Exam 1

Astronomy 105G: The Planets
Spring 2009

Date: Febraury 26, 2009

I Multiple Choice

Circle the letter of the correct answer to each question. Remember that it helps to eliminate the answers that are clearly wrong or do not make sense. Read each problem carefully. Draw a picture to help you with your answers if you can or need to. [3 points each]

The main cause of seasons on Earth is

(a)

Earth's orbit around the Sun causes the distance from the Sun, and hence the temperature on Earth, to vary.

(b)

Earth's tilt brings either the Northern or the Southern Hemisphere closer to the Sun as the Earth orbits.

(c)

Earth's tilt causes the directness of sunlight to vary as the Earth orbits.
If Earth's axis did not have a tilt, the weather in June in Las Cruces would be (a drawing will help you!)

(a)

warmer than normal

(b)

colder than normal

(c)

exactly the same as it always is in June

Figure: A phase of the Moon as observed from Earth. See Problem I.5.
Why do we see basically the same face of the Moon at all times?

(a)

Because the other face points towards us only during the new Moon phase

(b)

Because the Moon does not rotate

(c)

Because the Moon's orbital and rotational periods are equal

(d)

Because the Sun only illuminates one half of the Moon at a time

A planet named Aggie is discovered in a distant solar system orbiting a star with the same mass as our sun. Aggie's orbital period is 4 days. Mercury's orbital period is 88 days. Venus's is 226 days. Planet Aggie's average orbital radius must be

(a): less than Mercury's
(b): between Mercury's and Venus's
(c): greater than Venus's

**Figure:** The elliptical line (drawn to scale) denotes the orbit of a planet about some massive star. It is **not** a circle. The numbered circles are possible (or impossible) locations of the star. Refer to Problem I.7.

See Figure 1. This phase of the Moon as viewed from Earth is

(a)

waxing crescent

(b)

full

(c)

waxing gibbous

(d)

third quarter
If the Moon's orbit were not inclined with respect to the ecliptic, solar eclipses would occur

(a)

about twice a month

(b)

once a month

(c)

twice a year

(d)

once a year
Study Figure 2. Remember Kepler's First Law of planetary motion. For this particular orbit (the eccentricity is greater than zero), which point or points are acceptable locations for the star?

(a)

Points 1 and 2

(b)

Points 1, 2, and 4

(c)

Point 1 only

(d)

Point 2 only

(e)

Points 2 and 3
A kilogram is a measure of an object's

(a)

weight

(b)

force

(c)

mass

(d)

gravity
You are on the Moon. A bowling ball and a small marble will fall to the surface after the same amount time when dropped from a certain height because

(a)

the force of gravity from the Moon acting on each object is the same

(b)

the force of gravity for anything on the Moon is zero

(c)

the ratio of the force of gravity from the Moon on an object to that object's mass is the same for any object
Imagine you were standing on a planet with a mass 4 times that of Earth and a radius 2 times that of Earth. How much would you weigh on that planet? (Hint: Consult equation 4.)

(a)

four times your weight on Earth

(b)

the same as on Earth

(c)

one-half your weight on Earth

(d)

one-fourth your weight on Earth

(e)

two times your weight on Earth

II True or False

In the space provided, please write True or False. [2 points each]

The orbit of a planet in a highly eccentric orbit resembles very closely a circle. - FALSE
Ptolemy and the other Greek scientists were very strong supporters of an Earth-centered solar system. - TRUE
Gravity is a much weaker force than the electromagnetic force. - TRUE
Electrons are much larger than protons and neutrons. - FALSE
At the vernal (spring) equinox, there is more sunlight shining on the Northern hemisphere than on the Southern hemisphere. - FALSE
According to one of Newton's Laws, an object in motion will eventually come to a rest unless an external force is applied to it to keep it going. - FALSE
Humans on Earth have existed for about half of the lifetime of the Universe. - FALSE
A light year is the time it takes light to travel in one year. - FALSE

III Short Answers

Give a brief answer to the following questions. Show your work for any calculations for partial credit.

