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Answers: Assignments 2/3.

NB. I've included full working here, so this is much larger than the
answers I was expecting from you. These notes combine deductions from the
2nd and 3rd data sets.

What do we know?

We know the radii of A and B (9207 and 5413 km respectively). We know their
luminosities at 0.44, 2.1 and 10 micron wavelengths.

We know that the radio source changes in frequency between 1419.976 and
1420.024 MHz. The peak radio emission is around 7(10-16Wm-2, and we are
viewing it from a distance of half a light year. We also know that the
radio signal varies with a time-scale of 25500 sec.



What can we deduce?


Radio Power.

From the inverse square law, if the dot was emitting radio power in all
directions, its total luminosity L would be L=F/4(r2. R = half a light
year, F is the flux given above, so we deduce a total luminosity of
2.2(1017W.
This is an upper limit: clearly radio power is only being emitted at half
the sky, and typically at around half the peak flux. So the true luminosity
is probably around 25% of the calculated value (it might be 10%, it might
be 30%). Not very accurate, but quite accurate enough to deduce some very
interesting things!
What does this sort of number tell us? Firstly, it is worryingly big. This
corresponds to more than 10 H-bombs worth of energy per second! It is
vastly more than the energy output of all the power stations on Earth
combined.
It is interesting to compare it to the total radiation from Twinky falling
on A and B combined, as any natural mechanism (without some extra energy
source) cannot exceed this.
The luminosity of Twinky is 52,000 times that of the Sun, so
LTwinky=52000 ( 3.8(1026 W = 2(1031 W.
From the inverse square law, the amount of radiation from Twinky landing
per unit area at a distance of 360 AU away is
[pic]
The cross sectional area of A = (r2=2.6(1014m2, and B is of course smaller.
So the total power striking both is around 1017W. So the radio power coming
out at 1420 MHz is comparable to all the Twinky light falling on both
objects combined! So unless both objects are 100% perfect absorbers, and
100% of the observed energy is then broadcast as radio waves at this
frequency (both highly implausible) there must be an extra energy source: a
big one!
It is hard to imagine any natural power source producing so much radiation,
beaming it in a particular direction, and at a particular wavelength only.
So this is probably the result of intelligent life. Whatever they are, they
are clearly capable of sustained power generation at levels far beyond
Human technology!


Orbital Parameters.

From the variation in frequency of the radio source, we can deduce the
orbital velocity of the transmitted, using the Doppler Effect Equation.
The change in frequency is (f = ( 0.024 Hz. So the change in velocity (v is
given by the normal Doppler effect equation:
[pic]
where f is the frequency and c the speed of light. This comes out as 5.07
km/s. Multiplying this speed by the period, we can work out the
circumference of the orbit: divide by 2( to get the radius, which is
20,526km.
Where in the double system are the radio waves coming from? Imagine that
they were coming from one of the two planets. If both planets had the same
mass, they would orbit around a point mid-way between them, so the radius
of motion of the radio source would be around 14,000 km. It is much larger,
which indicates that it is on the less massive of the two objects.
This is probably B (it is the smaller) but we'd really like more evidence.
The short gaps in the radio emission seem to occur when the transmitter is
passing behind some object. They last 3600 seconds. At a speed of 5.07
km/s, the transmitter will move 5.07(3600 = 18252 metres. So whatever is
blocking the radio waves must have a diameter at least this big. Planet B
isn't big enough: it must be Planet A that is doing the blocking. This
means that the transmitter is on the surface of Planet B, facing Planet A.

[pic]
From this geometry, we deduce that the centre of A is 1101 km from the
centre of mass of the whole binary, while the centre of B is a whopping
25989 km away from it. As the whole system is rotating at the same angular
speed around the centre of mass, the velocity of each component must be
proportional to its distance from the centre of mass. We know that the
radio source is 20,576 km from the centre of mass, and is moving at 5.07
km/s. Planet A is 1101/20576 = 5.35% as far, and so is moving at 5.35%(5.07
= 0.271 km/s. Similarly, Planet B is (20576+5413)/20576 = .


Masses.

From these facts, we can deduce the mass of both planets. Consider Planet A
to begin with. Its centre is a distance rc from the centre of mass, and rab
from the centre of B, both of which we now know. To stay in its observed
nice circular orbit, centrifugal force must, of course, balance gravity.
So:
[pic]
MA cancels, of course, leaving us with an equation for MB. We thus deduce
that MA = 1.72(1025 kg, and MB=6.7(1023 kg.


