Astronomy 115

Homework Assignment #2

1. A.) Pick two of the lunar phases from the list below and for each, draw the arrangement of the Sun, Earth and Moon. Also draw what the Moon would look like during that phase. (Phases to chose from are: Waxing Crescent, First Quarter, Waxing Gibbous, Waning Gibbous, Third Quarter, Waning Crescent and New.)

B.)A Waning Crescent moon occurs when the Moon is between the Earth and the Sun, just BEFORE it reaches New Moon Phase.

A Waxing Gibbous moon occurs when the Earth is between the Moon and the Sun, just BEFORE it reaches Full Moon Phase.

2. The Moon has phases because we view it at different angles throughout the month. While the half of the Moon that faces the Sun is illuminated at all time, we only see this half at Full Moon. At other times of the month we see only part of the illuminated side of the moon.

3. Ptolemy described the solar system in terms of a Geocentric Model in which planets orbit the Earth on small circles called epycycles, which themselves ride on larger circles. When the motion of a plenet in its epicycle carried it in the opposite direction as its main motion, it would appear to move backward or retrograde.

4. What if ... Earth orbited at a different distance? ?

The length of the year is the time it takes a planet (e.g. Earth), to orbit. This orbital period is is given by Kepler's third law:

(semimajor axis in AU)3 = (period in years)2
or
a3 = P2

Note that a and P are not directly proportional. So if we double a, (the size of a planet's orbit), the planet's period, P, does not double. To see how changes in a affect P, used Kepler's 3rd law:

a=2 AU, then a3 =8. So P2 = 8 and P=the square root of 8 years = 2.8 years.

Hypothetically, if Earth orbited at twice the distance, our years would be 2.8 (not 2) times longer. At this larger distance we would also get less light, and Earth would be a very cold place.

Alternatively, if a=1/2, then a 3 =1/8. So P2 = 1/8, and P=the square root of 0.125 = 0.35 years
It would take only 0.35 years for Earth to orbit. In fact, a "year" would then be defined as 0.35 of our current years (just 127 days). So, by the time you had aged 35 of our normal years (of 365.25 days each), you would celebrate your 100th birthday, because Earth would have orbited the Sun 100 times!
It would also be much hotter as Earth would receive 4 times more light (inverse square law). The seasons would also last 2.8 times shorter. Winter would not be very cold, in either hemisphere.

5. What is special about the position and speed of the Earth when it is at aphelion? Which law or theory tells us this?

Kepler's 2nd law says that a planet sweeps through equal areas in equal times. To do this a planet must move fastest when it is closest to the Sun, (perihelion) and slowest when farthest from the Sun (aphelion) So on July. 4th, when Earth reaches aphelion, it is moving the slowest, and on Jan 4th, when it reaches perihelion, it is moving fastest, as predicted by Kepler's 2nd law.

6. If the period (P) of an asteroid is: 46.4 years, what is its average distance from the Sun?

A planet or asteroid's semi major axis is its "average distance from the Sun". This is a measure of how big its orbit is. It is related to the Period via Kepler's Third Law (see above). In this case:

(period in years)2 = 46.42 = 2153
So, (a)3 = 2153

The semi-major axis, a, is thus the cube root of: 2153
Looking for an integer whose cube is about 2153 we might guess: 12 3 = 1728. (too low) while 13 3 = 2197 is high, but pretty close. The semimajor axis of this asteroid's orbit is 12.9 AU.

7. A.) One wave of red light is 700 nm, or 700 x 10 -9 meters. One wave of violet light is 400 nm, or 400 x 10 -9 meters. So red light waves are almost twice as long as violet light waves.

B.) Both of these measurements are much smaller than a grain of sand, which has diameter of 0.2 millimeters (mm) or 200 micrometers. Each micrometer = 1000 nanometers, somewhat larger than 1 wave of red light. So a sand grain is about 200-500 times larger than red light and almost 1000 times smaller than violet light! This website has some nice perspectives: Scale of the Universe

8. Which types of electromagnetic radiation pass through Earth's atmosphere to reach the ground?

The two main types of radiation which reach Earth are radio waves and visible light. Almost all telescopes on Earth are designed to collected one of these two types. Some Ultraviolet light leaks through Earth's atmosphere to the surface, especially where the Ozone Layer is weakened. While greenhouse gasses absorb and block much of the Infrared light, small amount also leaks through.

9. Refraction: The bending of light as it passes through another medium, such as glass or water. Glass lenses can be used to bend, or refract light to a focus. This is the basis of refracting telescopes.
Reflection: The bouncing of light off a mirror. This is used in reflecting telescopes to form an image, by reflecting light off a curved mirror, usually in the shape of a paraboloid.

10. Compare the light gathering powers (LGP) of a 5 meter telescope and the human eye.

Light Gathering Power is proportional to the Area of the light collecting surface (eg the mirrors). Area is proportional to the diameter squared. A = PI * r2 = 3.14 * (d/2)2

The Palomar telescope is 5 meters, while the human eye is 5 mm or 5 x 10-3 m.
So Palomar's light gathering power is greater by a factor of: 52/0.0052 = 106 = 1,000,000. The 5m Palomar telescope can see stars one million times fainter than the human eye.