\sim\frac{R_P^2}{R_*^2}$

We can use mass-radius scaling relations to simplify this

$\delta\sim\frac{1}{R_*^2}$

$M\sim R$

$\delta\sim\frac{1}{M_*^2}$

c)  How does the transit probability, Prob$_{HZ}$, depend on stellar mass?

We learned on worksheet 11.1 that the probability that a planet will pass in front of its star is

$\frac{R_*}{a_P}

Now we have to remember both the mass-radius scaling relation and the relationship between a star's mass and $a_P$.

$M\sim R$

$a_P=\frac{M_*a_*}{M_P}$

Then we can rewrite the probability as

$\frac{M_*M_P}{M_*a_*}$

So the probability of a transit happening doesn't really depend on the star's mass at all.


d)  How does the number of transits (orbits) per year depend on stellar mass?

This wording of this question is a little unclear, but it reminds me of part (c) from problem 2 on this same worksheet.

If the Sun were half as massive and the Earth had the same equilibrium temperature, how many days would our year contain?

To solve this, we set up a proportion between the period of a habitable zone planet and the period of our Earth under the conditions specified in the problem.

$\frac{P_{HZ}}{P_\oplus}\sim\frac{a_{HZ}^{3/2}M_\odot^{1/2}}{M_*^{1/2}a_\oplus^{3/2}}$

If we drop the constants, we get

$P_{HZ}\sim\frac{a_{HZ}^{3/2}}{M_*^{1/2}}$

Now we can substitute in the equation for the semimajor axis of the habitable planet and simplify to get

$P_{HZ}\sim M_*$

But the problem asks for the relationship between a star's mass and the number of orbits of a planet, which is inversely proportional to its period, so

Transits$\sim\frac{1}{M_*}$

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?

From parts (a) through (d), we can see that the Doppler amplitude and number of transits are both inversely proportional to stellar mass and the depth of the transit goes as the inverse square of the stellar mass.   We want all of these quantities to be as large as possible.  A high Doppler amplitude makes a transit more noticeable on an RV graph; a deep transit is more noticeable on a light curve; the more transits there are, the more opportunities we have to gather data.

Based on this, we can see that lower-mass stars make better targets for finding habitable-zone planets.


In doing this, we ignore the fact that lower-mass stars are harder to detect in general because their luminosities are so low.

WS 11.1, Problem 3: What's in a Light Curve?

This blog post is loosely modeled after problem 3 on worksheet 11.1.  I'm not going to answer the specific questions from the problem to generalize to all sets of data, but I am going to follow the path that the problem sets to discover what can be learned from a transit light curve.  This should be fun.

The problem gives us three pieces of information.

• The system's period is 3.1 days (not necessary since I'm not actually answering the parts of the problem, but now we know we can find a lot of the system's properties with its period).  
• This light curve

• And this radial velocity graph

The first part of the problem asks us to qualitatively describe the brightness distribution of the star.  We can tell this by looking at the bottom of the light curve.  The dots along the curve represent the amount of light in each image taken of the system.  The dots stay fairly close to the line, so the star's brightness is relatively evenly distributed.  But because the dip is more of a curve than a straight line, so we know that the center of the star is slightly more luminous than the edges.  This is to be expected due to the limb darkening effect, which basically just says that the center of the star will appear brighter.  (Yay for self-explanatory scientific terms!)

The second part asks us to find the ratio of the radii of the planet and star.  This is a little bit trickier than the first part because it involves actual math, but it's not that bad.  The theory behind the solution to this problem is that that the depth of the light curve depends on the ration of the areas of the two objects.  Actually solving the problem is a pretty logical process.  If the star is really big, the light curve will be pretty shallow.  In the same vein, the curve will be shallow if the planet is small (because it's not blocking a lot of the star's light).  So the depth of the curve goes as 

$\frac{R_P^2}{R_*^2}$

When the problem asks us to find the ratio of the objects' radii, we can find the average depth of the light curve and take its square root.  

The third part asks us to find the ratio of the planet's semimajor axis with the star's radius 

$\frac{a_P}{R_*}$

To solve this problem, we look at the duration of the transit.  If you believe that velocity is distance over time (which I do), then you can also believe that time is distance over velocity.  The distance that the planet has to travel to complete a transit is twice the star's radius, and it's velocity is $v_P$.  Lucky for us, the velocity of the planet is dependent upon its semimajor axis in the following way:

$v_P=\frac{2\pi a_P}{P}$

which means the ratio of the semimajor axis to the star's radius should be 

$\frac{a_P}{R_*}=\frac{P}{\pi t}$

where t is the duration of the transit.

