More Final Exam Review Questions

1. What can you use the Virial Theorem to derive? Go ahead and do the derivations!
2. What is the temperature of a black marble placed in an orbit 2 AU away from the Sun?
3. What is the Jeans Mass and Jeans Radius for a giant molecular cloud, how does they relate to star formation, and how do you derive these quantities? HINT: there are two methods
4. What is the speed of a Jupiter-mass planet in orbit 1 AU from a 1 Msun star? How does this speed relate the the speed of the central star?
5. What is the approximate relationship between the location of the habitable zone around a star and the mass of the star? 
6. What is the transit duration of a Jupiter-size planet around a 1 Msun star? How about for a 2 Msun star?
7. What is the scale height of a planet's atmosphere? How does the scale height of a nitrogen-dominated atmosphere (like ours) compare to that of a pure hydrogen atmosphere?
8. Two stars are in orbit around each other separated by 1 AU. Star A has a mass of 2 Msun while Star B has a mass of 0.5 Msun. What are their relative speeds? What are their relative semimajor axes? What are their orbital periods? If they eclipse, how long does the eclipse last (keep in mind that both stars are moving)?
9. How can we use the Sun's spectrum, along with other observations of the Sun to measure the AU?
10. Two planets orbit a Sun-like star. One planet has a radius of 1 Rjup, and the other has a radius equal to the Earth's. Compare the transit depths.
11. How does the velocity of a star orbited by a planet depend on the mass of the planet, the semimajor axis of the orbit and the mass of the star?
12. How fast is a particle moving in a gas cloud of temperature T?
13. Astronomers often assume that the luminosity of a main-sequence star scales as $L \sim M^4$. Where does this come from?
14. The target field of the NASA Kepler Mission was at an RA of 18 hours and a declination of +30. When does the target field cross the meridian at midnight? Can we observe stars in the Kepler field from Cambridge tonight?
15. What is the flux at the surface of a star of radius $R_\star$ and temperature $T$? How does this flux change at a distance $d > R_\star$?
16. Check out the visual binary star Alberio by doing a Google images search. Compare the properties of the two stars.
17. Why are red dwarf stars such good targets for searching for habitable-zone planets?
18. What is the main sequence? How is luminosity related to effective temperature on the main sequence?
19. If a star is 4 magnitudes brighter than another star, what is the flux ratio of the two stars? Try this question for various values of the magnitude difference.
20. How does the flux of a star depend on its distance? How does its magnitude depend on distance?
21. What are the (approximate) transit parameters of the following transit light curves assuming that all of the planets have 3-day orbits?
    
22. Assuming the central star has a mass of 2 $M_\Sun$, what are the properties of this planet? Compare your properties to that of Pollux b.
    
23. How does the luminosity of a star depend on its temperature and radius?
24. How does the surface flux of a star depend on it's temperature and radius?
25. What is the speed of Jupiter compared to the speed of the Earth? Compare their momenta. Compare their kinetic energies.
26. What is the diffraction limit of the human eye observing at 0.5 micron?
27. What is the diffraction limit of a 100 meter telescope observing at 1mm?
28. What is the angular diameter of the Sun as viewed from Saturn (approximately 9 AU)?
29. What is the angular diameter of the Sun as viewed from alpha Centauri A (approximately 1 pc)?
30. How many AU away is alpha Cen A?
31. If the Sun were powered by gravitational collapse, how long would it shine at its current luminosity?
32. How does the lifetime of a star scale with its mass? Look up the stars in the alpha Cen triple system on Wikipedia. Compare their lifetimes.

Midterm review practice problems

It's that time of year again . . . finals.  After the midterm, I wrote some addtional problems on that material to help those who wanted to review it.  I figured I'd make these available as practice problems for those who want to revisit this material before the final. It should be emphasized that the final will focus on post-midterm material, but understanding the first half of the class well will undoubtedly help you both now and in your future endeavors!