1. What is the main difference between a geocentric solar system and a heliocentric solar system? [3 points]
  A geocentric solar system would have the Earth as its center, but in a heliocentric one the Sun would be the central object about which all planets orbit.
2. Which one of these models correctly describes our solar system? [2 points]
  Heliocentric
A planet orbits the Sun and its average orbital distance is 4 AU. What is this planet's orbital period in years? (You probably don't need a calculator!) [4 points]
We can use Kepler's third law p² = a³. This gives you p = $\sqrt{{4^3}}$ = $\sqrt{{64}}$ = 8 years.
1. Which astronomer first put forth the idea of planets orbiting the Sun in elliptical paths and derived other properties of these orbits including three main laws? [3 points]
  This was Kepler.
2. About when was this person alive? [1 points]
  About the late 1500's to early 1600's.
1. What three fundamental particles are the basic building blocks of atoms? [3 points]
  Electrons, protons, and neutrons.
2. What type of charge does each one have? [3 points]
  Electrons - negative, protons - positive, neutrons - no charge, neutral.
The closely-spaced vertical lines in the middle of the figure below denote the ``visible'' part of the electromagnetic spectrum. Label the figure (above the lines) at each end of the spectrum using the following 4 terms: long wavelength, short wavelength, high frequency, low frequency. There should be two terms above each line. Also, draw arrows and indicate with labels roughly where the infrared and ultraviolet regions are. [6 points]

Figure: See the .pdf version of this document to get the proper diagram here.
high frequencylow frequency

short wavelength long wavelength
$\rotatebox{270}{{\bf ultraviolet}}$ $\rotatebox{270}{{\bf infrared}}$
1. The earliest radio broadcasts were emitted about 100 years ago and have been traveling at the speed of light ever since. How far away are these radio waves right now (in any units you choose)? [3 points]
  The radio waves travel at the speed of light and are therefore 100 light-years away.
2. Have these radio waves possibly reached any star by now? Why or why not? [2 points]
  Yes they have reached stars by now, even the Sun. The next nearest star is only 4.4 light-years away and there are many stars within 100 light-years from Earth.
1. Write the following items in a sequence from smallest to largest. Earth's diameter, Pluto's orbit, Moon's orbit, the Sun's diameter. [3 points]
  Earth - Moon's orbit - Sun - Pluto's orbit
2. Draw a simple picture of the solar system containing only these 4 things (Sun, Moon's orbit, Earth, Pluto's orbit) and try to make it roughly to scale so that it agrees with your answer in part (a). [3 points]
  The main thing to remember is that the Sun is bigger than the Moon-Earth system.
Provide and explain one example of a situation where energy is converted between potential and kinetic forms. Draw a simple picture to help show this. [4 points]
One example is an object being dropped from a tall building. At the top before release, it has high potential (gravitational) energy. It speeds up and accelerates towards the ground, and right before impact is is moving very quickly and has a large kinetic energy, but less potential energy because it is closer now to Earth.

**Figure:** See the .pdf version of this document to get the proper diagram here.
high frequencylow frequency short wavelength long wavelength $\rotatebox{270}{{\bf ultraviolet}}$ $\rotatebox{270}{{\bf infrared}}$

IV Quantitative Answers and Extra Credit

You must complete any 4 out of the following 7 questions. Any work you do beyond 4 will count as extra credit, so give all of them a try. For each question you choose to do show all of your work and calculations to receive partial credit. A $\star$ denotes problems for which you'll probably need a calculator.

Pluto lies at a distance from the Sun of 40 astronomical units (AU). If light takes about 8 minutes to reach the Earth from the Sun, how long does light from the Sun take to reach Pluto? [4 points]
We know the Earth is 1 AU from the Sun. So light takes 40 times longer to reach Pluto, or 40 x 8 = 320 minutes, or about 5 hours and 20 minutes.
$\star$ The planet Neptune has a radius of 25,000 km and a mass of 10²⁶ kg. Calculate how much you would weigh on Neptune in pounds. Assume you're mass is 65 kg, which corresponds to a weight of about 143 pounds here on Earth. Are you heavier or lighter on Neptune? (Hint: find the gravitational force acting on you when you're on Neptune's surface in Newtons, then convert the units into pounds). [5 points]
For this we need Newton's Gravitation Law to determine the force, or weight, you would feel on Neptune.

F_g = Gm_youm_Neptune/r_Neptune² = (6.67 x 10^-11)(65)(10²⁶)/(2.5 x 10⁷)² = 693.7 Newtons.

