Order of Magnitude Problems for the Group (OMG)

Order-of-magnitude estimates enable us to grasp the basic ideas from simple principles without complicated calculations (the so-called back-of-the-envelope calculations), which is extremely helpful in building physics pictures and breaking down unfamiliar problems. These are important trainings for doing research work. We will solve some order-of-magnitude probloms mainly in astrophysics during our group meeting. In the list below, the problems starting with *'s are adopted from various places (e.g., the OSU Order-of-Magnitude Astrophysics), while others are mainly designed or proposed by Zheng. You are also welcome to suggest order-of-magnitude problems.

  1. 2013-01-08
    Space observatories like WMAP, Planck, Herschel, and the planned JWST are all based at the 2nd Lagrange point (L2) of the sun-earth system. How far is it from L2 to earth?

  2. 2013-01-15
    * How much U-235 is there in an atomic bomb?

  3. 2013-01-22
    There is evidence that at the center of almost every galaxy with a spheroidal stellar component (e.g., elliptical galaxies or the bulges of spiral galaxies), there is a supermassive black hole (SMBH), with mass ranging from millions to 10 billions of solar mass. A star around the SMBH may fall into the SMBH (e.g., through perturbations from other stars). At what SMBH mass, a sun-like star can be swallowed into the SMBH horizon without being tidally disrupted?

  4. 2013-01-29
    * Your starship accidentally gets transported into the middle of a globular cluster, and you have no control over the motion of the ship; it will continue in a straight line until you escape the cluster, unless you run into a star first. In order to assess your odds of escaping, you transport a lump of coal with an attached thermometer into empty space. The temperature of the lump of coal drops rapidly at first, then more slowly, asymptoting at a temperature of 25K. What are your odds of escaping the cluster without hitting a star?

  5. 2013-02-12
    * In 1994, VLBI studies of the source GRS 1915+105 showed a blob of radio emission that moved 0.5" in 4 weeks. The distance to GRS 1915+105 is 12 kpc. What is the minimum possible space velocity of this moving blob? If the blob has mass m, how much energy was required to accelerate it to this velocity?

  6. 2013-02-19
    As in the recent news, a meteor exploded above Chelyabinsk, Russia, on Feb 15, 2013, setting off blasts that injured nearly 1000 people. If, instead of fragmenting in the sky, the meteor crashed directly onto the ground, compare the resultant energy release to that of the Hiroshima atomic bomb.

  7. 2013-03-05
    During its 2007 return, Comet 17P/Holmes temporarily brightened by a factor of about half million due to an outburst. What fraction of the comet's mass ended up in the outburst?

  8. 2013-03-19
    * What fraction of the infrared photons emitted by a z=6 quasar make it to the earth?

  9. 2013-04-09
    * The quantity lambda = J|E|^{1/2}G^{-1}M^{-5/2} is a dimensionless measure of the angular momentum of a self-gravitating system. Suppose that a dark matter halo of radius R has lambda=0.05 (gravitational perturbation theory and numerical simulations imply this is a typical value acquired through tidal torques). If the baryons in the halo dissipate their energy while conserving their angular momentum, what will be the size of the resulting disk?

  10. 2013-04-16
    * A black hole of mass M accretes gas at a rate Mdot. The gas forms a geometrically thin, optically thick, steady-state accretion disk. What is the total luminosity of the disk? What is the radial temperature profile? What is the frequency distribution of the emitted radiation?

  11. 2013-04-23
    * At what mass does a moon become spherical?

  12. 2013-05-07 2013-05-14
    * The star HD 209458 has a transiting planet with a period of 3.5 days. The extremely precise light curve allows determination of the start of the eclipse to a precision of a few minutes.

    Suppose there is another planet in the system, at larger orbital radius, with an orbital plane inclined to that of the transiting planet by an angle i. What would be the period of precession of the transiting planet's orbit induced by the second planet? Would this effect be detectable?

  13. 2013-05-21
    * A galaxy initially has a constant density core with all stars on nearly circular orbits. A black hole grows (e.g., through gas accretion) in the middle of the galaxy on a timescale that is long compared to the central orbital time, and reaches a mass of M_BH. What is the final density profile of the stars near the center of the galaxy?

  14. 2013-06-04 2013-06-18
    What is the minimum black hole mass for a quasar of luminosity L? How long does it take to form such a black hole by accretion, if the radiative efficiency is \epsilon? What quasar lifetime can be inferred by comparing the black hole population in present day galaxies to the peak space density of high-z quasars?

