A comet is discovered in our solar system. A search through the historical records indicates that it last passed by in 1776! What is the semi-major axis of the comet's orbit (in AU)? If its perihelion is at the orbit of the Earth, what is the aphelion distance (in AU) from the Sun?
Suppose a new planet were discovered that was found to have a greatest elongation of 30 degrees from the Sun. What is the radius of its orbit (in AU), assuming it and the Earth are in approximately circular orbits?
The asteroid Ceres is found to have a synodic period of 1.278 years as observed from the Earth. What is the Sidereal period of its orbit?
Using the sidereal period of Ceres just derived, calculate the semi-major axis of its orbit in AU.
The eccentricity of its orbit is e=0.077. What are the perihelion and aphelion distances of Ceres (in AU).
The brightest Uranian moons, Titania and Oberon, were discovered by William Herschel in 1787. Oberon has an orbital period of 13.5 days and a semimajor axis of 583000 km. What is the mass of Uranus in Earth masses?
The escape velocity for a distance R from a body of mass M is given by
If you launch a projectile away from the mass with velocity v_esc from radius R, then at an infinite radius the projectile would come to rest (v -> 0). But reversal of the energy equation means that v_esc is the velocity an inward moving projectile started at rest infinitely far away would achieve when it reached radius R from the mass!
Calculate the escape velocity from the surface of Jupiter (in km/s).
In July 1994, the Comet Shoemaker-Levy 9 astounded astronomers by impacting the planet Jupiter. This comet was disrupted by a previous encounter with the planet, and perturbed into a catastrophic orbit. If we assume, for the want of a better estimate, that it was on approximately a marginally bound orbit, it should have impacted Jupiter at the escape velocity from Jupiter's surface. Calculate the kinetic energy in one metric ton (1000 kg) moving at this speed (in Joules).
If the explosion of one ton of TNT liberates 4.2 x 10^9 J of energy, how many tons of TNT is our one-ton cometoid impact equivalent to?
I would estimate that the chunks of SL-9 that hit Jupiter were around 1 km in size. Approximating it as a cube 1km on a side, with the density of water ice (1000 kg/m^3), estimate the total mass of a comet fragment (in metric tons). Then, compute the imact energy (in tons of TNT). To put this in perspective, a really big H-bomb produces 40 mega-tons of energy (40 x 10^6 tons of TNT worth).
The famous explorer Christina Columbo bas passed the Cape Horn and reached the Pacific Ocean in her vessel, the Pinto.
After sailing for some time, she notices that the South Celestial Pole is 40° altitude above the South horizon. What is the latitude of her current location?
At this time, she notices that the bright star Fomalhaut is just crossing her meridian. She looks up the coordinates of that star and finds them to be RA = 22 h 56 m and DEC = - 29° 45'. What is the current Local Sidereal Time (LST) at the location of the Pinto?
Is Fomalhaut North or South of her zenith when it crosses the meridian?
Before she left port in London, she set her sidereal chronometer to the Local Sidereal Time at Greenwich (Greenwich Sidereal Time = GST), whose meridian defines the zero of longitude. She notices that when Fomalhaut was crossing the meridian, the chronometer showed a time of 10 h 26 m GST. (Note: this means that at that moment, RA = 10 h 26 m was on the meridian at Greenwich.) What is her current longitude in degrees (west of Greenwich)?
Find a World Atlas, map of the World, or globe, and locate the Pinto's position there. What is the nearest significant landmass and where is she located relative to it?
Suppose NASA wants to send a probe to Venus from the Earth with the least expenditure of energy. The way to do this is to launch it slightly retrograde to the Earth's orbit, so that its velocity cancels some of the Earth's orbital velocity, and it will end up in an elliptical orbit with an aphelion of 1 AU and an perihelion at the orbit of Venus (0.723). What is the eccentricity e, semimajor axis a, and period P of this orbit?
Sketch the orbits of the Earth and Venus around the Sun. Indicate the least energy orbit and the relative directions that Earth, Venus and the probe move along their respective orbits to reach Venus? How long will this voyage take?
Use the energy equation for this orbit to calculate the aphelion velocity of the orbit. Compare this to the circular velocity of Earth's orbit. What is the burnout velocity that you want the rocket to have with respect to the Earth directed opposite to the Earth's motion in order to place the probe in this orbit?
Mars has a radius of 0.53 times the radius of the Earth and a mass of 0.107 times the mass of the Earth. What is the surface gravity of Mars compared to the surface gravity g of the Earth?
What is the escape velocity (in km/s) from the surface of Mars?
The rotational period of Mars (the Mars sidereal day) is 24 h 37 m 22.6 s (in standard Earth solar time units). At what distance from Mars would a circular orbit be ``synchronous'' with the rotation of Mars, and thus appear to stay fixed in the sky above a specific spot on Mars's equator?
Mars has an inner satellite Phobos, in an orbit with semi-major axis of 9370 km and period of 0.32 days, and an outer satellite Deimos, with semi-major axis of 23520 km and a period of 1.26 days. How are Deimos and Phobos situated relative to the synchronous point (radius calcualted above) and how would they appear to move in the Martian sky? (Hint: Mars, like the Earth, rotates in the same sense as its orbital revolution. The Sun will appear to rise in the East and set in the West on Mars also. Be careful and think about this problem - draw diagrams to help.)
If the Earth were to stop in its orbit, how long would it take to fall in to the Sun? You can consider this an orbit of eccentricity e=1 that reaches from aphelion at the Earth to perihelion at the Sun. This time to fall in to the center (assume a point mass for the Sun, is called the free-fall time.
Derive a formula for the free-fall time t_ff from distance R with mass of M assuming spherical symmetry.
Use this relation to calculate the time it would take the planet Jupiter to collapse (like in the movie 2010) if all the gas pressure were removed from it.
Write down a formula for the mass M of a spherical planet with
density 
and radius R. Also write down
a formula for the surface gravity g of a spherical planet of
mass M and radius R. Combine these two equations to get an equation
giving the surface gravity g of a spherical planet of density
and radius R.
What is the mean density of the Earth, in kg/m^3?
Scale the above equation to the surface gravity, radius, and average density of the Earth to find a relation between the gravity (in g for the Earth) and radius (R earth) with Earth density. Pay attention to the scaling with R, to see how g scales with radius.
