Part 1:
Write your answers CLEARLY and SHOW YOUR WORK.
1. Kepler's Laws : Do the worksheet on page 87 of the book ("Exploration: Kepler's Laws"). It asks you to use the interactive animation at https://astro.unl.edu/classaction/animations/renaissance/kepler.html
2. A Comet Orbit : a) Go to the Wikipedia list of periodic comets (https://en.wikipedia.org/wiki/List of periodic comets) and choose one that includes the following information on its page: orbit period, semimajor axis, eccentricity, and inclination. Record this information in your homework. Verify by calculation that the orbit period and semi-major axis follow Kepler's Third Law. (Keep significant digits in mind.)
b) Use the semi-major axis and eccentricity to calculate the perihelion and aphelion distance. Name the pair of planets that the comet is between when it is at perihelion and at aphelion (if it remains inside Neptune's orbit). The planet pairs may be different.
c) Draw a side view of Earth's and the comet's orbits (so that the orbits appear as lines through the Sun) properly showing how much the comet's orbit is inclined compared to Earth's orbit. Make sure i) the Sun is clearly marked, ii) Earth's orbit and the perihelion and aphelion distances have the proper scale, and iii) the angle between the orbits is correct (a protractor would be helpful). You will probably want to make sure your comet's orbit can fit on the paper first, then draw in Earth's orbit to scale.
d) Graph an ellipse with the same eccentricity as your comet, or use an online tool like https://astro.unl.edu/classaction/animations/renaissance/ellipsedemo.html or https://astro.unl.edu/classaction/animations/renaissance/kepler.html (If you can find a better website for the purpose, give the web address.) Measure the semi-major axis a and the distance from the center to a focus c on a printout, record the numbers, and show that the eccentricity is correct. Draw a properly scaled version of Earth's orbit on the printout as well (you can represent it as a circle).
e) What is the ratio of the comet's speed at perihelion compared to when it is at aphelion? How does this explain why comets tend to become bright for just a short part of their entire orbits?
3. Newton's Laws : Do the worksheet on page 115 of the book ("Explorations: Newton's Laws"). It asks you to again use the interactive animation at https://astro.unl.edu/classaction/animations/renaissance/kepler.html
4. Liftoff: The Space Shuttle typically had a mass of around 2 million kg, and the three attached rockets had a combined thrust of around 30 million newtons.
a) How much force would the rockets have to exert to have the whole thing hover just above Earth's surface? What would be the net force on the whole thing just after liftoff if the rockets were exerting maximum thrust?
b) How fast would the whole thing accelerate initially? How many "g" would the astronauts feel? (Remember that gravity is still pulling down on them!)
c) Most of the challenge of getting into orbit is accelerating to orbit speed. What is orbit speed for a shuttle orbiting 350 km above Earth's surface? (Put your answer in km / s.)
5. The Gravity Tractor : One method for diverting a threatening asteroid involves sending a spacecraft to the asteroid, and having it hover nearby, using gravitational force to change the asteroid's orbit.
a) What would your weight be on the asteroid Apophis (mass 2.7 × 1010 kg; radius 0.14 km)? Compare this to your weight on Earth, and also report the ratio of the forces (FApophis/FEarth).
b) How much force would a spacecraft's rocket exhaust need to exert to keep it hovering just above the surface of Apophis? (Assume that the spacecraft has a mass of 10 metric tons, where 1 metric ton = 1000 kg). Discuss the size of this force - is it unrealistically large to expect a spacecraft to generate this much force?
c) How long would it take to move Apophis a distance equal to Earth's radius (about the distance needed to miss Earth) with the force in part b)? (If the acceleration a is constant, the distance traveled is d =12at2.)
6. Nearby Stars : Alpha Centauri A and B are the second and third closest stars to the Sun. They orbit each other every 79.91 yr, and their average separation is 23.4 AU. Calculate their combined mass in units of solar masses. (Make sure your answer makes sense.) The two stars are roughly the same mass. Describe where the center of mass is for the two stars relative to their orbits. Is anything there? How is that different than planets orbiting the Sun?
Part 2:
Write your answers CLEARLY and SHOW YOUR WORK.