It's important to realize that this equation is only true for equatorial transits (transits that happen along the equator of the star).  A more general equation would account for the transit time of a transit that travels along any chord on the star.

$T=T_{equatorial}(1-b^2)^{1/2}$

where b is called the impact parameter, which is basically the place on the star where the planet transits.  For equatorial transits, b is 0.  For transits where the planet just barely grazes the star--so it doesn't really cause any dip in the light curve at all--b is 1.  

$b=1-\delta^{1/2}\frac{T}{\tau}$

where $\tau$ is the time of egress--the time from the moment the planet first touches the star to the moment the last point of the planet crosses the first edge of the star--and delta is the depth of the transit.  

One of the parts of the question asks us to show that the ratio found above is related to density.  I started with the equation for density

$\rho_*=\frac{M_*}{4R_*^3}$

assuming that $\pi$ is 3.

But we can simplify this further by using the mass-radius scaling factor found in this post, which says that mass scales proportionally with radius.  

$\rho_*=\frac{1}{R_*^2}$

Because of this relationship, we can say that the planet's semimajor axis is related to the star's density in the following way:

$a_P=\frac{PR_*}{\pi t}=\frac{P}{\pi t sqrt{\rho_*}}$

If we know the radius of the planet (which we do, because we know the ratio of the radii of the two objects and the radius of the star), we can also find its density based on the 

$\rho_*=\frac{1}{R_*^2}$

equation.  

Here we have it.  Armed with three simple (I say simple, but in actuality, those graphs are not too easy to get) pieces of information, we can find the following properties

• ratio of the radii of the two objects in the system
• semimajor axis of the planet (which we can then use along with the mass ratios to find the semimajor axis of the star)
• the densities of both the star and the planet
• the impact parameter

Like I said at the beginning of this post, light curves are fun!

  

WS 10.2, Problem 2: Wobble, Baby, Wobble

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 $v_*$ .

We started with the basic equation 

$a_*m_*=a_Pm_P$

We can also relate the semimajor axis and velocity in the following way

$v=\frac{2\pi a}{P}\Rightarrow a=\frac{vP}{2\pi}$

The first equation can now be rewritten and simplified as 

$\frac{v_*P}{2\pi}m_*=\frac{v_PP}{2\pi}m_P$

$v_*m_*=v_Pm_P$

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.  Convert your units so that your expression is given as a speed in meters per second, with P measured in years, $M_*$ in solar masses and planet mass in Jupiter masses. 

Again let's start with the last equation from part (a)

$v_*m_*=v_Pm_P$

We can rearrange this to find the equation for the velocity of the star

$v_*= \frac{v_Pm_P}{m_*}$

We also know the following relationship between period and velocity

$v_P=\frac{2\pi a_P}{P}$

So now we can find K, the speed of the star

$K=v_*=\frac{m_P}{m_*}v_P=\frac{m_P}{m_*}\frac{2\pi a_P}{P}$

c) 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. Specfi cally, 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?

To someone who's grown up thinking that the solar system revolves around the sun, this problem might be a little bit conceptually difficult, because if the sun is truly the center of the solar system, it shouldn't move.  The point of this problem, however, is to show that the sun is not the exact center of the solar system (as I'm typing this, a bunch of dead white men are probably rolling in their graves because this problem debunks the work they devoted their lives to).  Instead, the sun wobbles--it moves away from the system's center of mass as the planet moves.  

If a planet is moving away from our viewpoint, the sun--assuming it only moves in our visual plane, and not up and down--is moving toward us.  If the planet is moving toward us, the sun has to compensate in its wobble and move away from the center of mass, or away from out viewpoint.  The sun's radial velocity would appear to be 0 (moving neither towards nor away from us) when the planet is at a point in its orbit when it's not moving towards or away from us, so when it's on the side of its orbital path.

d)  Sketch the velocity of a star orbited by a planet as a function of time. Denote the maximum velocity as K, and express the x-axis in terms of the number of periods, in intervals of P/4.