From the back, we know the pounds-to-Newtons conversion. We finally get about 156 pounds. You would be just slightly heavier on Neptune than on Earch.
1. $\star$ You happen to come across a cricket ball in a field and find that it has a radius of 3.5 cm and a mass of 160 grams. What is its density? [4 points]
  From the labs, you know that density is mass divided by volume. The volume of a sphere is given at the end of the exam. Volume = 4/3 $\pi$ (3.5)³ = 180 cm³. The density is then 160/180 = 0.89 g/cm³.
2. Would it float in water? Why? [2 points]
  It WOULD float in water because as you know from lab, in these units water's density is 1 g/cm³.
$\star$ Let's say the Sun were the size of a big beach ball, with a radius of 0.6 meters (2 feet). On this scale, how large would Earth's radius be? Name some object you know that has approximately this size. The real radius values are: R_Sun = 6.95 x 10⁵ km and R_Earth = 6, 378 km. (Hint: find a scale factor to reduce the real size of the Earth to something smaller.) [5 points]
To solve this problem, realize that to get the Sun to be the size of a beach ball we had to divide its real size by some number. That number is 6.95 x 10⁸/0.6 = 1.16 x 10⁹. We now divide Earth's radius by this scaling factor: 6.378 x 10⁶/1.16 x 10⁹ = 5.5 x 10^-3 meters. To be able to obtain something more tangible, recognize that this is about 0.5 centimeters of radius. Thus, an object this size could be a small marble or a bean.
1. $\star$ The planet Uranus is 2.72 x 10⁹ km away from Earth and has a radius of 25,000 km. What would be its angular size from here on Earth? Draw a diagram of this situation and give your answer in either degrees, arcminutes, or arcseconds. [5 points]
  The angular size formula is given at the end. We just take the diameter divided by distance: 50, 000/2.72 x 10⁹ = 1.8 x 10^-5 radians, or 0.001 degrees by using the conversion factor given, or 0.06 arcminutes, or 3.8 arcseconds.
2. Can this planet be seen with the naked eye? Explain using the fact that Tycho Brahe's observations had at best a 0.5 arcminute resolution. [2 points]
  Since the best the eye can do is roughly 0.5 arcminutes, Uranus in most cases is quite far beneath that threshold at 0.06 arcminutes, and likely undetectable by eye.
$\star$ What is the value of the acceleration due to gravity on top of Mount Everest, 8,800 meters above the Earth's surface? How does this compare with the value at sea level? You need Earth's radius that has been given earlier. (Hint: Use Newton's second Law and equate with the law of gravitation). [7 points]
We can find the acceleration due to gravity on an object from Newton's second law, Ftot = ma. The only force acting in this case is the gravitational force, so we plug in the Graviation Law equation ofr F. By doing that, your mass cancels. We also have to add our height above the surface to Earth's radius, and so we have

a = $\displaystyle {\frac{{Gm_\mathrm{\;Earth} }}{{r_\mathrm{\;Earth} ^2}}}$ = $\displaystyle {\frac{{(6.67\times10^{-11})(5.97\times10^{24})}}{{(6.378\times10^6 + 8800)^2}}}$ = 9.76 m/ s².

We see that this is a little less than the sea-level value of 9.8 m/s².
Draw a picture illustrating the concept of using Earth's orbit as stellar parallax and describe from your drawing how this can be used to determine the distance to far away objects. [6 points]

Because the Earth orbits the Sun its parallax is greatly increased. To determine the distance to a far-away object, it is observed against the background of very far away ``fixed'' stars. At the opposite ends of the orbit as seen in the drawing, the distant object appears shifted with respect to the fixed background. The angle can be determined and then ultimately the distance from the Earth. This is all you needed to draw for the problem.

Useful Constants, Relations, and Equations
(you may or may not need to use all of this information)

Astronomical unit: 1 AU = 1.5 x 10⁸ km
Light-year: 1 Ly = 9.46 x 10¹² km
Speed of light: c = 3 x 10⁸ m/s
The unit of force, Newton, has units of: kg m/ s²
There are 2.2 pounds in 9.8 Netwons. These are forces (or weights).
Remember density = mass/volume, where volume for a sphere of radius R is ${\frac{{4}}{{3}}}$ $\pi$ R³.
Angular size:

$\displaystyle \theta$ = $\displaystyle {\frac{{\ell}}{{d}}}$ , (1)

where $\ell$ is the object's actual size (diameter) and d is the distance to the object. This formula gives the angular size in radians. To convert to degrees, 1 radian= 180/ $\pi$ degrees.
Kepler's Law of Periods:

p² = a³, (2)

where p is the period and a is the semi-major axis, or average orbital radius.
Newton's 2nd Law:

F_tot = ma, (3)

where F_tot is the total net force, m is mass, and a is acceleration.
The law of gravitation. The force of gravity between two masses:

F_g = G $\displaystyle {\frac{{m_1 m_2}}{{r^2}}}$ , (4)

where the constant G = 6.67 x 10^-11 ${\frac{{\mathrm{\;m^3} }}{{\mathrm{\;kg s^2} }}}$ , m₁ and m₂ are the two masses, and r is the separation of the two masses.