Astr 221 Homework #1 - due Sep 8th
It's the year 2020. You discover a new comet as it
crosses the orbit of Jupiter heading in towards the Sun. At this point,
your observations show that it is moving at a speed of 18 km/s.
You further calculate that the perihelion distance of
the comet will be exactly 1 AU!
- Demonstrate that it is
bound to the solar system.
- Calculate the semi-major
axis of its orbit.
Excited by the fact that it will come so close to the
Earth's orbit, you think of all the fame you will get for discovering
comet, which you decide to name Comet Mihos in honor of your
astronomy professor who started you on your career in astronomy. As you
begin writing your press release, old Dr. Blowhard comes up to you and
"I discovered that comet the last time it passed by Earth in 1973."
- What, then, is the
eccentricity and aphelion distance of the comet?
- What will its speed be as
it reaches perihelion?
Your observations also indicate that the comet is
approximately 5 kilometers across; assuming it is roughly spherical,
and knowing that comets
have an average density of about 1.5 g/cm3, what do you estimate the mass of the comet to be?
- Why don't you believe Dr.
You realize that the potential exists for a
this comet and the Earth, so you start to worry about how much damage
be done by an impact. If
comet hits the Earth, how much kinetic energy will be released? (Think carefully about the relative velocity between
the Earth and the comet) If a hydrogen bomb releases about 15
megatons of energy, how many H-bombs is this impact equivalent to?
Do you want to be around if
Comet Mihos hits?
2. Mission to Mars
|In this problem, we will
calculate the parameters of a Hohmann transfer
orbit which will take us from Earth to Mars. A Hohmann orbit is
an orbit which goes from one planet to another using the least amount
of energy. It is an elliptical orbit
whose perihelion is at the Earth's orbit, and whose aphelion just
reaches the orbit of the other planet (see figure to the right).
The trick, of course, is that the other planet
has to be at the right spot to rendezvous with the spacecraft!
So you need to calculate three things:
Calculate these three numbers, then use the
to Mars" JavaLab applet to see if your solution works.
- a, the
semimajor axis of the transfer orbit to Mars
- e, the
eccentricity of the transfer orbit to Mars
- theta, the
angle between Mars' position when the probe is launched and Mars'
position at rendezvous.
(For more fun with transfer orbits, you can
also play with the Grand
3. I want my MTV!
You want to put up a communications satellite so that it appears
fixed in the sky (so you don't need to keep repointing your satellite
dish!). What is the semi-major axis (a), eccentricity (e),
and period (P) of the orbit it needs to be on? How is that orbit
oriented with respect to the Earth's equator?
4. Weighing Saturn
Use the orbital data for Titan (Appendix D, Table D.2 of your
text) to calculate the mass of Saturn.
5. Units, units, units
Using astronomical units as the unit of length, years as time, and
the mass of the Sun as the unit of mass, the value for k in Kepler's
Law is 1.0. In these units, what is the value for Newton's constant, G?