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.