For this graph, we're starting at the point where the radial velocity is zero. The K is at it's peak hi and low at every other 1/4 of a period.  

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

This is another simple plug-and-chug problem where we take the equation from part (b) and substitute the sun's values where necessary.  

Thanks to WolframAlpha, I was able to find a K value of about 9.4 cm/s.  Earth's pretty freaking slow. 

WS 10.2, Problem 1: Center of Mass

In the first part of this problem, we were asked to draw a picture illustrating the relationship between the masses of a star and planet and their center of mass.

The relationship between the center of mass and and masses is

$x_{COM}=\frac{(-a_*m_*)+(a_Pm_P)}{m_*+m_P}$

If we set the center of mass to 0, like it is in the picture, the equation becomes 

$a_*m_*=a_Pm_P$

b) In a two-body orbital system the variable a is the mean semimajor axis, or the sum of the planet's and star's distances away from their mutual center of mass.  Label this on your diagram. Now derive the relationship between the total mass, , orbital period P and the mean semimajor axis a, starting with the Virial Theorem for a two-body orbit (assume circular orbits from here on).

So we start with the Virial Theorem like the problem told us to.  

$K=-\frac{1}{2}U$

$\frac{1}{2}(m_*v_*^2+m_Pv_P^2)=\frac{Gm_*m_P}{a}$

In order to conserve momentum, the periods of the star and planet have to be the same.  

$v_*=\frac{2\pi a_*}{P}$

$v_P=\frac{2\pi a_P}{P}$

The star is so much more massive than the planet, so it moves a lot more slowly.  When the two are compared, the velocity of the star is practically 0.  

The distance from the center of mass to the planet, $a_P$ is also a lot larger than the distance to the star, $a_*$.  Since $a=a_*+a_P$, $a=a_P$.  

We can now rewrite the Virial Theorem as 

$m_P\left ( \frac{4\pi^2a_P^2}{P^2} \right )=\frac{Gm_*m_P}{a_P}$

and we can rearrange it to say

$m_*=\frac{4\pi^2a_P^3}{GP^2}$

which is basically just Kepler's Third Law of Motion, so yay!

c) By how much is the Sun displaced from the Solar System's center of mass (a.k.a. the Solar System \barycenter") as a result of Jupiter's orbit? Express this displacement in a useful unit such as Solar radii.

This should be a pretty simple plug and chug problem.  

$a_\odot M_\odot=a_{Jup}M_{Jup}$

$a_\odot=\frac{a_{Jup}M_{Jup}}{M_\odot}$

Plugging in all of those values, we find that the sun is displaced $5\times 10^4$cm, or $7\times 10^{-7}$ solar radii.  

WS 10.1, Problem 2: White (I still don't know why the plural is spelled this way) Dwarfs

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?

This part is pretty much just a bunch of algebraic manipulation of the Virial Theorem.

$K=-\frac{1}{2}U$

The above equation, though, is only useful for relating the energies of one electron.  In order to find the relationship for the whole white dwarf, you have to multiply by the total number of electrons.  

$N_e\frac{1}{2}m_e v^2 = \frac{3}{10}\frac{GM^2}{R}$

but because of the relationship between momentum and velocity, 

$N_e\frac{1}{2}m_e \frac{p_e}{m_e}^2 = \frac{3}{10}\frac{GM^2}{R}$

$N_e\frac{p_e^2}{m_e} = \frac{3}{5}\frac{GM^2}{R}$

b)  Express the relationship between the kinetic energy of electrons and their number density n (Hint: what is the relationship between an object's kinetic energy and its momentum?)

So, we start with the last equation from part (a).  

$KE=N_e\frac{p_e^2}{m_e}$

In the problem, we're given the following information

• $\Delta p\Delta x>\frac{h}{4\pi}\Rightarrow \Delta p\sim \frac{1}{\Delta x}$

• $\Delta p\approx p$

• Volume\sim \Delta x^3$

If number density is number over volume, 

$n_e=\frac{N_e}{V}=\frac{N_e}{\Delta x^3}\sim \Delta x^{-3}$

We went through all of that so that we could find a relationship between the $p_e^2$ in the equation from part (a) and the number density.  

