Physics and Chemistry of the Solar System

John S. Lewis

Chapter 9 The Airless Rocky Bodies: Io, Phobos, Deimos, the Moon and Mercury - all with Video Answers

Educators

Chapter Questions

Problem 1

Some theories of the accretion of the terrestrial planets predict that each planet will sample preplanetary solids from a very wide range of heliocentric distances and formation temperatures, and that bodies up to the size of Mars may wander widely throughout the inner Solar System under the effect of planetary gravitational interactions during accretion. If such theories are correct in some planetary system, how would you expect Fig. IX. 1 to look for that system?

Sarah Mccrumb

Numerade Educator

01:58

Problem 2

From a point on the equator of Mars, how long does it take for Phobos to complete one circuit of the sky?

Andy Chen

Numerade Educator

01:58

Problem 3

From a point on the equator of Mars, how long does it take for Phobos to complete one circuit of the sky?

Andy Chen

Numerade Educator

01:40

Problem 4

What is the synodic period of Deimos as seen from Phobos?

Andy Chen

Numerade Educator

01:26

Problem 5

Among the few geochemically plausible materials that can provide Io's very high albedo are a number of soluble salts, including halite $(\mathrm{NaCl})$. Discuss the implications of a global surface layer of salt on a body with no presentday water.

Jheremiah Simon

Numerade Educator

06:01

Problem 6

a. If the entire surface of Io to a depth of several kilometers were subducted every 10,000 years, what is the rate at which new crustal material must be generated to maintain a steady state?
b. How much energy would be released per second by crustal solidification with such a recycling rate?
c. What would the effective temperature of the surface of Io be if resurfacing were the only source of energy at the surface of Io?

Abhishek Jana

Numerade Educator

04:26

Problem 7

Estimate the magnitude of Jupiter's magnetic field at Io's orbit [see Eq. (V.250)] and compare it to that calculated for Earth's magnetic field strength at the mean distance of the Moon. What does this comparison show regarding induction heating of the Moon?

Matthew Miranda

Numerade Educator

01:56

Problem 8

Prove that the slopes of the contours of constant $\mathrm{SO}_2$ pressure in Fig. IX. 12 have a slope of exactly -2 .

Khoobchandra Agrawal

Numerade Educator

01:44

Problem 9

Suppose that a mixture of pyrite $\left(\mathrm{FeS}_2\right)$ and magnetite $\left(\mathrm{Fe}_3 \mathrm{O}_4\right)$ were heated to the point at which a significant gas pressure is developed.
a. Prove that neither oxygen nor sulfur is buffered in this system.
b. Derive an expression for the slope of the pyrite-magnetite boundary on a diagram such as Fig. IX. 12 .

Teesta Dasgupta

University of Pittsburgh - Main Campus

14:04

Problem 10

Graph the depth-diameter relationship given in Eq. (IX.4) versus velocity for several different solid bodies (Phobos, the Moon, Mercury, Mars, and Earth) and comment on the differences in their behavior. A reasonable first approach is to set the impact velocity equal to the escape velocity.

Brandy Heflin

Numerade Educator

03:03

Problem 11

The Tunguska explosion of 30 June 1908 was caused by the aerial explosion of a small asteroid over central Siberia. The explosion, which had a yield of about 10 megatons of TNT, crushed and burned some $2000 \mathrm{~km}^2$ of boreal forest. If it had survived passage through the atmosphere and delivered its kinetic energy to the ground, how large a crater would it have produced?

Prabhu Ramji

Numerade Educator

01:07

Problem 12

What is the minimum angle that can occur between the pole of the ecliptic and the pole of the Moon's rotation?

Chris Johnson

Numerade Educator

01:37

Problem 13

How large is the gravitational attraction of the Sun on the Moon compared to Earth's attractive force at a time when the Moon is eclipsing the Sun?

Joseph Petrullo

Numerade Educator

01:41

Problem 14

Explain why the Moon sometimes completely eclipses the disk of the Sun (total eclipse), but at other times may leave a visible ring of the Sun's photosphere all around the periphery of the lunar disk (annular eclipse).

Rodger Claar

Numerade Educator

05:42

Problem 15

Use the Surveyor 5 data in Table IX. Ia to calculate the atomic abundances of the elements listed in the analysis and plot these data onto Figs. VIII. 5 and VIII.7. What conclusions can you reach regarding the composition of the lunar crust in comparison with the compositions of meteorite classes?

Carson Merrill

Numerade Educator

00:42

Problem 16

The abundance of glass in lunar regolith samples correlates with the abundance of hydrogen and helium in them. Explain why.

Nidhi Singhi

Numerade Educator

02:43

Problem 17

Suppose a layer of solid basalt 3000 m thick on the Moon is resting atop a region in which basaltic liquids are being generated. An impact opens a crack from the surface down to the liquid layer, and it rises until it achieves hydrostatic equilibrium with the overlying rock. To what height could the column of melt in the crack rise?

Mehrnaz Siavoshi

Numerade Educator

06:08

Problem 18

a. If the Moon originally had formed from a high-temperature condensate at 1000 K and $10^{-3}$ bar [see Fig. (IV.7)], what changes would occur in the abundances and compositions of the major (M) and minor (m) lunar minerals listed in Table IX.2?
b. How would the plot in Fig. IX. 22 be changed?

Dr. Satish Ingale

Numerade Educator

00:32

Problem 19

Suppose some future mission carries out active seismic sounding (using explosives, not waiting for rare moonquakes) of the deep interior of the Moon and finds that there is no lunar core. How would this discovery constrain chemical models for the origin of the Moon?

Zachary Warner

Numerade Educator

01:54

Problem 20

What is the upper limit on the fraction of the Moon's mass that is contained in a metallic core, based on the limit of a $360-\mathrm{km}$ radius?

