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 NAME:________________________ Homework #3 – Due March 18th in class Please write your answers on 8.5” X 11” paper, one-­‐sided, and written neatly or typed. STAPLE multiple sheets together and write your name on all sheets. Explain how you arrived at your answers and make sure all numbers have units. You are encouraged to work with other students, but your answers must be your own. You are also encouraged to look up information from outside sources, but when you do so note the source. A book or article title, author and year or a website address are sufficient. 1. a) (10 points) Alice discovers a new X-­‐ray binary system, and wants to determine if the compact object is a black hole. What are TWO properties Alice could look for to prove that is definitely NOT a black hole? There are several possible signals that would prove it is not a black hole, but instead a neutron star. The detection of a pulsar would indicate a spinning neutron star. X-­‐ray bursts, from nuclear burning on the neutron star surface, would also prove it is not a black hole. An extra thermal component, from gas hitting the neutron star, would also prove there is a surface, meaning it cannot be a black hole. b) (5 points) What types of telescope (i.e. what types of light) would Alice want to use to make each of these observations? Pulsars are most likely to be seen at radio wavelengths, though some are detectable as X-­‐ray pulsars. Surface nuclear burning is see in X-­‐ray emission. An extra thermal component would also most likely be seen in X-­‐rays. c) (5 points) What is one way to tell that the compact object (probably) is a black hole, and what type of observation would Alice need to make to determine this? The easiest way to tell if it is a black hole is if the mass of the object is measured to be larger than 3 solar masses. To measure this, Alice would need to know the orbital velocity of the companion star and it’s mass. The velocity can be determined from spectra take in the optical. The mass can also be determined from optical observations. An alternative would be to measure the properties of an accretion disk. The orbital velocity and size of the disk can be determined, the mass of the object could be measured. This could be done with optical observations, or with microwave observations of masers. 2. a) (5 points) If you count up the number of galaxies that appear to have active nuclei, it comes to about one percent of the total number of galaxies we see. But this doesn't mean that only one percent of galaxies have supermassive black holes in their nuclei. How so? Even though all large galaxies are though to have supermassive black hole, they need to be accreting material at a high rate to produce an AGN. At any given time, only about 1% of SMBHs will be accreting, but all have accreted material, and were likely and AGN, at some point in the past. b) (10 points) Except for a few very near-­‐by galaxies, even the best telescopes are not able to resolve the center of galaxies well enough to tell if a bright center is due to a black hole AGN or a dense cluster of stars. What are TWO possible pieces of evidence that an AGN is present in a galaxy with an unresolved bright center? Some possible pieces of evidence are 1) a non-­‐thermal spectrum (i.e. Bremsstrahlung or synchrotron radiation), showing the emission is not from stars, 2) Rapid variability, on the scale of months or years, indicating the emission is from a compact region, 3) Jets detected, indicating the emission is from a single source. An extremely large amount of energy from a small region may also indicate that a massive black hole is the only plausible power source, because stars enough star could not be in such a small area, but this is hard to determine from energy alone. 3. In class, we found that the for a star to be tidally disrupted by a supermassive black hole, the star must come within a distance of 𝑀!"
𝑅! = 𝑟!"#$ × 𝑚!"#$
!/!
a) (10 points) Using the equation above, how close would a sun-­‐like star have to get to the black hole in Sag A* for tidal disruption to occur? 𝑚!"#$ = 1 Solar Mass 𝑟!"#$ = Solar Radius = 7 × 10! m 𝑀!" = 4 × 10! Solar Masses !
𝑅! = 7 × 10 × ! × !"!
!
!/!
m = 1.1 × 10!! m So, a sun-­‐like star would have to get closer than 𝟏. 𝟏 × 𝟏𝟎𝟏𝟏 m to be tidally disrupted. b) (5 points) How many AU is that? How many light-­‐years? 1 AU = 𝟏. 𝟓 × 𝟏𝟎𝟏𝟏 m, so the tidal radius is 0.74 AU, about the distance from Venus to the Sun. I Light year = = 𝟏𝟎𝟏𝟔 m, so the tidal radius is 𝟏. 𝟏 × 𝟏𝟎!𝟓 Light years , or about one 100,000th of a light year. c) (10 points) Recall that the Schwarzschild radius, the size of the event horizon of a black hole, is 𝑅!
= !!!!"
!!
= 3𝑘𝑚 × !!"
!!"#
For a large enough black hole, the tidal disruption radius can be smaller than the Schwarzschild radius. Using the equations for Rt and Rs, derive the minimum size of a black hole for which this is true (Rt < Rs), for a sun-­‐like star. To find the minimum size of a black hole that the sun could fall into without being tidally disrupted, we want to find when Rt = Rs. 𝑅! = 𝑟!"#$ × !!"
!
!
!!"#$
= 3𝑘𝑚 × !!"
!!"#
= 𝑅! 𝑚!"#$ = 𝑀!"# = 1 Solar Mass 𝑟!"#$ = Solar Radius = 7 × 10! m !
𝑅! = 7 × 10! 𝑚 × !!" !
!!"#
!
7/3 × 10! × !!" !
!!"#
7/3 × 10! =
!!" !
!!"#
1.1 × 10! =
!!"
=
!
!!"#
= 3 × 10! 𝑚 × !!"
!!"#
!!"
!!"#
= 𝑅! So, a black hole larger than 110 million solar masses would be able to swallow the Sun whole, without first disrupting it.