Class description: In this "shotgun seminar", students will calculate order of magnitude estimates of astrophysical interest, such as the size of an active galactic nucleus, the height of tides on the earth, the lifetime of radiatively supportive star. The goal is to acquire broad knowledge of astrophysical phenomena and the ability to estimate quantities so that astrophysical scales become intutive.

Prerequisites: Good understanding of astrophysics.

Grades: Pass or fail.

References: to be listed

Problem sets:
Week 1 (Yoo): Advanced LIGO just announced the first ever direct detection of gravitational waves produced by a black hole binary system at r ~ 400 Mpc. Each black hole is of mass 30 Msun, and the gravitational wave frequencies range from 30~300 Hz.

Compute the gravitational wave strain and understand what it means. Also, think of the difference, compared to the Hulse & Taylor system.

Week 2 (Saha): This week's problem continues from the discussion we had last week. Lasers, as we know, are crucial to the detection of gravitational waves. But don't lasers themselves affect spacetime?

The metric around a laser is not simple, but since we are only interested in the order-of-magnitude of the dimensionless metric perturbation h, we can reason from a simpler albeit exotic system: a cosmic string.

Around a cosmic string, the azimuthal angle runs not over 2 pi but has a slight angle deficit or excess. The deficit/excess is proportional to the pressure. This suggests that h around a laser will be proportional to mass per unit length. We have to infer (or guess) the proportionality constant.

Concretely, what is the order-of-magnitude of h around an ordinary laser pointer?

Week 3 (Yoo): A massive elliptical galaxy like M87 contains a supermassive black hole (~10^9 M). Suppose a smaller galaxy with a black hole (~10^8 M) at the center merges, would these two black holes merge? If not, what would happen?

Week 4 (Saha): Estimate the mean density of the Sun, relative to the mean density of the Earth, using information that would have been available to Newton.

Newton did not have the value of G, nor the distance to the Sun.

Week 5 (Saha): The accompanying figure (generated on exoplanets.eu) shows exoplanet radii against mass. As we see, planetary radius increases roughly as R proptp M^(1/3) up to 1-2 Jupiter radii, and then levels off. In other words, planets have a maximum radius of about 10^5 km.

The reason for the maximum radius is that for very massive planets, the pressure becomes enough to ionise atoms. We can estimate this maximum radius by calculating the central pressure for a ball of frozen hydrogen, and equating it to the energy density in electron bonds. Or (more roughly) by equating the gravitational potential energy per particle to the ionisation energy of hydrogen.

The little Python script has the values of the Plankian units. They don't have standard names, but are denoted here as marble, lap, tick, and therm. [Expressions in terms of c, G, hbar, k_B are easy to find online (or derive) but we won't need them.] Also given are the electron and baryon masses in Planckian units, and the dimensionless fine-structure constant (alpha).

1. The Bohr radius is given by 1/(alpha me). Using the given values, verify that you get about 0.5 A.

2. The ionization energy of hydrogen is (1/2) me alpha^2 -- we can imagine an electron moving with alpha times the speed of light. Verify that the equivalent temperature is about 160 000 K. [Hydrogen can actually ionize much below this temperatures, we'll explore the reason for that another time.]

3. In solids, the typical inter-atomic separation is about 4 Bohr radii. Verify that this gives about the right density for water.

4. Show that the threshold radius for pressure-ionisation in a ball of solid hydrogen is propto 1/(me mb). Estimate the numerical coefficient and see how your answer compares with 10^5 km.

Week 7 (Yoo): As we discussed, the optical depth represents a measure of how far we can see. The CMB optical depth is tau = 0.08 (Planck). What is the reionization redshift? Simplify the calculation and discuss the assumptions.

Quasars at high redshifts provide a powerful way to probe the reionization epoch. Compute the Gunn-Peterson optical depth and why is it so different from the CMB one?

Week 8 (Yoo): Intensive repeat imaging over numerous stars in LMC led to detection of microlensing events. The microlensing optical depth is a direct observable of such surveys, defined as the probability that a star gets magnified.

Assuming that microlensing events are produced by a compact object of mass 10 Msun and our halo is composed of such objects, compute the microlensing optical depth and compare to the observed value ~ 1e-7 (need to assume the density of our halo and distance to LMC; simplify the situation).

Why is gravitational lensing useful for probing dark objects?

Week 9 (Saha): This week's problem, as discussed, is to estimate the size of an old supernova remnant.

The ingredients are: 1. The size of the blast wave in the first nuclear explosion -- see the picture in https://en.wikipedia.org/wiki/Trinity_(nuclear_test) and the accompanying information.

2. The Taylor-Sedov scaling argument.

3. The energy output of a supernova (excluding the part in neutrinos) and a plausible ISM density (say 1 atom per cubic cm.)

Answer can be size in ly for an age in yrs.

Week 10 (Saha): This week's problem, as discussed, is to estimate the error on an 18th-century speed-of-light measurement.

The measurement had two ingredients: 1. Timing of a Jupiter satellite to get eclipsed by the Earth. This gave the speed of light in AU/s.

2. The time for a Venus transit to cross the Earth. This measured an AU. For this derive the crossing time

R/(pi AU) P Pv / (P - Pv)

where P, AU, R are the Earth's orbital period, orbital semi-major axis, and radius, while Pv is the orbital period of Venus. Then make some kind of error estimate for timing the beginning/end of a transit.

Assume the timing error from clocks was negligible.

Week 12 (Yoo): Consider a small object orbiting a parent star. Its circular motion balances the gravitational pull, and radiation pressure from the star, if you like. The object absorbs the incident radiation and immediately re-emits away the energy isotropically but in its rest frame. This takes away some of its angular momentum.

Understand the physics behind this mechanism and compute how long it would take the object fall into the parent star through this process. Would we on Earth be worried about it?

Week 13 (Saha): Imagine a space elevator consisting of a single cable, anchored on the ground at the equator. Calculate the tension as a function of height, assuming the tension is zero at the ground base and at the top.

- How high is the elevator?

- Where is the tension maximal?

- What is the molecular binding energy (energy per unit mass) needed to withstand the tension?

Week 14 (Yoo): Technical advances in spectroscopy have been crucial in detecting the extra solar planets. Stability and precision in spectroscopic measurements allow one to measure the change in wavelength as small as caused by perturbation by the planet to its host star.

The current technology is already capable of measuring the perturbation of a five earth-mass planet at the Mercury orbit. What is the radial velocity of the Sun in this case? What would that be for the Earth to the Sun?

The Universe expands, also causing the change in wavelength of light emitted in the past. The redshift of an object will be different if measured in two different epoch, say, a century apart, which is a direct measurement of the cosmic expansion. How large is the signal in terms of radial velocity?