Class description: In this semeser, students form four groups to tackle the problem sets. Representatives of each group will present the solution to the problem introduced a week ago. The goal is to acquire broad knowledge of astrophysical phenomena and the ability to estimate quantities order-of-magnitude.

Prerequisites: Good understanding of astrophysics.

Grades: Pass or fail.

References: to be listed

Problem sets:
Week 1 (Saha): "StarShot" project conceives of a gram-scale spacecraft accelerated to "100 million miles an hour" by pushing on it with lasers on or near the Earth. Estimate the cost of the electricity needed to accelerate a 1-gram spacecraft to such a speed.

Week 2 (Feldman): "Save the Earth!"
4179 Toutatis is a ~4 km big, Apollo-class asteroid that frequently approaches Earth, the next time on Nov 9 2069 to within about 8 lunar distances (wikipedia). It goes without saying that an impact by Toutatis would be extremely catastrophic. Let us assume it were to hit Earth in 2069 and that Humanity's only chance is to deflect the asteroid before it hits the planet!

You may assume the following parameters for Toutatis: Aphelion 4 AU, perihelion 1 AU, mass ~ 5x10^16 gram. To simplify the math, assume that the impact would hit Earth dead-center and at the perihelion of Toutatis' orbit. To deflect the asteroid a spaceship ('Serenity') will be launched on intercept course with the asteroid. Serenity will reach Toutatis during its final aphelion before hitting Earth. Your job is to develop a first assessment of this rescue mission. Hopefully there will be enough time to implement it in the next decades!

What is the necessary change in Delta_v for Toutatis to avoid hitting Earth?
Discuss the advantages / disadvantages of the following strategies: gravity tractor, pushing with rocket engines, high velocity kinetic impact, and which of these minimizes the required mass of Serenity (including fuel)? How large is the ratio of rocket mass (including fuel) to the mass of Toutatis required to do the job for a conventional chemical rocket (exhaust velocity 5 km/s) and a hypothetical nuclear propulsion rocket (exhaust velocity 1000 km/s)?

Week 3 (Yoo): Understanding the gravitational wave physics and its detection (pdf)

Week 4 (Saha): Infinite grid of ideal one-ohm registors in xkcd #356, but in our astrophysical thinking class, we assume m and n >>1. What is the resistance between the two nodes? What about the case with 3D grid?

Week 5 (Yoo): Dynamical friction formula and its applications (pdf)

Week 6 (Helled): M-R relation for the polytropic equation of state. For n=2 of the polytropic equation of state, show that the radius of objects in hydrostatic equilibrium does not depend on its mass.

Week 7 (Feldmann): Masses of galaxies (pdf)

Week 8 (Helled): Small hydrogen-helium planets (pdf)

Week 9 (Yoo): Acoustic peak and sound horizon (pdf)

Week 10 AIM week

Week 11 PhD Jamboree

Week 12 (Saha): Accretion around stellar-mass black holes produces radiation that peaks in x-rays. From accretion around supermassive black holes, the spectrum peaks in the ultraviolet. The task is to derive an order-of-magnitude formula relating the peak wavelength (or frequency) to the mass of the accreting black hole.

Radiation comes from near the inner edge of the accretion disc. For this problem, let us drastically approximate that region by a sphere with radius corresponding to the ISCO, and radiating as a blackbody. If we can estimate the blackbody temperature, the result will follow.

If the temperature is too high, the radiation pressure will blow away accreting gas. So the expected temperature will be such that radiation pressure times gas-particle cross-section equals the weight of the gas particles. The gas will certainly be ionised, hence the cross-section will be that of Thomson scattering.

Feel free to use h = c = G = 1 for the calculation. You can always put the factors back in the last step, using dimensional arguments.