  15. 2013-08-13
    What is the slope of the white dwarf sequence in the color-magnitude diagram shown here (black points, WDs in globular cluster 47 Tucanae, taken from Hansen et al. 2013)? Why that of the main sequence (green, SMC stars) is steeper?
    (N.B. the axis labels are masked on purpose. Once you figured out an answer, you can check the unmasked diagram.)

  16. 2013-10-02
    * You discover an object in the ecliptic, at opposition, moving with angular speed of 1.4 arc-seconds per hour, with apparent visual magnitude of 18.8. What is its distance from the Sun, and what is a lower limit to its size? Hint: it is beyond Pluto.

  17. 2013-10-09
    Estimate the (comoving) scale R_{BAO} of the Baryon Accoustic Oscillation feature (bump) seen in the galaxy two-point correlation function.
    The comoving scale of the first peak in the CMB angular power spectrum corresponds to 2*R_{BAO}. How large an angular separation does the first peak correspond to?
    For the above estimation, let's assume spatially flat universe with Omega_m=0.25. If you need the radiation component, you can assume Omega_r=8.4x10^{-5}.

  18. 2013-10-30
    The mass of neutron stars is peaked around 1.4Msun (see this plot from arXiv:1309.6635). For the estimation below, let's take the mass as 1Msun.

    (1) What is the radius of a neutron star?
    (2) What is the size ratio of a 1Msun neutron star and a 1Msun C/O white dwarf?

    You may do part (2) first since the ratio is easier to figure out.

  19. 2013-12-04
    * Consider a gravitational potential of the form Phi = - GM/(r-rg). What is the innermost stable circular orbit (ISCO) in such a potential?

    If a parcel of gas of mass DeltaM spirals in to 3rg as part of a thin viscous accretion disk, then spirals from 3rg into the event horizon with no further interactions, how much radiation does it emit?

    [N.B. This potential is called Paczynski-Wiita potential, which is a very good approximation for that of a Schwarzchild black hole and is widely used in black hole accretion calculations/simulations. The quantity rg is just the Schwarzchild radius (gravitational radius, radius of the event horizon).]

  20. 2014-01-14
    [Based on arXiv:1401.0535 "A millisecond pulsar in a stellar triple system"]

    A stellar triple system consisting of an inner binary and an outer binary is discovered. It has a pulsar, and the discovery relies on the monitoring of the arrival time of the pulsar's pulses.

    You are given the following information: The two binary systems are co-planar and are both on nearly circular orbits. One object of the inner binary is the pulsar (with mass 1.44Msun) and the inclination of the orbit plane is 39 degrees.

    Use the top two pannels in Fig.1 to figure out the masses of the other two stars.

  21. 2014-01-30
    Tidal disruption of a star around a supermassive black hole

    A star had a close encounter with a super-massive black hole at the center of a galaxy, and the star was tidally disrupted. This provides materials to feed the black hole. Show that the late-time light curve of such a tidal disruption event follows t^{-5/3}, where t is the time measured since the disruption.

    For simplicity, you can make the approximation that the star was on a parabolic orbit and the disruption occured at the pericenter.

    [An example of the light curves of a tidal disruption event is shown in this plot, from Gezari et al. (2012; Nature 485, 217).]

  22. 2014-02-27
    Tidal disruption of a binary star system around a supermassive black hole
    (one mechanism to produce hypervelocity stars)

    Last time, we worked out the late-time light curve from tidal disruption of a star around a supermassive black hole (also Enrico's talk).

    Now let's change the above star to a binary star system. When this system travels close to the supermassive black hole, it gets tidally disrupted. As a consequence, one star becomes bound to the black hole, and the other becomes unbound (ejected). Estimate the velocity of the unbound star when it travels to a large distance away from the black hole.

    Again for simplicity, you can make the approximation that the binary is on a parabolic orbit and that the binary is made of two stars of equal mass on circular orbit. Make other simplifications as needed.

    [Hint: You can use the following notations (N.B. some quantities are not independent)

    m - mass of each star
    a - 2a is the orbital separation of the two stars
    v_orb - orbital velocity of each star in the binary system

    M - black hole mass
    r - distance from the black hole to the binary system at disruption
    V_inf - infall velocity of the binary system at disruption

    The final result can be put in terms of (M,m,v_orb) or (M,m,a), which would be instructive.]

