How far from the Sun would the Earth have to be (in AU) for it to be able to have retained molecular hydrogen (H2) in its atmosphere since its formation? (Assume that the greenhouse factor is equal to its albedo.) What would the ``Earth'' be like if it had formed at this place?
The solar magnetic field has a strength of 3 x 10-9 Tesla at the Earth's orbital distance of 1 AU. Estimate the solar magnetic field at the orbit of Jupiter, and estimate the distance of the magnetopause (assume this is where the two fields are equal) from Jupiter (in units of Jupiter's radius). The surface magnetic field of Jupiter is 4.3 x 10^-4 T and a dipolar field falls off like r^-3.
The outermost Galilean satellite of Jupiter, Callisto, has an orbital period of 16.69 days. The most distant known satellite of Jupiter is the tiny Sinope, with a period of 753 days (retrograde). What are their orbital semimajor axes in units of Jupiter's radius, and are they inside or outside of Jupiter's magnetosphere (at least on the sunward side)?
When Voyager 2 passed by Neptune in 1989, it beamed back many startling images of this planet, its rings, and satellites. Neptune's moon Triton was found to have a very thin atmosphere of molecular nitrogen (N2). This is one of the larger satellites in the solar system, with an equatorial radius of 1350 km and a mass of 2.14 x 10^22 kg. Calculate the expected mean temperature of Triton and its surface gravity.
Calculate the escape velocity from Triton's surface, and the rms thermal velocity for molecular nitrogen gas. Would you expect Triton to be able to retain this gas? If so, estimate the scale height of nitrogen in Triton's atmosphere (Triton has an albedo of A=0.7).
Calculate the mean density of Triton. From this mean density, what is the likely composition of this satellite? Explain. (Hint: Assume that typical silicate mantle material in a planet has a mean density of 3300 kg/m^3, and when compressed like deep in the Earth can be as high as 5000 kg/m^3. Iron-nickel metal has density of 8000 kg/m^3, and up to 10^4 kg/m^3 when compressed as in the Earth's core. Gas and ice have mean densities of 1000 to 1400 kg/m^3. And it doesn't have to be made of only one thing.)
The nitrogen atmosphere results from ``geysers'' of liquid nitrogen shooting out from the crust of Triton! Some evidence of this can be seen in Voyager images. At the temperatures exhibited on Triton, nitrogen will liquify at a pressure of about 1 bar (the Earth's atmospheric pressure, about 10^5 Pa). At approximately what depth in Triton's crust (depth below the surface) will this pressure be found? (Hint: you can use the hydrostatic equilibrium just like we did in the atmosphere.)
The gravitational tidal acceleration corresponding to the
difference in force 
F over distance
R
can be determined by taking the derivative of the gravitational force
F 
( dF/dr ) · R
as long as R is much smaller than r, the distance from the mass
doing the damage.
Derive an equation for the differential tidal acceleration F/m between the center and surface of a body
of radius R=R, at distance r from a
spherical planet of mass M.
The Martian moon Phobos has a mean radius of 11.1 km, and a mass of 9.6 x
10^15 kg. The orbit of Phobos is nearly circular with a semimajor axis of
9370 km. Use the equation you derived above to calculate the tidal
acceleration induced on Phobos by Mars over the radius
R = R of the moon. Compare this to the surface
gravitational acceleration g of Phobos.
At what distance from (the center of) Mars would Phobos be in danger of being pulled apart by the tidal effect (the tidal acceleration over the Phobos radius equal to or greater than the surface gravitational acceleration g)? Express this in km and in units of Mars radii.
Speculate on the fate of a small satellite like Phobos should it stray too close to its planet. Take into account that the rings of Saturn span from 71000 to 140000 km from the center of the giant planet (how many Saturn radii is this?)
Jupiter has a mass of 1.9 x 10^27 kg and is 5.2 AU from the Sun. Assuming it is in a circular orbit, what is the angular momentum (in kg m^2 / s) contained in its orbit around the Sun?
The rotational angular momentum of a spinning sphere of mass M and radius R is given by
/5 ) M R2
/ P
The Sun rotates with a period of approximately 24.67 days. What is the angular momentum contained in its rotation?
