Problem Set 1

1. Astrophysical lasers/masers are very common phenomena. An example is a molecular cloud W43A in the Aquila constellation that has an observed −22 2 flux of Fν = 3 · 10 W/m Hz at the frequency ν = 22 GHz. An emitter with such a huge flux coming from only a region of size 0.100 must be a very hot object. In this exercise we will first derive some general properties of the Planck spectrum and then try to compare it to observations of the molecular cloud.

(a) Consider the limit hν kBT and Taylor expand the Planck spectrum to lowest order in ν to derive the Rayleigh-Jeans law. This is basically what you would ﬁnd for the Planck spectrum if you wouldn’t take into account the quantum nature of the photons. Explain the presence of a UV catastrophe.

(b) Next, consider the opposite limit kν kBT to derive the Wien law. Notice the presence of a exponential cut-off. Use the results obtained so far to sketch the Planck spectrum as function of frequency on a log-log plot. Without doing any computation, at which frequency do you expect to find the maximum intensity approximately? (c) Prove that curves of higher temperature lie entirely above a blackbody curve with a lower temperature. (d) Derive more precisely at which frequency the blackbody curve peaks. You should find the Wien displacement law

hνmax ≈ 2.82kBT

With the knowlegde of the last two questions, draw another blackbody curve in your sketch with a higher temperature. (e) We can repeat the same exercise for the Planck spectrum as function of wavelength. First, express the Planck spectrum in terms of intensity

1 per unit wavelength Bλ(T ). You should ﬁnd

2hc2/λ5 Bλ(T ) = . ehc/λkB T − 1 Consequently repeat the previous exercises. For the Wien displacement law you should ﬁnd

λmaxT = 0.290 cm K.

Can you understand why λmax 6= c/νmax? In particular, explain why λmax < c/νmax. (f) Now we turn to the molecular cloud. Compute the temperature that a blackbody should have to produce the measured ﬂux. (g) Compute the corresponding wavelength at which the intensity then peaks. We don’t observe X-rays from this source. What can we conclude?

2. Upon reaching a planet, part of the central star radiation will be reﬂected and the rest will be absorbed and re-emitted as a cooler blackbody. In the spectra of planets both of these components are observed. Consider the 9 case of Jupiter, with radius RJ = 7.1 × 10 cm and mean orbital radius 13 aJ = 7.8 × 10 cm. Assume that the spectrum of the Sun is a perfect blackbody.

(a) Suppose that Jupiter perfectly reﬂects 10% of the light coming from the Sun. Calculate its reﬂected luminosity. At which wavelength does it peak? In which spectral band is it observed? (b) At which wavelength does the re-emitted luminosity peaks? In which spectral band is it observed?

3. A protostar represents the ﬁrst step in the formation of a star inside a molecular cloud. This object can be idealized as a hot core, surrounded by a colder, accreting envelope. Assume that gas from the envelope can accrete onto the core only spherically. The spectra of these objects show several molecular lines on top of the continuum emission. Assume that the hot core emits black body radiation at temperature Tc and that the envelope has a temperature Te and emits thermal radiation. Finally, consider that the envelope is opaque only at a frequency ν1 (i.e., τ(ν1) > 1) and transparent for every other (i.e., τ(ν 6= ν1) = 0), that would correspond to an envelope containing only one single kind of molecule with only one characteristic transition. (a) Consider the envelope in hydrostatic equilibrium (no accretion). What would be the total spectrum, if we observe directly at the core through the envelope? Consider speciﬁc brightness as a function of the fre-

quency. Sketch the spectrum in the case τν1 > 1 and τν1 1. (b) How would the above spectrum be modiﬁed if: • the envelope was hotter than the core? • the envelope was accreting onto the core with a line of sight velocity va? • the envelope corresponds to a wind moving away from the core with a line of sight velocity vw? (c) Now let’s consider that the envelope can accrete spherically into the core. During the accretion process gas loses energy and emits radiation. The emitted radiation interacts with the surrounding medium −25 2 mainly through the electron scattering process (σT = 6.7×10 cm ), and exerts a pressure. In order to keep the structure in equilibrium the emitted luminosity cannot be arbitrarily high, but has a limit, so-called Eddington limit. Derive its expression.

4. Consider an accretion disk around a supermassive black hole, in a galaxy at a distance D from Earth. The disk is geometrically thin but very opti- cally thick at all frequencies, so that radiation and matter are in thermal equilibrium. In these disks, the energy is transported vertically (towards the surface) by radiative diffusion. The surface (τ = 2/3) is at a height H, while the optical depth integrated from z = 0 to z = H is τc 1. The disk extends from an inner radius Rin to an outer radius Rout. Assume 2 a radial temperature profile T = Tin(Rin/R) . Consider that the vertical flux is constant with z.

(a) At a given R, derive the relation between the temperature at the midplane and the effective temperature at z = H. (b) Assume that we are observing directly at the disk and that the disk is seen with an inclination angle i with respect to its normal. Calculate the total specific flux that we observe from the whole disk. Consider a frequency in the Rayleigh-Jeans regime. Tc Teff

z = H

z = 0

Rin Rout

Figure 1: Accretion disk around supermassive black hole.