Other Properties

Now we know the masses and radii, we can work out the densities of both
objects (dividing the masses by the volumes - 4/3(r3). Planet A turns out
to have a very reasonable density: 5260 kg m-3: similar to that of most
rocky planets in our solar system. Planet B, however, turns out to have a
very low density similar to that of water: 1010 kg m-3. This might be
typical of a gas giant, but this planet is far too small to be a Gas Giant
- its gravity is too low to trap hydrogen. Perhaps it is artificial (a
Death Star?) The only plausible natural solid material with a low enough
density would be ice, but it should melt when it comes close to Twinky
during its elliptical orbit.
From these masses and radii we can work out the surface gravity g, using
the equation
[pic]
For Planet A, this comes out at 13.5 m s-2: a little higher than on Earth
(9.8 m s-2) but probably not uncomfortable. For Planet B, however, this
comes out at a tiny 1.5 ms-2 - slightly less than on the Earth's moon.
This means that any mountains on Planet A will be comparable to those on
the Earth, while Planet B could have much larger mountains.


Temperatures.

Now comes the tricky bit: working out the temperatures, and seeing if they
match the theoretical equilibrium temperature of around 220K we worked out
from the first data release.
The light we see from both planets will come from two sources:
1. Reflected Twinky light
2. Black body emission from the planet itself.
Now Twinky is very blue (as you could tell from its spectrum). If all the
light we were seeing came from reflected Twinky light, the spectra of the
planets should also appear very blue - ie. the flux should drop as the
wavelength became longer.
Instead, we see the reverse - the 10 micron flux is both objects is
actually greater than the 2.1 micron flux. This suggests that the 10 micron
flux, at least, is mostly black body radiation from the planets themselves.
The Planck Law tells us that the radiation per unit area, per unit
wavelength, from a black body with temperature T, is given by:
[pic]
where ( is the wavelength. This is the flux per uit (1 metre) of
wavelength. The fluxes quoted in the data release are per nanometre, so we
must divide by 109. We know the surface area of both planets. Set (= 10
micrometres (10-5m). Substituting the various numbers in, we find that the
predicted luminosity per unit area at this wavelength is:
[pic]
Dividing the total luminosity by the surface area, and solving for T, we
find that Planet B has a temperature of around 215K - almost exactly what
you'd expect from thermal equilibrium with the radiation from Twinky.
Planet A, on the other hand, is much hotter: 540K! You get exactly the same
answer from both the K and L-band fluxes, suggesting that it really is
black body radiation that you are detecting.
So Planet A is far hotter than it should be. This might be due to a strong
greenhouse effect. On the other hand, it is being bombarded with an
enormously powerful radio signal from Planet B: a signal depositing
significantly more energy on its surface that all the light from Twinky.
This may well account for some of the excess heat: perhaps Planet B and its
radio transmitter are a planet-sized central heating system for Planet A.


Atmospheres.

Now that we know the temperatures, masses and radii, we can use the
equation
[pic]
to work out the minimum mass of gas atoms or molecules that can stay in
their atmosphere. For Planet A, it turns out to be 9(10-27kg, so it will
loose hydrogen and helium, but nothing else. It won't be a gas giant, but
it could, for example, have an atmosphere like the Earth or like Venus.
For Planet B, m > 5(10-26kg. This is more than the mass of a nitrogen atom:
a nitrogen atmosphere like the Earth's would thus be impossible. It is
marginally less than an oxygen molecule (O2), so oxygen may just hang
around. CO2 can hang around fine. Needless to say, Hydrogen and Helium are
history.


Conclusions.

So what advice would you give to the captain based on all this? It looks
like there must be aliens there, and they probably live on Planet A. Planet
B is very strange, and may be a giant artificial moon. This alone suggests
extreme caution: a race that can build an artificial moon is a very capable
race. The incredible power of the radio signal emanating from the "moon" is
further evidence that we are looking at a really advanced civilisation -
far ahead of the Earth in its energy generation capabilities. It is
possible that this incredible radio signal is a heat source, designed to
keep Planet A warm while it is in the further reaches of its orbit around
Twinky.
If so, the aliens enjoy a much warmer climate than humans do, as they heat
their planet to a steamy 267 Celsius. This indicates that they are unlikely
to be water-based life-forms.