  23. 2014-04-10
    On Apr 1, 2014, a 8.2 magnitude offshore earthquake took place in Chile. It may be able to trigger tsuinami in Japan (fortunately this time it did not). If a similar earthquake like this one happens again in Chile, how much prepartion time does Japan have for the possible tsuinami?

  24. 2014-05-15
    We talked a little bit about maser galaxies at this week's AstroCoffee. The plot here (taken from Reid et al. arXiv:1207.7292) shows the observations of water maser in an (edge-on) accretion disk of a supermassive black hole at the center of galxy UGC 3789. From the bottom two panels of the plot, answer the following questions based on your estimation.

    1. What is the distance to the galaxy?
    2. What is the mass of the supermassive black hole?
    3. What is your estimate of the Hubble constant?

    [If you'd like to correct for the Milky Way's peculiar motion along this observation direction, we have V0=VLSR+60km/s, where V0 is the velocity measured in the CMB frame at our position. Let's worry about this correction later.]

  25. 2014-05-29
    * A globular cluster has 10^5 stars and a 1-d velocity dispersion of 5 km/s. What is the covering fraction of stars?

  26. 2014-06-05
    * During the lifetime of the sun, what is the probability of having an encounter with another star close enough to perturb the earth from its orbit? What if the sun were in a globular cluster?

  27. 2014-06-12
    Kepler-10 is a Sun-like star. This figure shows the observations of the star's radial velocity variation caused by a transiting planet orbiting around it and the transit light curve. Based on the information in the figure, estimate the density of the planet.

  28. 2014-06-19
    * The central surface brightness of a globular cluster is 17 mag/arcsec^2. What is the covering fraction of stars?

  29. 2014-06-26
    How many neutrinos from the Sun pass through your body per second? [Compare this to the number from cosmic neutrino background.]

    * Estimate the number of neutrinos emitted by the sun in its lifetime and that emitted by a Type-II supernova.

  30. 2014-07-03
    An explosion of energy E in a small, localized volume occurs in a medium of density rho. What is the radius R of the blast wave as a function of time t?

    From the pictures of the first atomic bomb (Trinity), estimate its energy release (yield). [famous story on how G.I. Taylor did the estimation with unclassified pictures].

    The Crab nebula has a radius of 1.7pc. Estimate the (kinetic) energy release from the corresponding supernova.

  31. 2014-07-10
    * The globular cluster M13 is barely visible as an extended blob to the naked eye in a dark site. If you traveled in a rocket at 10,000 km/s aimed at the edge of the cluster, what are the chances that you would hit a star on your way through? How does the answer change if your velocity is 100 km/s? 10 km/s?

  32. 2014-07-17
    Einstein ring

    When the source, lens object (e.g., a point mass), and observer are perfectly aligned, the source will appear to be a ring, as a result of the gravitational lensing of the source's light by the lens object. This is called Einstein ring. For a lens of mass M, source distance D_s, lens distance D_l and lens-source distance D_ls, what is the angular radius of the Einstein ring, theta_E?
    (Hint: A light ray from the source is deflected by the gravity of the lens. Try to first figure out the deflection angle.)

    If the source is a bulge star and the lens is a star midway to the Galactic center, how large is theta_E?

    If the source is a galaxy at z=0.2 and the lens is a galaxy cluster of mass 10^15Msun at z=0.1, what is theta_E?

  33. 2014-07-25, 2014-08-06
    One earlier model for the Milky Way's dark matter halo is that it is composed of MAssive Compact Halo Objects (MACHOs). If we observe the Large Magellanic Cloud (LMC), what is the chance for a star in the LMC to be gravitationally lensed by MACHOs?

  34. 2014-08-26, 2014-09-02, 2014-09-09
    Continue with gravitational lensing.
    (1) Figure out the lensing equation for the general case of a points source, a point lens, and observer that are not perfectly aligned. Put it in terms of the the angular position of the source theta_S, the image theta_I, and the Einstein ring radius theta_E.
    (2) Find the solution for the positions of the two images.
    (3) What are the magnifications of the two images?
    (4) What is the total magnification of the two images?

  35. 2014-09-22
    same as 2013-02-12

  36. 2014-10-07
    same as 2013-04-16

  37. 2014-10-21
    Estimate the number of dark matter particles in your body at each instant (assuming they are WIMPs with mass 100GeV).

