29:52 Exploration of the Solar System

Lecture 4

The Motion of the Moon, Orbits and Kepler’s Laws and Planetary Motion

## The Motion of the Moon in the Night Sky

This is a good time to talk about the motion of the Moon;  you can check it out over the next several nights.  This week, the Moon is in the night sky,  approaching the full phase. Look at your textbook (Chapter 9) with one of the illustrations of the orbital motion of the Moon about the Earth.

The following are the main points of the properties of the Moon’s orbit about the Earth.

• The moon orbits the Earth-Moon Barycenter , which is the center of mass of the Earth-Moon system.
• The average distance of the Moon from the Earth is 384,000 kilometers (240,000 miles, or 60 Earth radii.
• The orbit is not  circular;  at its closest, the Moon is only 356,400 km, and at its most distant, 406,700 km.  This means that the angular size of the Moon changes during the month.
• From Earth we only see the illuminated part of the Moon, which varies during the course of the month from 0 % to 100 %. These are called the phases of the Moon.
• There is a correspondence between the place the Moon is in its orbit and the phase we see it. You will never see the full moon on the meridian at sunset.
• The Moon undergoes synchronous rotation.  This means that it turns on its axis once in the time that it orbits the Earth.  As a result, we always see the same hemisphere of the Moon.  This is illustrated in Figure 9.2 of the textbook.
• The sidereal period  of the Moon’s orbit, or time it takes to complete a path across the sky relative to the background stars, is 27.32 days.  The period determined by the phases, or times between successive full (or new) moons is 29.53 days.

Think about why it is that these two periods are different.

The plane of the Moon’s orbit is nearly the plane of the ecliptic. The inclination angle of the Moon’s orbit to the plane of the ecliptic is 5 degrees.  This means that the Moon also moves along the ecliptic, and is seen only in the constellations along the ecliptic.

However, from simple observations, and your SC1 chart, you can easily see the 5 degree inclination.  The orbital inclination is illustrated in Figure 9.4 of the book. Eclipses

The properties of the Moon’s orbits discussed here allow us to understand the properties of eclipses.

There are two kinds of eclipses,  lunar eclipses like we saw last November

http://antwrp.gsfc.nasa.gov/apod/ap031121.html

and solar eclipses.  A total eclipse of the Sun has not been seen in Iowa in my lifetime

http://antwrp.gsfc.nasa.gov/apod/ap010408.html

Both of these phenomena are due to shadows cast into space by another object.  The geometry of eclipses is shown in Figure 9.12 of the book. In a lunar eclipse, the Moon moves into the shadow cast into space by the Earth.  In a solar eclipse,  the Moon moves between the Sun and Earth, and cuts off the light of the Sun.

Why don’t eclipses happen every month?  Given Figure 9.12 of the book, you would think that a lunar eclipse would happen every month at the time of full moon,  and an eclipse of the Sun would occur every month at new moon.  But that doesn’t happen.  We typically see a lunar eclipse every 18 months of so,  and an eclipse of the Sun happens somewhere on Earth about as often.

The reason for the relative scarcity of eclipses is the 5 degree inclination of the Moon’s orbit.  Most of the time at full moon or new moon, the Moon is above or below the planet of the ecliptic,  and the shadow of the Moon misses the Earth (new moon) or the shadow of Earth misses the Moon (full moon).  This is illustrated in Figure 9.14 of the book. For an eclipse to occur,  the Sun, Earth, and Moon must be in the right geometry,  and the Moon must be at one of the nodes of its orbit, that is ,  it must be in the plane of the ecliptic, too.

If you would carefully watch the position of the Moon in the sky, and plot it on your SC1 chart, you would be able to predict lunar eclipses several months in advance.  Solar eclipses are harder.

### Kepler’s Laws of Planetary Motion

To really understand how the objects are moving in the solar system, and also to understand about the trajectories spacecraft take in interplanetary travel, we have to talk about the properties of orbits.

Although they are called Kepler’s Laws of Planetary Motion they actually apply to all objects, planets, asteroids, comets, etc, that are in the solar system.

There are three of Kepler’s Laws.

(1)   All planets move on elliptical orbits, with the Sun at one focus.

An ellipse is the sort of Figure shown in Figures 4.13 and 4.14 of your textbook.

Characteristics  of an ellipse are as follows,  which may be understood via reference to those figures:

(1)    An ellipse is defined in terms  of distances from the foci.

(2)   The Major and Minor axes are important in establishing the size of the ellipse.  Half of the major axis is the semimajor axis.

(3)   The degree of oblateness of the ellipse is determined by the eccentricity. The eccentricity  e is in the range 0 to 1.0.

(2)   Equal Area law

A line from the Sun to the planet sweeps out equal areas in equal time intervals.

Þ An object on an eccentric orbit will move through a large angular range when it is close to the Sun, and a small angular range when it is far from the Sun.

Þ An object on an elliptical orbit spends forever near Aphelion, and very little time near Perihelion.   The equal area law is illustrated in Figure 4.15 of your textbook.

(3)   The Harmonic Law

There is a relationship between the size of an orbit and the period of an orbit. If we let

The variables be: a= semimajor axis (units of astronomical units) and P = period in years, the Harmonic Law is

A3 = P2

With Kepler’s Laws we can carry out some neat calculations.

A Trip to Mars

How long does it take for a spaceship to travel to Mars? What kind of orbit do we follow?

Since the spaceship is moving in the solar system, it has to obey Kepler’s Laws. The minimum energy orbit is one which has the Earth on the major axis at the point of closest approach to the Sun (perihelion) and Mars at the most distant point on the major axis (aphelion).

If we make a drawing of the orbit, we see that the major axis = 1.0 + 1.523 = 2.523 astronomical units.

The  semimajor  axis therefore has to be 1.262 astronomical units. We can apply Kepler’s Laws to figure out the period of the spaceship.

P2 = a3  = (1.262)3 = 2.008

P = 1.417 years (total period).

The time for the cruise from Earth to Mars is obviously half that, = 0.708 years=8.5 months.

The cruise back to Earth would obviously take the same amount of time, so the crew would be in interplanetary space for 1.4 years. Add to this the time spent on Mars waiting for a favorable alignment of the planets, and you have a lot of time drinking Tang.