Andrew Duncan

Numerade Educator

02:30

Problem 21

Let us take the mean heat flux out of the Moon's interior to be $2 \times 10^{-6} \mathrm{~W} \mathrm{~cm}^{-2} \mathrm{~s}^{-1}$. If
the thermal conductivity of basalt is $2 \times 10^5 \mathrm{erg} \mathrm{cm}^{-1} \mathrm{~s}^{-1} \mathrm{~K}^{-1}$, what would you expect the vertical temperature gradient to be in the basalt fill of a mare basin?

Mayank Tripathi

Numerade Educator

08:07

Problem 22

If most of the radionuclides in the Moon were concentrated in the outermost 30 km , how would you modify Eq. (VIII.15) to estimate the time scale for conductive cooling? Note the similarity to what happens during T Tauri phase solar wind heating of a fairly good conductor.

Khoobchandra Agrawal

Numerade Educator

04:53

Problem 23

The Moon at present raises open-ocean tides with amplitudes of about 1 m . If the Moon was once close to Earth's Roche limit, what would the approximate tidal amplitude have been at that time?

Willis James

Numerade Educator

08:01

Problem 24

The inclination of the Moon's orbit plays a major role in theories of the history of the Earth-Moon system.
a. Assuming conservation of momentum and a perfectly inelastic collision, how large an impact would be required to change the inclination of the Moon's present orbit by $20^{\circ}$ ?
b. How large a crater would such an impact produce? Explain your assumptions regarding the velocity of impact.

Molika So

University of North Florida

01:06

Problem 25

Estimate the aberration angle of the solar wind at Mercury (the angular offset in its apparent direction caused by Mercury's high orbital velocity).

Tyler Moulton

Numerade Educator

01:12

Problem 26

a. Write a program that calculates the elevation of the Sun as viewed from any fixed point on Mercury's equator, assuming zero axial tilt. You should start at local noon and assume any location on Mercury's orbit.
b. Present the results of that program for two Mercury years.

Michael Twiton

Numerade Educator

07:27

Problem 27

Assume the rocky component of a fully differentiated Mercury has a density of $3.2 \mathrm{~g} \mathrm{~cm}^{-3}$ and the dense (metallic) component has a density of $7.5 \mathrm{~g} \mathrm{~cm}^{-3}$.
a. Neglecting self-compression, what fraction of the mass of Mercury is metal if the overall density of the planet is $5.43 \mathrm{~g} \mathrm{~cm}^{-3}$ ?
b. What is the fractional radius $\left(r_{\mathrm{c}} / r_{\mathrm{s}}\right)$ of the core in this model?
c. What is the coefficient of rotational moment of inertia of this model?
d. Qualitatively, how would including self-compression effects change the answers to these three questions?

Ronald Prasad

Numerade Educator

02:08

Problem 28

Suppose a family of small (uncompressed) planets is made of a mixture of two components: metal with a density of 7.8 and silicates with a density of 3.5 .
a. Show how to calculate the volume percent of each component in one of these planets given its bulk density.
b. Show how to calculate the mass percent of each component of such a planet if its bulk density is given.

Jonathon Brumley

Numerade Educator

07:38

Problem 29

If the entire silicate mantle of Mercury were stripped off by impacts, how would the escape velocity of the planet be changed? Use the "weak consensus" model of Mercury in Fig. IX. 30 as your point of departure.

Donald Albin

Numerade Educator

Problem 30

a. Recall our earlier discussions of the relationship between the energy density of the solar wind with velocity $v$ and its dynamical (ram) pressure. Show that the solar wind incident on the equatorial regions of a planet with a symmetrical dipole field of surface strength $B_0$ reaches pressure balance with the planetary magnetosphere at a distance
$$
L=r / r_{\mathrm{s}}=\left(B_0 / v\right)^{1 / 3}\left[1 / 4 \pi n m_{\mathrm{p}}\right]^{1 / 6} .
$$
b. If the early T Tauri phase solar wind was $10^6$ times as intense as the present wind, how much stronger would $B_0$ have to be to
stand off the solar wind at the same distance from the planet?
c. If you should calculate that $r / r_{\mathrm{s}}<1$ for the early solar wind approaching Jupiter, what factor would then be responsible for stopping the solar wind before it hits Jupiter?

Check back soon!

01:59

Problem 31

Using standard sources of physical data, calculate the temperature at which elemental sodium would have a vapor pressure sufficient to provide the observed gaseous atomic sodium abundance on Mercury.

Ronald Prasad

Numerade Educator

03:46

Problem 32

A flat crater floor at $85^{\circ} \mathrm{N}$ on Mercury may contain ice on the permanently shadowed crater floor. How important is it to consider the effects of indirect illumination by light scattered from the crater walls? Choose a reasonable depthdiameter ratio for large craters on Mercury and a reasonable albedo and do a simple calculation.

Krista Gresham

Numerade Educator

18:33

Problem 33

A spacecraft is launched from a circular heliocentric orbit at 1.000 AU from the Sun into an elliptical heliocentric orbit that grazes Earth at aphelion and grazes Mercury's orbit at perihelion. Its mission is to match speeds with Mercury for the purpose of entering orbit about that planet. If the orbit is designed so that it grazes Mercury's orbit at Mercury's perihelion distance, the spacecraft will require a larger injection $\Delta V$, but it will overtake Mercury when Mercury is traveling faster, and therefore need a smaller second $\Delta V$ to match speeds. If it is instead targeted to intercept Mercury at aphelion, the launch from 1 AU circular orbit will be easier, but Mercury will be traveling slower when the spacecraft overtakes it. Which mission has the lower total $\Delta V$ requirement?

Khoobchandra Agrawal

Numerade Educator