  38. 2014-10-28
    Fossil shells and deposits in certain sandstones can reveal the change in the length of the earth day over the past a few hundred million years. It is found that one earth day becomes longer by 2.4 ms per century. Based on this, estimate the rate that the Moon drifts away from the Earth.

  39. 2014-11-04
    * How many Type Ia supernovae at z=0.5 are needed to distinguish dark energy with an equation of state parameter w=-0.9 from a true cosmological constant (w=-1)?

    Assume a flat universe with Omega_m=1/3 and that Type Ia supernovae have an rms dispersion of 0.1 magnitude in peak luminosity. We would like a 3-sigma detection of the difference.

  40. 2014-11-11
    * What is the mass of a black hole that evaporates via Hawking radiation over the age of the universe?

  41. 2014-11-18
    [Poynting-Robertson effect]
    For a dust particle of radius 10 micrometer initially in a circular orbit at 1AU from a Sun-like star, it will slowly spiral in toward the star. Why? How long does it take for it to fall into the star (i.e. to be cleaned)?

  42. 2014-11-25
    [Inspired by the movie Interstellar, even though I haven't watched it.]

    Stars around a super-massive black hole (SMBH) at the center of a galaxy can be perturbed to fall into the SMBH. At what SMBH mass, a sun-like star can be swallowed into the SMBH horizon without being tidally disrupted?

    Now if you would like to fall into a black hole for fun (not sure about this) or for space/time travel, what's the minimum black hole mass you should choose for you to "safely" cross the horizon?

  43. 2014-12-02
    * Suppose that structure grows by gravitational clustering in an Omega_m=1 cosmology with an initial fluctuation power spectrum P(k) proportional to k^n. Consider the halo mass function \phi(M)=dn/dM (number density per unit mass). If the halo mass function is \phi_1(M) at time t_1 (redshift z_1), what is the halo mass function \phi_2(M) at time t_2 (redshift z_2)?

  44. 2014-12-09
    The electric dipole and quadrupole radiation powers are given by
    L=2|\ddot{d}|^2/(3c^3)
    and
    L=1/(20c^5) |\dddot{Q}|^2, respectively.
    For a system of charged particles, the dipole d (vector) is defined as the sum of q\vec{r} over all particles and the quadupole element Q_ij is defined as the sum of q [x_i x_j - 1/3\delta_ij r^2] over all particles.

    (1) If we were to make an analogy to obtain the gravitational radiation, what would be your guess of the corresponding formulae?

    (2) For a self-gravitating system, does the gravitational diople radiation exist?

    (3) The gravitational quadrupole radiation power is in fact 4 times higher than our naive guess (any intuitive reason for why?). Consider a binary system of two point masses m1=m2=m on circular orbit with separation a. What is the luminosity of the gravitational radiation?

    (4) If the binary is composed of a pair of neutron stars with separation 0.01AU, how long does it take for the two stars to merge?

    (5) If the binary is composed of a pair of supermassive black holes with m=10^8Msun, what's the maximum separation for the system to merge within a Hubble time?

  45. 2014-12-16
    @ An elliptical galaxy with a stellar velocity dispersion of sigma=200 km/s and a supermassive central black hole (M1~10^9 Msun) merges with a smaller elliptical galaxy with a smaller supermassive black hole (M2~10^8 Msun). The smaller supermassive black hole is dragged in towards the center of the larger galaxy by dynamical friction.
    (1) Approximately at what radius, M2 starts to get bound with M1?
    (2) How long does it take for M2 to reach that radius from a radius of 1kpc?
    (3) What process could happen at that radius? Will M2's orbit continue to shrink due to dynamical friction so that the two black holes get close enough to radiate away their remaining orbital energy in gravitational waves (within a Hubble time)? Why?

  46. 2015-01-13
    Continuted with 2014-12-16

  47. 2015-01-20
    [Density profile of a singular isothermal sphere]
    (1) A self-gravitating, spherically symmetric gas cloud of particles of mass m has a constant temperature T. If the density profile rho(r) follows a power law, fingure out rho(r) as a function of r.
    (2) A self-gravitating, spherically symmetric stellar system has a constant 1D velocity dispersion sigma. If the density profile rho(r) follows a power law, fingure out rho(r) as a function of r.
    [One can make an analogy with (1) to obtain the results. But let's also derive it from considering the distribution function.]
    (3) What is the velocity for an object moving in system (2) along a circular orbit at radius r?