On the mass conservation equation of stellar structure

In class, this perhaps was not completely transparent.  Let's try again.  Consider a spherical shell of thickness dr located at radius r from the center of a sphere.  It will have surface area \(4\pi r^2\) and we can, since dr is very small, get its volume just by taking it to be a rectangular prism with area \(4\pi r^2\) and thickness dr, so its volume is
$$dV=4\pi r^2 dr.$$

At a radius r, the star has density \(\rho (r )\), so the differential mass enclosed by this shell is just
$$dM=\rho( r)dV=4\pi r^2 \rho (r ) dr.$$

Dividing both sides through by dr, we have the mass conservation equation:
$$\frac{dM}{dr}=4\pi r^2 \rho (r ).$$

Note that if we integrate from zero up to radius R, we get the total mass within radius R:

$$M (<R )=\int_0^R 4\pi r^2 \rho ( r) dr.$$

Note also that differentiating this integral with respect to R will give
$$\frac{dM}{dR}=4\pi R^2\rho ( R)$$
by the fundamental theorem of calculus!

In class, Professor Johnson argued that we have
$$M=\frac{4}{3}\pi r^3\rho ,$$ so we can differentiate to get
$$\frac{dM}{dr}=4\pi r^2 \rho .$$

This is not quite right.  Mass only equals \( \frac{4}{3}\pi r^3 \rho\) if \(\rho \) is the average density of the star, and that is not a function of radius: it is just total mass divided by total volume.  So we do not recover that we must evaluate the density at a specific radius in the mass conservation equation if we approach it this way.  Also, if we did make \(\rho \) a function of radius, then differentiation would lead to two terms by product rule, as Jimmy pointed out in class.  So this derivation is not the best way to think about mass conservation.

Earth's atmosphere: how good is our model from class?

