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You might find the Planetary Calculator useful for these problems: http://janus.astro.umd.edu/astro/calculators/pcalc.html.
1. Build a Planet.
The density of water in standard SI (or MKS) units
is 1000 kg per cubic meter (1000 kgm-3). Rock and iron have
densities 3 and 8 times as high as water, respectively. Mixtures of
two or more materials have densities intermediate between them.
a) Find the densities of the planets using the above website or
another source (include Pluto) and write them down. If you
knew only of these three materials and nothing else about the planets,
what would you predict that each of them would be made of (rock, iron,
water, or mixtures of these)? Be specific when you talk about
mixtures, for example: "Earth is mostly X and some
Y", "Mars has about 1/2 X and 1/2 Y".
b) Which planet or planets absolutely must be made of something
other than water, iron, or rock given your results from part a)? What
sort of extra material is needed?
c) Extreme pressure inside planets increases a material's
density, and the largest planets have the highest internal
pressures. Inside Uranus, Neptune, Saturn, and Jupiter, the pressure
is so large that all materials are compressed to many times their
normal densities. How does this change your interpretations of the
four giant planets?
2. Go to the Inner Solar System
Viewer.
a) Kepler's Third Law: Look at Mercury,
Venus, and Earth. Earth (the blue dot) goes around the Sun in one
year. Venus and the Earth start out together near the right - about
how many years does it take Venus to lap the Earth? About how many
times does Mercury lap Earth in one year? How does this illustrate
Kepler's third law? Hit "Reload" to restart the simulation.
b) Kepler's First Law: Toggle the asteroid orbit to
a=1AU,e=0.0, and w=0 degrees. Now increase the eccentricity (e) of the
asteroid to 0.3 in increments of 0.05. What do elliptical orbits with
low eccentricity look like?
c) Kepler's Second Law: Now toggle the asteroid orbit
to a=1AU,e=0.9,and w=40 degrees. Watch the asteroid for a full orbit;
where is it fastest? Draw a picture showing how this relates to
Kepler's second law. Finally, make a prediction for the period of the
asteroid's orbit using Kepler's third law. Now check your prediction
by waiting until the asteroid goes through its closest approach to the
Sun (its pericenter) and note Earth's position - where is Earth when
the asteroid returns to pericenter?
3. Travel to Mars!
a) Sketch a diagram of Earth's orbit and
Mars' orbit around the Sun treating the orbits as circular. The radius
of Mars' orbit is 1.5 AU, where the AU is the average Earth-Sun
distance. Pick a direction and call it the x-axis; put Mars along the
positive x-axis and Earth along the negative x-axis.
b) What are the closest and furthest distances that Earth and
Mars can be separated in AU?
c) The most fuel efficient way to get to Mars is not the shortest
one! Instead, it is 1/2 of the elliptic orbit that connects the orbits
of Earth and Mars. In your diagram, connect Earth (shown at launch)
and Mars (shown at arrival) with a smooth curve.
d) From you picture, the distance that the spacecraft travels is
more than 1/2 of Earth's orbit but less than 1/2 of Mars'. Estimate
the distance in AU by taking the average of these distances.
e) In a similar way, estimate the spacecraft flight time to
travel to Mars as the average of the time Earth and Mars take to go
1/2 way around the Sun. Give your answer in months.
f) Finally, light (and radio waves) takes 8 minutes to travel
from the Earth to the Sun. How long does it take a signal to reach a
spacecraft at Mars when the planet is closest and furthest from Earth?
This is why rovers on Mars need to be somewhat autonomous.