  48. 2015-01-27
    There is a maximum mass for a stable white dwarf, i.e., Chandrasekhar limit.
    (1) For white dwarfs with masses below this limit, how does the radius scale with mass?
    (2) Why is there an upper limit in mass?
    (3) Estimate the Chandrasekhar limit (at an order of magnitude level).

  49. 2015-02-10
    [Reionization Epoch from CMB Observation (How the number changed over the past ten years)]

    In 2003, from the observation of the anisotropy in the Cosmic Microwave Background (CMB) polarization on large angular scales (more exactly the TE cross power spectrum), WMAP determined a Thomson scattering optical depth of tau=0.17 (+/-0.04) to the last scattering surface. Last week, Planck 2015 results give the latest determination, tau=0.066 (+/-0.016).
    (1) What are the implied redshifts of reionization from these numbers, respectively?
    (2) In each case, how old was the universe at the epoch of reionization?

  50. 2015-02-17 2015-02-24
    Thermal Sunyaev-Zeldovich effect (tSZ effect)

    The blackbody spectrum of the Cosmic Microwave Background photons (with a temperature T_CMB) will be slightly distorted along the line-of-sight direction of a galaxy cluster, as a result of the inverse Compton scattering by thermal electrons in the cluster. Consider a galaxy cluster of mass 10^{15}Msun in the local universe.

    (1) Estimate the temperature T_e of the thermal electrons in this cluster (in K and in eV).
    (2) For each scattering with the electron, the mean fractional energy change of a CMB photon is 4kT_e/(m_e c^2) [Figuring out this is not straightforward, but you could have some simple argument or estimate to get the order-of-magnitude result]. Taken this as given, figure out the mean fractional energy change of a CMB photon as it passes through the cluster.
    (3) At the frequency in the Rayleigh-Jeans regime of the CMB spectrum, does the CMB along the line-of-sight of the cluster appear to be hotter or colder than T_CMB? At an order-of-magnitude level, estimate the effective temperature change \Delta T/T_CMB?
    (4) The tSZ observation, in combination with X-ray observation of galaxy clusters, is proposed to constrain the Hubble constant. What is the underlying principle of this method?

  51. 2015-03-03
    Same as 2013-04-09

  52. 2015-03-10
    A paper published in Nature by Wu et al. last week presents the discovery of an ultraluminous quasar at redshift 6.3. The inferred mass of the underlying supermassive black hole is 1.2x10^{10}Msun. Let's assume that
    (a) this supermassive black hole grows through accreting gas, starting with a seed black hole of mass M0 at z_ini=15;
    (b) the quasar has been always shining at the Eddington luminosity;
    (c) the radiation efficiency is \eta = 0.1, i.e., 10% of the accreted mass is converted into radiation (i.e., radiating away).

    (1) What mass M0 do we need for the seed black hole?
    (2) What if z_ini=24?
    (3) If it was less radiatively efficient, e.g., \eta=0.05, how would the required M0 change?

  53. 2015-03-24
    Same as 2013-01-29

  54. 2015-03-31
    Silk Damping and Silk Mass (proposed by Ethan Lake)

    Small-scale fluctuations in the initial dark matter density field are smoothed out by the free-streaming of the dark matter particles. A (not exactly) similar process (called Silk damping) occurs for Baryons as well. Before decoupling, photons can diffuse from high density to low density regions, dragging the baryons along with them. This process wipes out small-scale fluctuations in the baryon density field.
    (a) Estimate the (comoving) length scale l_s of Silk damping.
    (b) What angular scale does it correspond to in the CMB anistropy? What redshift interval does this scale correspond to at the time of recombination?
    (c) What is the approximate mass M_S corresponding to this length scale?

  55. 2015-04-14, 2015-04-21
    21cm Cosmology Basics

    Consider a patch of gas with uniform density and temperature, with optical depth across it as tau_\nu. A ray of light with intensity I_\nu(0) passes through it. The radiative transfer equation is dI_\nu/d\tau_\nu=-I_\nu+S_\nu, where S_\nu is the source function and in our case is the Planck function.

    (a) What is the intensity after the ray passes through the cloud?

