Solar System Homework #2 Answers

1.   Phobos, the moon of Mars, orbits Mars at an average distance of 9378 kilometers, making one revolution in a mere 7 hours, 59 minutes. The mass of Phobos is so small compared to the mass of Mars that we may neglect it. Use Newton's Version of Kepler's Third Law to compute the mass of the planet Mars in grams.

Part 1: Data

Everything we need to know is in the problem, and we can make a data table:

Semi-major axis of Phobos' orbit
   A    
9378 km
Orbital Period of Phobos
   P    
9hr, 59 min
Mass of Phobos
   MPhob   
essentially 0
Mass of Mars
   MMars   
?

Part 2: Equation

We must use Newton's Version of Kepler's Third Law: MMars + MPhob = (A3 / P2)

Part 3: Unit Conversion

Nothing is in the right units! We must convert the semi-major axis into Astronomical Units:

A = 9378 km X (1 AU / 1.5 X 108 km) = 6.25 X 10-5 AU

We must convert the Period into years:

P = 7 hrs, 59 minutes = 7.983 hours X (1 day / 24 hours) = 0.3326 days X (1 year/365.25 days) = 9.11 X 10-4 yrs

Part 4: Computation

MMars + MPhob = (A3 / P2)

MMars + 0 = (6.25 X 10-5)3 / (9.11 X 10-4)2

MMars = 2.44 X 10-13 / 8.29 X 10-7 = 2.94 X 10-7 solar masses

Now to convert from solar masses to grams, we just multiply by the number of grams in a solar mass:

MMars = 2.94 X 10-7 solar masses X (1.989 X 1033 grams / solar mass) = 5.85 X 1026 grams

Part 5: The Answer

We compute a value for the mass of Mars of 5.85 X 1026 grams, close to the book's value!

2.   If impacts by objects from space occurred at a uniform rate over the entire history of the Moon, and the maria have only one-fourth as many craters (per square kilometer) as the rest of the Moon, how much younger than the surface of the moon would the maria be? Is this consistent with what we believe the actual ages of the maria to be compared to the age of the Moon? What does this imply about the rate of impacts over the Moon's history?

ANSWER:

The maria generally have one-fourth the craters that the highlands have. It stands to reason that this implies that the maria have been hit with one-fourth as many craters as the highlands have. Now if we assume that the rate of meteor impacts on the moon's entire surface has been constant over the eons, this in turn means that the maria are one-fourth as old as the highlands. That is, if the highlands are 4 billion years old, the maria are "only" one billion.

If the cratering rate has NOT been constant, however, it becomes more difficult to guess the maria ages from that alone. Dating of lunar rocks indicates that the maria are not much older than the highlands; the maria are all of 3 billion years old. This means that the cratering rate was indeed NOT constant, that there were many, many impacts during the first billion and a half years or so (pockmarking the highlands), then the rate tapered off. Any impacts that occurred in the lowlands during the first billion and a half years was erased by the formation of the maria, and the few craters we see are from the later, calmer period.

3.   Recall that what we call "weight" is merely the force of gravity between a person and the planet he/she/it stands on. Use this fact to compare the weight of a person on Earth to the weight of a person on the Moon. If I weigh 240 pounds on Earth, how much would I weigh on the Moon?

Part 1: Data

When we are concerned about weight, we worry about masses and radii:

Mass of Earth
   ME    
5.97 X 1027 grams
Radius of Earth
   RE    
6378 kilometers
Mass of the Moon
   MM   
7.35 X 1025 grams
Radius of the Moon
   RM   
1738 kilometers
Weight on Earth
   WE   
240 pounds
Weight on Moon
   WM   
? pounds

Part 2: Equation

We shall use the weight equation, in its "final form" in the handout:

WM / WE = (MM / ME) / (RM / RE)2

Part 3: Unit Conversion

No unit conversions are needed, since the units of mass agree, and the units of radius agree.

Part 4: Computation

MM / ME = 7.35 X 1025 / 5.97 X 1027 = 0.0123

(RM / RE)2 = (1738 / 6378)2 = (0.2665)2 = 0.0743

WM / 240 lbs. = 0.0123 / 0.0743 = 0.165

WM = 240 lbs. X 0.165 = 39.75 lbs

Part 5: The Answer

The person who weighs 240 pounds on Earth will weigh about 40 pounds on the Moon, about one-sixth of Earth weight.
Updated 8/16/99
By James E. Heath
  
 
 

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