The Physics
Hypertextbook
Opus in profectus

# Orbital Mechanics I

## Problems

### practice

1. Geosynchronous/Geostationary Earth Orbiting Satellites
There is a special class of satellites that orbit the Earth above the equator with a period of one day.
1. How will such a satellite appear to move when viewed from the surface of the Earth?
2. What type of satellites use this orbit and why is it important for them to be located in this orbit? (Keep in mind that this is a relatively high orbit. Satellites not occupying this band are normally kept in much lower orbits.)
3. Determine the orbital radius at which the period of a satellite's orbit will equal one day. State your answer in…
1. kilometers
2. multiples of the Earth's radius
3. fractions of the moon's orbital radius
2. Locate the L1, L2, and L3 Lagrange points for the Earth-sun system using dynamical principles. State your answers as distances…
1. from the sun and earth in meters
2. from the Earth as multiples of the moon's orbital radius
3. from the sun as multiples of the Earth's orbital radius
3. Dark Matter
1. The orbital speed of the planets decreases with distance from the sun. Why does this happen? Derive a formula that shows the relationship.

2. The orbital speed of the stars remains roughly constant with distance from the center of the Milky Way. (This is true for other galaxies as well.) What does this tell us about the distribution of mass in galaxies? Derive a formula that shows the relationship.

3. Calculate the mass of the Milky Way given a typical orbital speed of 220 km/s and a radius of 50,000 light years. Give your answer in solar masses (m = 2 × 1030 kg) and compare it to the approximate number of stars in the Milky Way (1011).
4. Dwarf galaxies, star clusters, and gas clouds beyond the edge of the visible galaxy have nearly the same orbital speed as the stars within visible galaxy.There is evidence that rotational speeds remain roughly constant at 220 km/s out to distances of 300,000 light years or six times the radius of the Milky Way. What is so amazing about this observation and what does it imply?
4. TRAPPIST-1 is a planetary system 39 light years from our solar system in the constellation of Aquarius. The star at the center is much less massive than the sun and only slightly larger than Jupiter. The table below provides the period and semimajor axis for six of the planets in this system that were reasonably well known in 2016. Convert these values to their SI equivalents and use them to determine the mass of the TRAPPIST-1 star. (These data are also available as a tab delimitted text file.)
Six exoplanets of the star TRAPPIST-1 source: Gillon, Triaud, Demory, et al.
planet

period
(days)
semimajor
axis (au)
b 01.51087081 0.01111
c 02.42182330 0.01521
d 04.04961000 0.02144
e 06.09961500 0.02817
f 09.20669000 0.03710
g 12.35294000 0.04510

### conceptual

1. If objects in earth orbit are weightless, why can't astronauts throw objects like baseballs or screwdrivers into outer space? Since they're weightless, it should be possible to heave them to the moon, planets, or distant stars. What's wrong with this thinking? In addition, why would casually discarding junk overboard from a space station or space shuttle be a bad idea?
2. One way to send a spaceship to the planet Mars would be to point it in the general direction of the Red Planet, ignite the rocket engines, and let it go. This method won't work, however. Give two reasons why this procedure would never result in a successful mission, no matter how precisely the spacecraft was aimed.
3. Spacecraft in extreme near earth orbit are subject to small but (in the long run) non-negligible amounts of aerodynamic drag from the upper regions of the Earth's atmosphere.
1. What happens to the altitude and speed of such a satellite over time?
2. Sketch the path of a satellite in such an orbit.
4. Pluto was discovered in 1930, but it's mass wasn't known with any accuracy until 1978 when Pluto's moon Charon was discovered. What was it about Charon's discovery that enabled astronomers to finally determine the mass of Pluto?
5. trajectories-satellite.pdf
The accompanying pdf file shows a satellite in a circular orbit about the Earth. Sketch the new path that the satellite would take if its speed were changed abruptly in the ways described.

### numerical

1. Satellite Motion
1. Calculate the speed needed for the space shuttle to travel around the Earth in a circular orbit at an altitude of 350 km above the Earth's surface.
2. Calculate the period of the space shuttle at this same orbit.
2. Black holes are formed when massive stars exhaust their nuclear fuel and collapse. The gravitational field near a black hole is extremely intense. Within a radius known as the event horizon nothing can escape, not even the speediest thing known — light. (We will discuss the event horizon on another day.) Inside the event horizon there is another special radius called the photon sphere. A beam of light directed at a tangent to the photon sphere will be trapped in a circular orbit around the black hole. A black hole may be black on the outside, but inside it is filled with light — light that is locked forever in orbit about the black hole.
1. Determine the radius of the photon sphere of…
1. a small black hole with a mass about three times the mass of the sun
2. a supermassive black hole (like the one at the center of the Milky Way galaxy) with a mass about three million times the mass of the sun
2. Complete the following table where you compare your answers to the radius of the sun (r = 695,500 km) and the radius of mercury's orbit (r = 58,000,000 km).
black hole r (km) r r
small
supermassive

### statistical

1. The table below gives the orbital period in days and orbital radius in millions of meters for Jupiter's four largest satellites (named the Galilean moons in honor of their discoverer, Galileo Galilei). Use this data to determine the mass of Jupiter
moon period (days) distance (106 m)
Io 1.769137786 422
Europa 3.551181041 671
Ganymede 7.154552960 1070
Callisto 16.68901840 1883