    (b) Now consider the application to 21cm cosmology. The 21cm radiation originates from the hyperfine transition (F=1 -> F=0; spin flip) of the neutral hydrogen. I_\nu(0) is that of the CMB, and S_\nu is related to the so-called spin temperature T_s (determinging the relative population at the F=1 and F=0 states, with number densities n1 and n0).
    In the Rayleigh-Jeans limit, express the result in (a) in terms of the CMB temperaure T_gamma, the spin temperature T_s, and the surface brightness temperature T_b.

    (c) The interesting quantity is the difference between the observed intensity and the expected CMB intensity. For radiation from the gas at redshift z, with corresponding CMB temperature T_gamma(z) and spin temperature T_s(z), what is the observed intensity difference? Express the results in terms of \Delta T_b, that between the surface brightness temperaures, and assume that the gas is optically thin.

    (d) In general, the spin temperature T_s is determined by the CMB temperaure T_gamma and the gas kinetic temperature T_k, beacuse of stimulated absroption/emission of CMB photons, collisions with hydrogen atoms, electrons, and protons, and scattering of UV photons.
    During the dark age, baryons can still be strongly coupled to CMB through the residual electrons, leading to T_k=T_gamma. As the baryon and electron density drops due to cosmic expansion, this coupling becomes more and more inefficient and disappears around z~150. The baryons evolves independently thereafter.
    What is the IGM temperature T_k at z~100?

    (e) As the first stars and galaxies start to form around z~15-20, the IGM is heated to a much higher temperture. Schematically draw the the evolution of T_gamma and T_k as a function of redshift, from z~200 to z~10.
    The spin temperature T_s in the mean IGM closely tracks the baryon temperature T_k before z~100 due to collision coupling, then starts to be more closely track the CMB temperaure T_gamma. Then around z~15-20, as a result of UV pumping (N.B.), T_s becomes more closely tracking T_k again. Schematically draw the the evolution of T_s in the above plot.
    Schematically show the corresponding, expected mean 21cm signal \Delta T_b that will (hopefully) be observed by 21cm experiments.

    (f) Consider the mean IGM undergoing Hubble flow. Estimate \tau_\nu for 21cm radiation at redshift z, with mean neutral hydrogen fraction x_HI and spin temperature T_s. The Einstein A coefficient for the hyperfine transition A_10 = 2.85x10^{-15} s^{-1}.

  56. 2015-05-12
    Estimate the magnitude of the full moon.

  57. 2015-06-02
    Estimate the height of the highest possible mountain on Earth. [The only number you are given is the latent heat of rock, which is about 30 cal/g.]

    Use the result to estimate at what radius (or mass) a moon becomes spherical.

  58. 2015-06-09
    [Motivated by McQuinn&Sanderbeck (arXiv:1505.07875)]

    Cosmological hydrodynamic simulations show that the z~2-4 intergalactic medium (with density less than ~10 times the cosmic mean) exhibits a tight power-law relation between temperature and density, T∝ργ-1 (a relation commonly adopted to produce Lyman-alpha forest mocks). Figure out the value of γ.
    Note that the gas has been reionized long time ago, but has kept photoionized (photoheated) by a nearly uniform ionizing UV background.

    (a) Consider a patch of IGM gas in a Lagrangian volume. Write down the equation governing the evolution of its temperature and density.
    (b) Put the equation in (a) in a form that the LHS=dT/dt and the RHS has terms related to T, density contrast Δ, and the ionizing background. Explain the meaning of each term on the RHS.
    (c) Some previous work argues for a balance between photoheating and cosmic-expansion-caused cooling. In such a case, what is the value of γ?
    (d) McQuinn&Sanderbeck(2015) argue that the argument in (c) is incorrect. Try to solve the equation to find the approximate value of γ. (The recombination rate coefficient can be approximated as ∝T-2/3)

  59. 2015-06-17
    Relaxation time in stellar systems
    [In Raphael's presentation about simulations a few weeks ago, relaxation time is mentioned (and proposed to be a possible OMG problem).]

    In a stellar system, relaxation time is the time for a star to be significantly perturbed by others in the system.

    (a) For a self-gravitating stellar system made of N stars, estimate the ratio of the relaxation time trelax to the crossing time tcross.
    (b) Estimate trelax and tcross for a globular cluster.
    (c) Estimate trelax and tcross for a big elliptical galaxy.

