Solar System Homework #3 Answers

1.   Why could the mass of Venus not be deduced from Earth-based observations in the same manner as the masses of most other planets? What other planet has this problem? How can we overcome this?

ANSWER:

It is of course impossible to take celestial objects and weigh them on a scale, so we must measure the masses of such objects by indirect means. The most accurate of these methods is the use of Newton's Version of Kepler's Third Law. Kepler's Third Law applies to any two objects orbiting a common center of mass, and uses orbital data to compute the sum of the masses of the orbiting bodies. Herein lies the quandary: Venus has nothing orbiting it! Venus's lack of a satellite disqualifies it from analysis by Kepler's Third Law, a trait that it shares with its fellow inner planet, Mercury. It is true that Venus orbits the Sun, and we could in theory use that orbit to compute Venus's mass. However, the Sun is almost a million times more massive than Venus, so our calculations would have to be VERY accurate (to six decimal places!) to separate the mass of Venus from the mass of the Sun. Errors in the measurement of Venus's period and semi-major axis prohibit this kind of accuracy. We have measured the mass of Venus by "giving" it artificial satellites in the form of the Venus space probes. Measuring the periods and semi-major axes of these satellite orbits gives us estimates for Venus's mass. Before the Space Age, we could only guess at the mass of Venus based on the tiny gravitational "tugs" it applies to the other inner planets.

2.   Would you expect Venus to have belts of charged particles surrounding it like the Van Allen Belts around Earth? Explain.

ANSWER:

The Van Allen Belts of Earth are caused by the fact that the Earth's magnetic field exerts a force on charged particles (electrons, cosmic rays) that come from the Sun. The magnetic field "herds" the particles into the region of the Van Allen Belts. Venus has, for all practical purposes, no magnetic field, we believe due to its slow rotation. No magnetic field, no Van Allen Belts.

3.   Using the information in the book, compute the density of the planet Mars in grams/cubic centimeter. Watch units carefully! Which is Mars closer to in density, the Earth or the Moon? Given the knowledge that the surface of Mars is low density (about 3 gm / cm3), what can you say about Mars interior?

Part 1: Data

Looking in the chapter on Mars, we see that the mass of Mars is 6.42 X 1026 grams. The radius is given 3394 kilometers.

Part 2: Equation

We must use the density formula: density = mass / volume

Part 3: Unit Conversion

Mass is in the right units, but the radius of Mars is not. We must convert the radius into centimeters:

RMars = 3394 km X (1000 m / km) X (100 cm / m) = 3.394 X 108 cm

Part 4: Computation

So V = (4/3)(pi)R3 = (4/3)(3.14)(3.394 X 108 cm)3

V = 1.64 X 1026 cm3

And Density = M / V = 6.42 X 1026 gm / 1.64 X 1026 cm3

density = 3.92 gm/cm3

Part 5: The Answer

The average density of Mars is 3.92 grams per cubic centimeter. This is closer to the density of the Moon than the density ofthe Earth. This is larger than the density of material found on the surface of Mars, but not excessively so. This tells us that while there are some heavy, dense materials to be found inside of Mars, there isn't a lot. Mars is a slightly differentiated planet. Although Mars has a great amount of iron on its surface (in the form of rust), the planet as a whole is iron-poor.
Updated 8/16/99
By James E. Heath
  
 
 

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