Homework 6 Due Thurs Dec 8 by class time
1) Imagine alien astronomers a long distance away wish to
detect our Jupiter, but they are not near the ecliptic plane
so that cannot use the transit method.
What wavelength resolution would their spectrometers need
to use the radial velocity method if the sine of their
inclination was 1/2? (Assume their spectrometer would need
suitable wavelength resolution to be able to resolve the Sun's
Doppler shift.)
answer: Jupiter orbits at 13 km/s, and its mass ratio to the
Sun is close to 0.001, so to have zero net momentum in the
center of mass frame, the Sun's orbital speed around Jupiter
is 13 m/s. That is the speed that modulates over Jupiter's
10-year orbit, the signal the aliens wish to resolve. The
projection into their line-of-sight is 1/2 that speed, or about
6 m/s, so they need an instrument that can well resolve
6 m/s, so the width of each bin in their spectrometer would ideally
be not much more than 6 m/s. (They can do better by co-adding lots
of lines moving all together, but this resolution provides a
benchmark of what is desired.) Resolution can be expressed
as a ratio of the wavelength width to the central wavelength,
which is also the velocity width relative to the speed of light
(given the formula for first-order Doppler shift), so that's
6 m/s divided by 300,000 km/s. Normally it is the inverse of this
ratio that is reported, because we want good resolution to look
like a big number, so in this case that is about 5 x 10^7.
(This is extremely difficult to do, so we generally co-add lots
of lines instead, to improve on this dramatically.)
2) Let us say another set of aliens wish to detect Jupiter,
but they are situated perpendicular to our ecliptic plane,
so they cannot use either the transit nor the radial velocity
method. Let's say they expect the Sun to emit a blackbody
spectrum at 5800 Kelvin, and Jupiter to emit a blackbody
spectrum at 130 Kelvin, and they try to detect Jupiter by
seeing excess emission in some spectral band.
a) In what approximate spectral band should they expect to
see the greatest contrast?
answer:
We know the solar blackbody spectrum peaks in the visible range, around
0.5 micrometers for T near 6000 K, so Jupiter's 130 K would peak at
wavelengths a factor of 40-50 longer wavelength, say around 20 micrometers.
That would maximize the contrast in Jupiter's emissions.
b) Let's say their distance to the Sun, and the size of their
largest telescope, is such that they can obtain 100 solar
photons per second, across the band from part (a).
Let us also say that to detect Jupiter, they need to see
5 times more photons from Jupiter than is the Poisson noise
from the solar photons. (That means a "signal to noise" of 5,
where Poisson noise is random noise equal to the square root
of the number of photons observed. Hence if they looked for 1 second,
they would see 100 solar photons, with a noise of 10 photons,
so they would need to see 50 photons from Jupiter to clearly
detect its excess in that key band.) For how long would they
need to observe to detect Jupiter this way? (Hint: along the way,
you will need to compute the ratio of Jupiter's photon emissions
to the Sun's in that band, so you can use the size of Jupiter
and the Sun. Also, you would have to replace the Stefan-Boltzmann
law, which applies for the full energy flux, with the flux just in
the key band, and you will need to convert from energy flux to photon
flux by dividing by the energy per photon.)
answer:
The signal-to-noise ratio is the number of photons seen from Jupiter,
divided by the square root of the number of photons seens from
the Sun, and we want that to exceed 5. Let's square it, and say
we want to get at least 25 when we square the number of photons they
get from Jupiter and divide by 100*t, for t the observing time.
The number of photons from Jupiter is t times the ratio of the rate
of Jupiter photons to the Sun's photons, so that requires looking
at the solar Planck function around 20 micrometers, and also the ratio
of the cross sectional area of Jupiter to the Sun. The ratio in
blackbody emission rate at 20 micrometers for 130 Kelvin vs. 5800 Kelvin
can be approximated by just using the Rayleigh-Jeans tail for both
(an excellent approximation for the Sun but also not too bad for
where Jupiter is peaking), and in that tail, the brightness is
proportional to T, so here the photon flux ratio would be about
130/6000 ~ 1/50 or so. Hence the aliens get about 2 photons
per second from Jupiter in this band, so our equation for
the square of the signal-to-noise ratio says the aliens need
25 = (2t)^2/(100t) = .04*t, or t = about ten minutes.
(That is a very reasonable time to get a detection, whereas in class
I did not realize the 100 photons per second was given to be in the
infrared band, not the total rate from the Sun across all bands.)
3) According to the "Grand Tack" hypothesis about Jupiter's
migration in our solar
system, both Jupiter and Saturn have assisted life on Earth.
a) Consider how Jupiter has helped us by asking how things might
have been different had there been no large gas giants in our solar system.
answer:
Jupiter's gravity may have helped clear the solar system of asteroids
and comets, producing more hits early on which may have helped
bring water to the planet, and fewer later that could have caused more
extinctions.
b) Consider how Jupiter could have hurt us had there been no Saturn, at
least in the "Grand Tack" scenario.
answer:
In the "grand tack", the gravitational effects of Saturn
caused Jupiter to cease migrating inward as it was clearing out the
region known today as the asteroid belt, and turn Jupiter's migration
around, like "tacking" a sailboat.
Had Jupiter's migration not been ceased, Mars and the Earth would
have been its next victims.
What actually happened is not clearly known, but this is one possibility.
4) Assume the solar wind carries a total mass flux of 7 X 10^(-14)
solar masses per year, at a constant temperature of 1 million Kelvin
(it eventually cools down via expansion, but let's ignore that for simplicity),
and a speed of some 300 km/s.
a) Is its kinetic energy mostly due to bulk motion
due to its rapid speed, or mostly due to thermal energy associated
with its high temperature?
answer:
Multiplying the mass-loss rate by 1/2 the square of the wind speed
gives the kinetic energy flux, and by 1/2 the square of the thermal
speed gives the thermal energy flux, so the ratio between them
is the ratio of the square of those speeds. If the temperature is
1 million Kelvin, then 3kT divided by the average mass per particle
(about half a proton mass) gives the square of the thermal speed
to be about 1.2x10^4 (km/s)^2, while the square of the wind speed is
about 9x10^4 (km/s)^2, so the kinetic energy flux in the bulk motion is
some 7 or 8 times more than the thermal energy flux.
b) Why is it not a coincidence that those two numbers are within
the same general order of magnitude?
answer:
The wind is driven by the gas pressure, which is related to the
thermal motions.
So one can view a wind driven by gas pressure as a gradual conversion
of thermal energy into bulk flow energy, so the two are of the same
order of magnitude while that conversion is happening. (Once it is
fairly complete, say by the orbit of the Earth, the thermal energy
has been almost completely converted into bulk flow energy, meaning
that the solar wind has by then become highly supersonic.)
c) Use the above to find to order unity
the ratio of the kinetic energy flux in the solar wind
to the luminosity of the Sun.
answer:
In cgs units, the solar luminosity is 3.8x10^33 erg/s, which compares
to 2x10^(27) erg/s if we take the above numbers for the solar wind
kinetic energy. So about 1 part in a million of the solar luminosity
goes into accelerating the solar wind.