  60. 2015-07-01
    Same as 2013-05-21

  61. 2015-07-08
    Same as 2013-03-05

  62. 2015-09-15
    Based on Verbiest et al. (2008)

    PSR J0437-4715 is a millisecond pulsar with a white dwarf companion with orbital period of P_b=5.74104646 +/- 0.00000200 days. Analysis of 10 years of high-precision timing of pulse arrival time reveals an orbital period derivative \dot{P_b}=(3.73 +/- 0.06)x10^{-12}. The proper motion of the pulsar is measured to be mu=140.914 +/- 0.016 mas/yr.

    The orbital period change can be caused by the orbital decay of the binary due to gravitational wave radiation, but this is neglectable, at the level of -4.2x10^{-16}. Another source of the orbital period change is acceleration in the Galactic gravitational potential, but it is also too small (-2.3x10^{-14}).

    The remaining source of the orbital period change is the so-called kinematic effect from the common motion of the binary (a.k.a Shklovskii effect, 1970), which induces an apparent line-of-sight acceleration.

    (1) Figure out how this effect works.
    (2) Given the data, estimate the (kinematic) distance of the pulsar and the uncertainty.

  63. 2015-11-03
    Motivated by D'Orazio, Haiman, & Schiminovich (2015; arXiv:1509.04301)

    Consider a binary supermassive black hole system at the center of a galaxy, composed of two black holes with masses M and m (M+m=2x10^8Msun and q=m/M=0.1), on an edge-on circular orbit with separation 0.01pc. Suppose that the emission mainly comes from the accretion disk associated with the lower-mass one. We observe the system in V band, over which the spectrum can be approximated as F_\nu \propto \nu^\alpha with \alpha=1. The motion of the lower-mass black hole causes a variability in the light curve. What is the amplitude of the variability?

  64. 2015-11-10
    One possible channel to produce Type Ia supernovae comes from a binary system composed of a white dwarf and a companion star. The white dwarf accrets mass from the companion star. As the white dwarf reaches the Chandrasekhar limit, thermonuclear explosion will lead to a supernova.

    Compare the total energy from the explosion to that of the gravitational binding energy of the white dwarf. Will the exploding white dwarf leave any remnant?

  65. 2015-11-24
    Estimate the number of white dwarfs, neutron stars, and stellar mass black holes in the Milky Way Galaxy, respectively.

  66. 2016-01-28
    Estimate the fraction of baryons that are locked in present-day super-massive black holes.

    Estimate the fraction of baryons that are locked in present-day stellar-mass black holes.

    Suppose that a M=10^8Msun black hole (BH) was ejected as a result of the kick from BH-BH merging, traveling in the present-day intergalactic space with a speed V=300 km/s. What would be the mass growth rate of this black hole, caused by accreting dark matter?

  67. 2016-03-31
    The unusual orbital configuration of a group of trans-Neptunian objects indicates the presence a large planet, Planet Nine, in the outer Solar system. Suppose that Planet Nine is an ice giant with twice the earth radius, at a distance 600AU from the Sun. Estimate its apparent visual magnitude and the maximum angular speed (arcsec/hour) w.r.t. background stars observed by us.

  68. 2016-04-14
    On Apr 12, 2016, the Breakthrough Starshot initiative was announced (with Stephen Hawking, Yuri Milner, and Mark Zuckerberg as board members; see more details here). Hundreds of gram-scale robotic spacecrafts (each with gram-scale StarChip plus meter-scale Lightsail) will be sent to space. Laser beam from a ground array will be used to accelerate each spacecraft to one-fifth of light speed within minutes. It will take the spacecrafts about 20 years to travel to Alpha Centauri and another 4 years to send data back to the Earth. With about 20 years of development before the first launch, this would be a half-century long project.

    (1) Estimate the laser energy needed to accelerate each spacecraft to the target speed.
    (2) Compare this energy to the daily electricity consumption of Salt Lake City.
    (3) Discuss the challenges.

  69. 2016-05-26
    [Extension to 2013-05-21; Reionization Epoch from CMB Observation (How the number changed over the past ten years)]

    In 2003, from the observation of the anisotropy in the Cosmic Microwave Background (CMB) polarization on large angular scales (more exactly the TE cross power spectrum), WMAP determined a Thomson scattering optical depth of tau=0.17 (+/-0.04) to the last scattering surface. Planck 2015 results give the latest determination tau=0.066 (+/-0.016) and the Planck 2016 (HFI) result is tau=0.055(+/-0.009).
    (1) What are the implied redshifts of reionization from these numbers, respectively?
    (2) In each case, how old was the universe at the epoch of reionization?

