ASTR 100 PS #3
October 21, 2002
Q: What evidence do we have that the climate on Mars
has changed?
A: Here we are talking about climate change over long periods of time, not plate tectonic activity. We have evidence that water once existed on the Martian surface, though it clearly is not there in liquid form any longer: past flooding would produce the outflow channels we see, and dried-up riverbeds were once runoff channels. The Mars Global Surveyor analyzed some samples from the surface that indicate that water flowed on Mars for quite a long time: perhaps a million years. Sedimentary layers in the soil that have been exposed by wind erosion appear to have been deposited by water 1-3 billion years ago. Due to Mars’ small size and lack of ability to maintain a thick atmosphere, the water eventually photodissociated due to incident UV rays. The hydrogen evaporated into space, while the remaining oxygen was incorporated into the soil and rock on the Martian surface. Water is thought to remain in the permafrost layer underneath the outer layers of soil, and also in the polar caps of Mars, which are composed of a solid CO2 layer with a solid H2O layer underneath. Over time the carbon dioxide layer has sublimated, shrinking the caps such that only the ice layer remains, accumulating dust and leaving a layered terrain around the permanent caps that has been deposited by wind-borne dust. This dust builds up in layers as the caps shrink, and roughly 10,000 years of climate history is evident in each successive layer. The differences in dust content between the layers are indicative of periodic changes in the frequency of dust storms due to changes in the Martian orbit, affecting the rate at which dust is deposited. Consequently, the dust layers found near the polar caps on Mars can tell us a lot about changes in wind speed and temperature on the planet.
Q: Why is Jupiter so much richer in hydrogen and
helium than Earth?
A: Jupiter formed further out in the solar nebula than Earth, and due to the temperature gradient in the nebula, its immediate environment was quite a lot cooler than that of the Earth. As such, Jupiter formed from a conglomeration of icy planetesimals and grew rapidly from gravitational collapse, enabling the planet to imbibe gas directly from the nebula itself since it had the self-gravity to hold lighter molecules like hydrogen and helium. Large self-gravity means a large escape velocity, so it is much more difficult for molecules to reach the kinetic energy needed to escape the gravity of Jupiter than that of Earth. Jupiter therefore has a much greater concentration of hydrogen and helium than Earth does.
Q: What is the maximum angular diameter of Jupiter as
seen from Earth? Repeat the calculation
for Saturn and Pluto seen from Earth.
A: Because the angular size will be quite small since we are dealing with astronomical distances much greater than a planet’s diameter, we can use the small angle approximation: sin θ ~ tan θ ~ θ. Using the tangent formula, then, and making sure that the radius of Jupiter and the distance from Earth to Jupiter are both in the same units (1 AU = 150,000,000 km), θ ~ Jupiter’s radius/Earth-Jupiter distance for Earth at maximum distance from sun, Jupiter at minimum distance from sun. θ times 2 will yield the maximum angular diameter for the whole planet in radians, which can be converted to arcsec using 206,265 arcsec/radian.
Jupiter: 50 arcsec
Saturn: 20.7 arcsec
Pluto: 0.11 arcsec
Q: How can most meteors be cometary if all meteorites
are asteroidal?
A: Meteors are pieces of the nuclei of comets that fall through our atmosphere whenever the earth’s orbit takes us through a cometary tail. Since comet nuclei are mostly made of ice and other volatiles, they easily burn off due to friction with the earth’s atmosphere, producing the bright trails we associate with meteor showers. Asteroids, however, are made of mostly rock, which is much more robust than ice and won’t tend to burn up as easily when the asteroid falls through our atmosphere. Thus one doesn’t see the bright streaks when an asteroid falls to the ground, and more of the asteroid tends to survive its fall and become a meteorite.
Q: What is the mass of the Oort cloud in Earth masses?
A: We are given that the average mass of a cometary nucleus is ~1012 kg, that the Oort cloud has roughly 200 x 109 nuclei in it, and that the mass of the earth is about 6 x 1024 kg. Multiply the typical mass of a nucleus by the number of nuclei to get the total cloud mass, then divide by the mass of the earth to get this value in Earth masses: 0.033 Earth masses.