Astronomy 102 Problem Set #4 Solutions
Problem #1: When a massive star explodes as
a supernova, it shines with a luminosity ten billion times greater than
that of the Sun for a short period of time. If such a supernova explosion
were to appear in our sky as bright as the Sun, how far away from us must
it be located?
Solution: This is another problem where
there's an easy way and a hard way to do it. The hard way is to calculate
the flux of the Sun, and then use that value as the flux of the supernova
to calculate its distance. This method will work fine, but it involves
a lot of unnecessary multiplication.
First, make sure you understand what you're told in the problem. The
first sentance tells you that the luminosity of the supernova is equal
to 10 billion times the luminosity of the Sun; that is,
Secondly, we're told that the supernova and the Sun appear equally bright
in the sky. What's that mean? Well, objects that are equaly bright have
the same flux, since you eye is a flux-measuring machine. So,
Now, we have flux and luminosity (albeit in ratio from), so we can use
the relationship between flux, distance, and luminosity that we're now
used to:
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flux = luminosity/(4 x pi x distance2).
However, instead of calculating the actual flux of the Sun as a numerical
value, let's just call it fSun. Then,
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fSun = LSun/(4 x pi x DSun2).
where LSun and DSun mean the luminosity and distance
to the Sun, repsectively. We can write the same relationship for the supernova,
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fSN = LSN/(4 x pi x DSN2).
Now we're told in the problem that the flux of the supernova equals the
flux of the Sun, or fSN = fSun. That also means
-
LSN/(4 x pi x DSN2) = LSun/(4
x pi x DSun2).
We can immediately gid rid of the 4 x pi's on either side of the equation,
and then solving we DSN we get
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DSN = (LSN/LSun)1/2 x DSun
where as always, (x)1/2 means "the square root of x." At this
point we can plug in the values from the problem and use our calculator.
We're told that the luminosity of the supernova is 1 x 1011
L0, and since the luminosity of the Sun is 1 L0,
the fraction LSN/LSun = 1 x 1011. The
distance to the Sun is 1 Astronomical Unit, or 1 A.U. So,
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DSN = (1010)1/2 x 1 A.U.
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= 105 A. U.
This are perfectly reasonable units for a distance, but if you'd rather,
we can convert to more common units.
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105 A. U. x (1.495 x 1011 m/A.U.)
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= 1.495 x 1016 m
or
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105 A. U. x ( 1 pc/ 206265 A.U.)
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= 0.48 pc
Personally, I find the last set of units most meaningful for this problem,
since I have a feel for the distance to stars in parsecs. The star nearest
to the Sun, Proxima Centauri, is about 1 parsec away. It's not a massive
star, and there aren't any massive stars that close to us, so it appears
that we won't be seeing supernovae as bright as the Sun anytime soon. That's
a good thing, since a supernova that nearby would be really unhealthy.
Problem #2: Radio wavelength observations
of a gas cloud near the Galactic Center show spectral line emission at
a
frequency of 1421.65 MHz (1 MHz = 106 Hz). Astronomers have determined
that this emission line is generated by the
spin-flip transition of atomic hydrogen, which has a rest frequency
of 1420.406 MHz (Electrons can come in two "flavors", spin
up and spin down. Transitions between such state is much less energy
intensive than, say, a transition from the ground state to
the first excited state in hydrogen). Based on these observations,
calculate the speed of the gas cloud toward or away from the
Earth.
Solution: This ia a pretty straightforward
Doppler application. The photons you detect have a frequency which is different
from the rest frequency you expect for atomic hydrogen emission. Why? Because
the source of these photons (a cloud of gas near the galactic center) is
moving toward or away from you.
Well which way is it? Toward you or away from you? The photons you detect
have been shifted to a higher frequency, so the waves muct have been squished
together, and so the cloud must be moving toward you. To figure out how
fast the cloud is moving, we need to measure the change in frequency due
to the Doppler effect. The change in frequency is the difference between
the frequency you detect and the rest frequency, or
-
change in frequency = frequency detected - rest frequency
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= 1421.65 MHz - 1420.406 MHz
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= 1.244 MHz
Now let's insert that quantity into the Doppler folmula,
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(change in frequency)/(rest frequency) = (speed of emitter)/(speed of wave)
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1.244 MHz/1420.406 MHz = (speed of emitter)/ 3 x 108 m/s
where you no doubt recall that radio waves, like any electromagnetic waves,
travel at the speed of light. Note that I'm using units of MHz on the left
hand side of this equation. Since the left side is a ratio between the
frequency shift and the frequency, I'm free to use any units provided
I use the same units top and bottom. If I instead used units of Hz,
both the top and bottoms numbers would be a factor of 1 million greater,
but the ratio would turn out the same.
We can solve for the speed of the emitter as follows
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speed of emitter = 3 x 108 m/s x (1.244 MHz/1420.406 MHz)
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= 3 x 108 m/s x 8.75 x 10-4
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= 2.63 x 105 m/s or 263 km/s. towards you.
Problem #3: a) If the interstellar medium
has an average density of 4.5 x 10-18 kg/m3, what
volume of interstellar medium contains enough mass to make a 1 Mo
star?
b) If this volume were spherical in shape, what radius would
it have? (Hint: the volume of a sphere is 4/3 x pi x radius3.)
Solution: The density of anything is simply
the amount of mass that is contained in a unit of volume of the material.
In this case, the density of the interstellar medium is such that there's
4.5 x 10-18 kg of stuff in a cubic meter. How much mass is in
2 cubic meters? Why 9.0 x 10-18 kg, of course. Clearly, if we're
going to collect one solar mass of material, which is 2.0 x 1030
kg, we're gonna need a lot of cubic meters.
Density is just mass over volume, and for this probelm we know the mass
we want to get and the density of the medium, so
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density = mass/volume
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4.5 x 10-18 kg/m3 = 2.0 x 1030 kg/(volume)
and, solving for volume,
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volume = (2.0 x 1030 kg)/(4.5 x 10-18 kg/m3)
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= 4.4 x 1047 m3
and the units come out as cubic meters, which makes sense for a volume.
Now we're asked to calculate the radius of a sphere which has the above
volume. We're given the equation for the volume of a sphere
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volume = 4/3 x pi x radius3
so,
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4.4 x 1047 m3 = 4/3 x pi x radius3
and solving for radius
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radius = (4.4 x 1047 m3/(4/3 x pi))1/3
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= 4.7 x 1015 m
where (x)1/3 means "the cube root of x." This answer is perfectly
correct, ande the units make sense, but I like it more in units of parsecs,
because I understand stellar distances in parsecs better.
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4.7 x 1015 m x (1 pc/3.08 x 1016 m)
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= 0.15 pc
It might not surprise you to learn that this size is rather typical for
the dense star-forming cores in molecular clouds.