If the angular momentum in the orbit of Jupiter were transferred into the rotation of the Sun, how fast would it rotate?
Sirius (Alpha Canis Majoris) is the brightest star in the sky (other than the Sun). The peak of its blackbody spectrum is at a wavelength of 3222 Angstroms. What is its temperature?
Sirius has a luminosity of 23 times that of the Sun. What is its radius (in units of the Sun's radius R_sun)?
We showed that the equilibrium blackbody temperature for a uniform spherical planet 1 AU from our Sun is 277K. How far away would a blackbody planet be from Sirius to have this temperature?
The ``habitable zone'' is defined to be the range of distances from a given star where the equilibrium blackbody temperature is between 173K and 373K. Find the habitable zone around Sirius (in AU).
The surface temperature of Venus is around 700K, but the upper cloud deck (at an altitude of 60 km) is at a temperature roughy equal to the equilibrium blackbody temperature you would calculate from the solar radiation and assuming A=G. The atmosphere of Venus is predominantly (96 %!) carbon dioxide. What are the pressure scale heights at the surface and in the cloud decks?
The surface pressure on Venus was measured by the Venera lander to be around 95 atm. At what altitude above the surface of Venus would you have to go to find a pressure of 1 atm? (You will have to make some guesses about some things here, be sure to justify your assumptions.)
In the late 19th century, the astronomer Percival Lowell fanned the public and academic fancy for the prospect of life on Mars by reporting on his observations of an extensive network of channels on the planet, presumably of artifical origin. However, other astronomers at that time were unable to see these ``canals''. Using the eccentricities of the orbits of Mars and Earth, calculate the maximum apparent angular size of the disk of Mars as seen from the Earth at opposition, in arcseconds. What linear size (in km) would a canal have to be it subtended an angular size of 1'' (the best atmospheric seeing that Lowell is likely to have had)? Could Percival Lowell have resolved 1-10 km wide canals as he claimed?
The mean albedo of Mars is 0.16. Since it has no appreciable atmosphere to transport heat, its local surface temperature responds directly to sunlight. Calculate the noontime temperature on the Martian surface when the Sun is directly overhead (subsolar), at perihelion and aphelion. Compare this to the temperature range 210 -- 300 K reported in the textbook.
Consider the temperate latitudes 40 N and 40 S (roughly corresponding to Philadelphia and Melbourne, Australia on the Earth). The axial tilt of Mars to its ecliptic is 25.2 degrees. What are the midsummer and midwinter noontime zenith angles (at what angle down from the local zenith does the Sun cross the meridian) of the Sun at these latitiudes? (Note: these two latitudes give the same answer, so you need only give one.)
When discussing seasons on the Earth, we found that the solar surface flux on a planetary surface varies as the inclination of the Sun to the normal for the surface
where ZA is the zenith angle of the Sun, and F_in is the incident flux of sunlight. Calculate the midsummer and midwinter noontime surface temperatures for these two latitudes on Mars, first neglecting the effect of the eccentricity of Mars. You can then compare this to the change in temperature induced by the eccentricity of the orbit and comment on the relevant importance of these effects.
Pretend that you are the head of the science imaging team for NASA, working on a space probe that is now exploring a distant solar system.
The spacecraft is now passing by the outermost planet in that system. Examination of the images returned show a small terrestrial planet with a single satellite. From a distance of 200,000 km, the disk of the planet appears to be 59.2 arcminutes in diameter. From the same distance, the satellite has an angular diameter of 4.33 arcmin. What are the physical diameters of the planet and its satellite (in Earth radii)?
The probe is then put in orbit, and spends a long time observing the system. The semimajor axis of the orbit of the satellite around the planet is found to be 32000 km, with a period of 7.605 days. Detailed analysis of the data shows that the barycenter of the planet-satellite system was approximately 4.2 km from the center of the planet in the direction of the satellite. What is the mass (kg), average density (kg/m^3), and probable composition of the planet? What is the mass, average density, and probable composition of the satellite? Explain your calculations and reasoning.
