ASTR 1010 – Spring 2014 – Prof. Magnani
Answer Key – Homework 9
Question 1 – Ch. 10; #26
Giant planets are significantly larger and more massive than the terrestrial planets.
They have smaller average densities than the terrestrial planets (700 – 1800 kg/m3
vs. 3900 – 5400 kg/m3). They have stronger magnetic fields. They are composed
primarily of gas and ices instead of rocks and metals. They have lots of satellites
whereas the terrestrial planets have few or none. Giant planets all have ring
systems.
Question 2 – Ch. 10; #28
Jupiter’s originally resembled the Earth. The big difference was that it was located
beyond the frost line, the place where hydrogen compounds from the originally hot
Solar nebula could condense into ices. The Earth-sized Jupiter accreted a lot of
these ices and grew bigger and more massive. When it was 5-10 times more
massive than the Earth, it began to accrete large amounts of hydrogen and helium
gas (remember, these were the most abundant items in the Solar nebula). Thus,
Jupiter ended up being composed primarily of hydrogen and helium, like the Sun.
Question 3 – Ch. 10; #30
Because they are made primarily of low-density materials and because they spin on
their axes so quickly, the jovian planets flatten a bit.
Question 4 – Ch. 10; #31
Jupiter’s atmosphere has several cloud layers at different depths (see Figure 10.13).
Impurities in the ice crystals that compose the clouds produce the colorful hues we
see.
Question 5 – Ch. 10; #32
Uranus and Neptune look the way they do (deep blue color) because they have
relatively few clouds in their upper atmosphere and a decent amount of methane.
The methane absorbs red, orange, and yellow wavelengths of light so that which is
reflected is mostly blue-green light. The lack of clouds allows you to look deep into
the atmosphere so that most of the red, orange, and yellow light is gone so that what
is left is the deep blue color.
Question 6 – Ch.10; #37
The question asks about conservation of energy. Jupiter, Saturn, and Neptune give
off more energy than they take in from the Sun. For energy to be conserved, this
must mean that they are producing the “extra” energy. The details are still
somewhat controversial, but the basic mechanism is Kelvin-Helmholtz contraction;
that is, the conversion of gravitational potential energy - as matter contracts
towards the center of an object - into kinetic and, hence, thermal energy. Basically,
they are shrinking a bit, or some of their interior materials is “raining” to deeper
levels.
Question 7 – Ch. 10; #38
As alluded to above, Saturn may have a liquid helium “rain” that basically moves
some liquid helium closer to the core. This releases energy via Kelvin-Helmholtz
contraction (see above).
Question 8 – Ch. 10; #39
Synchrotron radiation. Basically, electrons spiraling in Jupiter’s strong magnetic
field.
Question 9 – Ch. 10; #40
The auroras in the polar regions of Jupiter and Saturn are caused by charged
particles spiraling in the magnetic fields of these two planets, reaching the polar
regions where the fields plunge into the atmosphere, and crashing into atoms and
molecules in the atmosphere. This excites the electrons in these atoms and
molecules to higher energy levels. When they de-excite, they emit the auroral light.
Question 10 – Ch.10; #45
Don’t get confused by the Sun-Moon brightness comparison. Basically, the question
is, how far away must the Sun move from the Earth (compared to its present 1 AU)
for it to look 400,000 times dimmer? Since the apparent brightness of something
decreases as the square of the distance (something that is moved twice its current
distance looks one-fourth as bright or four times fainter, something that is moved
three times farther is nine times fainter, etc.). Thus , for something to look 400,000
times dimmer, it has to be moved the square root of 400,000 times from its current
distance. The square root of 400,000 is 632.2. So, if you move the Sun 632.2 times
further away than it is now, then it will look 400,000 times dimmer…in other words,
it will then look as bright as the full Moon. Neptune’s orbit is about 30 AU, so you
would have to move the Sun more than 20 times farther than Neptune’s orbit. So,
that’s what this problem is looking for. But, if you think about it, the Sun moved to
632 AU would not look like the full Moon in SIZE, only in brightness. If you want to
calculate how big it would be in arcseconds, use the small angle formula. Think of it
this way: The Sun at 1 AU is about half a degree in size. Let’s call that 1800
arcseconds. If you move it about 600 times farther away, then it would have an
angular size 600 times smaller, or, about 3 arcseconds. So, to the naked eye, it
would look like a point of light (because the angular resolution of the eye is about an
arcminute), in other words, star-like. But it would be as bright as the full Moon
(which casts shadows). That would look pretty cool.
Question 11 – Ch. 10; #49
About 11 Earths could fit across Jupiter’s equator, so Jupiter’s diameter and radius
are both 11 times that of Earth’s diameter and radius. Thus, since the volume goes
as the radius cubed, you could fit more than 1300 Earth’s inside Jupiter.
Density is mass/volume. Since Jupiter’s mass is 318 times that of Earth, but its
volume is 1300 times that of Earth, Jupiter’s density is then 318/1300 that of Earth
or about ¼ the density of Earth.
Question 12 – Ch.10; - #52
Assume that the planet can be considered a thermal, or blackbody radiator. So you
can use the flux formula: F = T4 . This gives you the energy per second per unit
area (Joules/s/m2). You don’t have to calculate the energy coming out of the whole
planet, comparing one square meter at 82 K to one square meter at 95 K will do just
fine. So the flux at 82 K is just F = (82)4 , at 95 K, it’s just (95)4 . So, a square
meter at 95 K will radiate (95/82)4 more energy than one at 82 K. That’s 1.80 times
more at 95 K than at 82 K.
Question 13 – Ch. 10; #53
This is like problem #34. (Ttrue/Tequilib)4 = 2.6
or, (Ttrue/47)4 = 2.6
(Ttrue/47) = 2.6¼ = 1.27
Ttrue = 60 K