$p_e^2\sim n_e^{2/3}$

The total number of electrons, $N_e$, is equal to the number of protons, which is equal to 

$\frac{M_*}{m_p}$

After all of that, we can write 

$KE=\frac{M_*n_e^{2/3}}{m_p m_e}$

c) What is the relationship between $n_e$ and the mass M and radius R of a WD?

We kind of started solving this part in part (b).

$n_e=\frac{N_e}{V}$

We can get rid of a lot of constants in this equation to end up with 

$n_e\sim \frac{M}{R^3}$

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

First, let's recall the the equation from part (a).

$N_e\frac{p_e^2}{m_e} = \frac{3}{5}\frac{GM_*^2}{R_*}$

Based on all of the relationships we've found in parts (b) and (c), we can rewrite this equation as 

$\frac{M_*n_e^{2/3}}{m_p m_e}=\frac{3}{5}\frac{GM_*^2}{R_*}$

First, let's get rid of the constants we don't really need to have a basic understanding of the relationships between these quantities 

$M_*=\frac{M_*^2}{R_*}$

Finally, after simplifying, we find that the mass and radius of a White Dwarf are inversely proportional 

$M\sim R^{-1}$

WS 10.1, Problem 1: 'Til the Fusion Ends

We've talked about the birth of stars in molecular clouds. We also briefly discussed the main
sequence, on which stars are in hydrostatic equilibrium owing to energy generated by nuclear fusion in their cores. Now let's investigate what happens when a star like the Sun can no longer support itself via nuclear fusion.

a) At what rate is the Sun generating energy?

At first, we were confused by this question.  We knew it couldn't possibly be as simple as we were making it.  Then we realized we were being typical Harvard students and overthinking things, and that luminosity is given in ergs per second.  So, the sun is generating energy at a rate of

$L_\odot=4\times 10^{33}$ ergs/s

b) If fusion converts matter into energy with a 0.7% e fficiency, and if the Sun has 10% of its mass available for fusion (in the core only), how long does it take to use up its fuel supply? What is the general relationship between the mass of a star and its main-sequence lifetime?

The first step was to find the mass of the sun available for fusion.  

$(.1)(M_\odot)=(1\times 10^-1)(2\times 10^{33}g)= 2\times 10^{32}g$

The second step was to find the energy produces by fusion in the core using the equation for energy that I always knew, but never really knew when to use.  

$E=M_{fus} c^2=(2\times 10^{32}g)(3\times 10^{10}cm/s)$

But, the question states that the sun's fusion is only 0.7% efficient, so we have to multiply the energy by the efficiency.  

$E=M_{fus} c^2=(7\times 10^{-3})(2\times 10^{32}g)(3\times 10^{10}cm/s)=1.4\times 10^{51} ergs$

Finally, using dimensional analysis, we were able to find the final equation for the time.  

$t=\frac{E}{L_\odot}=4\times 10^{17}s\approx 1\times 10^{10} yrs$

d) The core will collapse until there is a force available to hold it up. One such force is supplied by degeneracy pressure. The pressure inside of a white dwarf star is provided by the motion of electrons. The electrons are in a tough situation: they can't occupy the exact same state, but there's not much room for them to coexist easily inside of a dense white dwarf. As a result, they must always be in motion to avoid other electrons (roughly speaking). This e ffect becomes important when the inter-particle spacing is of order the de Broglie wavelength $\lambda$, which is related to the momentum via 

$\lambda=frac{h}{v}$  

For a stellar core of a given temperature, which particles reach this critical density fi rst: electrons or protons?

The foundation of our solution to this problem was the equation for momentum:

$p=mv$

We had to compare the momentums of the electrons and photons in the star.  

$p_e=m_e v_e$

$p_p=m_p v_p=1800m_e v_p$ 

Because protons are just so much more massive than electrons, the momemtum of a proton is a lot bigger than the momentum of an electron.  Therefore, protons have a shorter de Broglie wavelength, so the protons reach their critical density first.   

e) If a typical white dwarf has roughly half the mass of the Sun and the radius of the Earth, what is the typical density of a white dwarf in grams per cubic centimeter? What is the volume of white dwarf material that weighs as much as a car?

This was pretty simple. 

$\rho_{WD}=\frac{m}{v}=1.5\times 10^6 g/cm^3$

We then figured that the average car is about 1 metric ton, so about $1\times 10^6$g.  

$v=\frac{m_{car}}{\rho_{WD}}=0.67$ cm^3