In class, we worked out how the density would scale with radius in the Earth's atmosphere using 2 assumptions: 1) taking the gravitational acceleration to be constant and equal to \(g=\frac{GM_{\oplus}}{R_{\oplus}^2}\) and 2) taking the atmosphere to be isothermal (all at the same temperature T). Here, I'll discuss what happens when you relax those assumptions, one at a time, and then go on to discuss a more realistic model of Earth's atmosphere. Let's first relax assumption 1) but keep assumption 2). Set up force balance on a parcel of mass with mass m, area A, and thickness \(\Delta r\) at a radius r above the center of the Earth: $$F_g+A(-P(r+\Delta r)+P(r))=0.$$ Now before we had used \(F_g=mg\) for the force of gravity, but now let's account for the fact that the mass is at some radius \(r>R_{\oplus}\). In other words, \(mg\) is the gravitational force at the Earth's surface; but we want the force at some distance \(h=r-R_{\oplus}\) above the surface. We have $$F_g=-\frac{GM_{\oplus}m}{r^2}.$$ We can then write $$-\frac{GM_{\oplus}m}{r^2}=A(P(r+\Delta r)-P(r )).$$ Now note that for a parcel with density \(\rho( r)\), the mass is \(m=A\Delta r\rho(r )\). Note that we have let the parcel's density be a function of \(r\) here because the density is changing as you go up in the atmosphere. We don't yet know \(what\) function of \(r\) \(\rho\) is, but we'll find out! Rewriting our previous result by replacing the mass and then dividing both sides by area \(A\) and \(\Delta r\), we get $$-\frac{GM_{\oplus}\rho(r )}{r^2}=\frac{dP}{dr}.$$ Note that, just as we did in class, I've rewritten \( (P(r+\Delta r)-P( r))/\Delta r \) as \(dP/dr \). We now use the ideal gas law, which is a good approximation for our atmosphere, provided we use the right average particle mass from the atmosphere's measured composition, to replace \( P(r )=\rho( r) k_B T/\bar{m} \), where we remember assumption 2) is still in force so \(T\) is not a function of \(r\). \(\bar{m}\) is the average mass per particle, which we used to write the number density in the ideal gas law as a mass density over a mass. We now have the differential equation $$ -\frac{GM_{\oplus}\rho(r )} {r^2}=\frac{d\rho}{dr}\frac{k_B T}{\bar{m}}.$$ We can rearrange this to read $$-\frac{GM_{\oplus}\bar{m}}{k_B T r^2}dr=\frac{d\rho}{\rho}.$$ We can now integrate both sides of this equation to find \(\rho (r )\), but before we do so, let's look at the physical meaning. The left-hand side (lhs) compares the gravitational energy due to a fractional increas in radius \(\Delta r/r)\), with the thermal energy, \(k_B T\). To make that clearer, remember, gravitational PE is $$PE=-\frac{GM_{\oplus}m}{r},$$ so we can rewrite $$-\frac{GM_{\oplus}m}{r^2}dr=-\frac{GM_{\oplus}}{r}\frac{dr}{r}=PE\times\frac{dr}{r}.$$ So this is saying, if we go up by a fractional radius \(\dr/r\), then the potential energy will change by PE times that fraction. How much the PE changes relative to the thermal energy determines the fractional change in density, \(d\rho/\rho\); that's what the right-hand side is telling us. Now let's solve: $$-\int_{R_{\oplus}}^r \frac{GM_{\oplus}\bar{m}}{k_B T r^2}dr=\int_{R_{\oplus}}^r \frac{d\rho}{\rho}.$$ We find $$\frac{GM_{\oplus}\bar{m}}{k_B T}\frac{1}{r}\bigg|_{R_{\oplus}}^r=\ln \rho \bigg|_{R_{\oplus}}^r,$$ which can be rearranged and simplifed to give $$\rho ( r)=\rho(R_{\oplus})\exp\bigg[\frac{GM_{\oplus}\bar{m}}{k_B T}\big[\frac{1}{r}-\frac{1}{R_{\oplus}}\big]\bigg].$$ To make sense of this, let's write \(r\) in terms of \( R_{\oplus}\) and height \(h\) above the Earth's surface, as $$r=R_{\oplus}+h=R_{\oplus}\left(1+\frac{h}{R_{\oplus}}\right).$$ We can then Taylor expand \(r^{-1}\) about \(h=0\), meaning \(h/R_{\oplus}\ll 1\). Doing so we have $$r^{-1}\approx \frac{1}{R_{\oplus}}\left(1-\frac{h}{R_{\oplus}}+\left(\frac{h}{R_{\oplus}}\right)^2+\cdots\right).$$ Let's see what this does in our expression for \(\rho\), specifically in the term $$\big[\frac{1}{r}-\frac{1}{R_{\oplus}}\big].$$ We can see the 1 in the expansion will be canceled out by the \(\frac{1}{R_{\oplus}}\) above, and at lowest order in \(h\) we recover the result from class for constant gravitational force, \(\rho \propto e^{-h}\). We have $$\rho (h )=\exp\left[\frac{PE}{E_{th}}\left[-\frac{h}{R_{\oplus}}+\left(\frac{h}{R_{\oplus}}\right)^2\right]\right].$$ Note that I've rewritten this in terms of the ratio of potential energy to thermal energy, for the physical reasons discussed earlier (before we integrated). You can see that accounting for the variation of the force of gravity with \(r\) slightly increases the density, because the term in \( \left(h/R_{\oplus}^2\right) \) is positive. Note that it's also small, because typical heights would be much less than the Earth's radius. Using that idea, we can even Taylor expand the whole exponential, this time by factoring out \(\frac{h}{R_{\oplus}}\) to have $$\exp\left[-\frac{PE}{E_{th}}\frac{h}{R_{\oplus}}\left[1-\frac{h}{R_{\oplus}}\right]\right]=\exp\left[-\frac{PE}{E_{th}}\frac{h}{R_{\oplus}}\right] \times \exp\left[-\frac{PE}{E_{th}}\frac{h}{R_{\oplus}}\frac{h}{R_{\oplus}}\right].$$ Approximating the second exponential, we have . . .