Download Homework 4: Special relativity, synchrotron sources

RADIO ASTRONOMY 29:186
Spring 2014
Homework 4: Special relativity, synchrotron sources
Due: Wed day Mar 5, 2014 at lecture
Useful physics/astronomy constants (double click arrow to make visible)
1. [2pts] (Review of special relativity). Consider an event consisting of the firing of a
high-speed bullet at v=0.99c over a distance of 1 km (speed and distance measured in the
lab frame). An observer moving at 0.9c in the direction of the bullet also observes the
event.
a. What is the spacetime interval [squared 4-length] of the event as determined by the lab
observer? (express in m2)
b. What is the spacetime interval of the event for the moving observer (m2)?
c. What are the spacetime intervals and separations of the event for lab and moving
observers (μsec)? Is there time dilation? lengtht contraction?
d. Finally, calculate the time interval and length for an observer comoving with the bullet.
How does the spacetime interval compare with the those for the lab and moving
observer?
2. [1pt] (Emission pattern modified by relativistic motion). A source of radiation with a
power pattern sin(θ)2 moves at velocity V along the x axis. (N.B. θ is measured in
direction of motion).
a. Plot the power pattern of the radiation in the lab frame for V values 0.9c and 0.99c.
[polar plot]
b. Show that the power pattern angular width is appx 1/γ in both cases (most easily done
graphically).
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3. [1 pt] Consider a plasma with a uniform magnetic field B = 0.01 Gauss The energy
distribution of the (relativistic) electrons obeys a power-law with a index p = 2.5 [N() ~
p] from min = 100 to max = 104 .
(a) Calculate the energy density of the magnetic field [erg/cm3].
(b) Assuming energy equipartition between fields and particles, and that the ratio of ion to
electron energy is = 40, calculate the electron density [cm-3]. Hint: You will need to integrate
over the energy spectrum to obtain the mean γ. Assume all electrons are in the power-law
distribution, which extends only from γ min to γ max]
4. [2pts] The naked-eye supernova that Tycho observed visually in 1570, now
known eponymously as Tycho's supernova (a.k.a. 3C10), is a bright radio emitter
of synchrotron radiation (see VLA image below). Look up the distance, angular
size, and spectrum in the literature. Assuming the SN is a uniform sphere (obviously
incorrect, since it is highly edge-brightened,but useful for simple order-of-magnitude
estimates), estimate: (a) the radio luminosity [W] over the frequency range 10 MHz
to 100 GHz. (Hint: you need to integrate the power-law spectrum), (b) the mean
magnetic field [G] using minumum energy argument, (c) total energy in the field
and particles [J], and (d) synchrotron lifetime [yr] (assume ratio of ion to electron
energy  ~40). Compare with age of the supernove, and estimate how long it will be
as bright as is seen currently.
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5. [2pts] A black hole at a distance of 1 pc is observed to have a thermal spectrum. The
measured luminosity is 100 solar luminosities. Assume that that the thermal radiation
arises from an spherical accretion disk with radius R = 3 Rs (Rs = Schwarzschild radius)
and that the black hole is radiating at the Eddington limit. Note: You will need to
research the capabilities of the Chandra instrument to answer parts (d) and (e)
(a) Estimate the central mass of the black hole.
(b) Estimate the temperature of inner accretion disk.
(c) At what wavelength will the radiation be maximum?
(d) Could the Chandra Xray space telescope detect this radiation (i.e., is the energy
within the Chandra energy range)?
(e) Could Chandra resolve the accretion disk?
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