  70. 2016-09-13
    Same as 2013-12-04

  71. 2016-11-01, 08
    [Slightly modified from 2013-04-16.]

    * A black hole of mass M accretes gas at a rate Mdot. The gas forms a geometrically thin, optically thick, steady-state accretion disk.

    (a) What is the total luminosity of the disk?
    (b) What is the radial temperature profile?
    (c) If the total luminosity is 10% of the Eddington luminosity, estimate the typical temperature around the inner edge of the disk for (1) a black hole of 10Msun and (2) a black hole of 10^8 Msun.

  72. 2017-05-25
    [Slightly modified from 2013-10-09]

    Estimate the (comoving) scale R_{BAO} of the Baryon Accoustic Oscillation feature (bump) seen in the galaxy, Lyman-alpha forest, or quasar two-point correlation function. [N.B. first BAO measurement with quasars from eBOSS; arXiv:1705.06373]

    (With a higher baryon density parameter, would R_{BAO} increase or decrease?)

    How is the comoving scale of the first peak in the CMB angular power spectrum related to R_{BAO}? How large an angular separation does the first peak correspond to?

    For the above estimation, if needed, you can assume spatially flat universe with Omega_m=0.3 (matter), Omega_b=0.045 (baryon), and Omega_r=8.4x10^{-5} (radiation).

  73. 2017-08-31
    [modified from 2014-12-09]

    According to rumors (Nature News), the aLIGO (plus VIRGO) has detected gravitational waves from the merger of two neutron stars and the electromagnetic (EM) counterparts have also been located and observed (in the galaxy NGC 4993). Let's study some basic results related to the gravitational waves and those from a neutron star binary system.

    (1) From charged particles, we usually can have electric dipole EM radiation (with the power depending on the square of the second time derivative of the electric dipole). For a self-gravitating system, however, we do not have dipole gravitational radiation. Why?

    (2) Consider a binary system of two neutron stars with masses m1=m2=m on circular orbit with separation r. The gravitational wave luminosity is given by L=G/(5c^5) \sum_ij |\dddot{D_ij}|^2, where D_ij is the ij element of the quadrupole moment tensor. Use simple arguments to derive the dependence of L on m and r.

    (3) The exact formula of L is L=32/5 G^4/c^5 (m1*m2)^2(m1+m2)/r^5 = 64/5 G^4/c^5 m^5/r^5. For the above binary system of two neutron stars with separation r=0.01AU, how long does it take for them to merge?

    (4) For the above system, what is the rate of the orbital period change? Express the result in second per year. (The famous Hulse-Taylor double pulsar system provides indirect evidence for gravitational waves, and Hulse and Taylor was awarded the Nobel prize in physics in 1993 for the discovery of such a system.)

  74. 2017-09-14
    (1)* Estimate the number of neutrinos emitted by the sun in its lifetime.
    (2)* Estimate the number of neutrinos emitted by a Type-II supernova.
    (3) On Feb 23, 1987, Kamiokande II detected a burst of neurtinos associated with SN1987A. The burst lasted for about 10 seconds with the 12 detected neutrinios in the energy range of ~10-40MeV. Derive an upper limit for the neutrino rest mass.

  75. 2017-09-28
    If dark matter is made of WIMP of mass 100GeV, estimate the number of dark matter particles passing through your body per second.

    During your whole life time, how many dark matter particles are expected to interact with atoms in your body?

  76. 2017-10-19
    Earlier this week, on Oct 16, 2017, LIGO-VIRGO announced the first detection of gravitational waves (GW) produced by a binary neutron star inspiral (GW170817). The electromagnetic signals in multiple wavebands associated with the neutron star merger were also detected.

    In the GW detection, the GW frequency evolves with time (a.k.a chirp). It is essentially this phenomenon that leads to a tight constraint on a certain combination of the masses (m1 and m2) of the two neutron stars, i.e., the so-called "chirp mass". As an example, see Fig.1 and Fig.4 of the GW170817 paper.

    (1) Figure out the form of the chirp mass (i.e., how m1 and m2 are combined).
    (2) If the binary system is in a galaxy at redshift z, how would the measured chirp mass depend on z?

    You can consider a binary system of two neutron stars with masses m1 and m2 on circular orbit. At a separation r between them, the gravitational radiation luminosity is L=32/5 G4/c5 (m1 m2)2(m1+m2)/r5.