1. Clues in the Solar System: Do the worksheet on page 213 of the book ("Explorations: Formation of the Solar System").
2. Orbits and Rotation in the Solar Nebula (16 pts.) This problem asks you to examine the rotation clues we have that tell us how the solar system formed.
a) The time it takes for part of a cloud 100,000 AU in radius to collapse to form a new star turns out to be about half the time it would take an object to orbit the star on an elliptical orbit with a semi-major axis of 50,000 AU. Use this information and Kepler's Third Law to find the collapse time, assuming the star has the same mass as the Sun, and put your answer in years. What fraction of the age of the Sun (4.5 billion years) is this?
b) In an example in the text, an interstellar cloud having a diameter of 1016 m and a rotation period of 106 yr collapses to a sphere the size of the Sun (1.4×109 m in diameter). If all of the cloud's angular momentum stayed in that sphere, verify that the sphere would have a rotation period of only 0.6 s.
c) Because angular momentum is a quantity that is conserved, we can get an idea how the angular momentum of the original gas cloud was shuffled around. Look at the equations in Math Tools 7.1. Which planet has the second largest amount of orbital angular momentum, and how does it compare to Jupiter's? If you look at the planet characteristics in Appendix 4 and you understand what makes orbital angular momentum large, you should be able to narrow things down so that you don't have to calculate Lorbital for all of the planets.
d) Which planet has the most spin angular momentum? How does it compare to the Sun's spin angular momentum? Again, you don't have to calculate the Lspin for every planet if you can use the equation for Lspin and Appendix 4 to narrow your search.
3. Heating, Condensation, and Accretion:
a) An object that starts from rest and falls toward the Sun from a great distance will reach escape velocity by the time it gets to where it is going. What is escape velocity at Earth's orbit?
b) How much kinetic energy would an ice cube (mass about 30 g) have at that speed? (Put your answer in J.) How does this compare to the energy carried by a 2 metric ton SUV (mass of 2 × 103 kg) speeding down the highway at 90 km/h? Would you expect the inner solar system (close to the Sun) or outer solar system to have been heated more by this material falling inward? Explain.
c) Find the condensation/boiling temperatures (at which they turn from gas to liquid, or vice versa) for the following materials: iron, hydrogen, methane, silicon dioxide, helium, water, and ammonia. Put temperatures in units of Kelvin (K), and put the materials in order from lowest condensation temperature to highest. Which material could exist closest to the Sun in solid form?
d) It's estimated that there are a million asteroids 1 km across or larger. If a million asteroids (each 1 km in diameter) were all combined into one spherical object with the same density, what would its radius be? How many 1 km asteroids would it take to make an object the size (NOT mass) as the Earth? (The expression for the volume of a sphere is 4 3πr3.) How many Moon-mass objects would it have taken to make Earth?
4. Average Density :The average density of a planet is one of the most important ways we have of understanding what materials make up a planet. Consider the planet Kepler 34 b - the first extrasolar planet discovered by SDSU astronomers. The size (radius 0.76 Jupiter radii or RJ ) and mass (0.22 Jupiter masses or MJ ) of this planet can be measured from Earth.
a) What is the mass of the planet in kg?
b) What is the planet's radius in m?
c) What is the planet's volume?
d) What is the planet's average density? How does this compare to the density of water (1000 kg / m3 )? Is the planet likely to be rocky or gaseous?
Part 3:
REMEMBER: You can get information on the planets and Sun from the book's appendices.
Write your answers CLEARLY and SHOW YOUR WORK.
1. Dating Lunar Rocks : You are analyzing Moon rocks that contain small amounts of uranium 238, which decays into lead with a half-life of 4.468 billion years. See Math Tools 8.1 for help on radioactive decay.
a) In a rock from the lunar highlands, you determine that 55% of the original uranium 238 remains, while 45% has decayed into lead. How old is the rock?
b) In a rock from the lunar maria, you find that 63% of the original uranium 238 remains, while the other 37% has decayed into lead. Is this rock older or younger than the highlands rock? By how much?
2. Crater Counting on the MOON!: We are going to use the number of craters in a certain range of sizes from an area of the moon to estimate its age. We will use medium size craters (not too small that we can't see them, and not so large that they are rare).
a) Take a ruler and identify the craters with diameters between 1/16th and 2/16ths of an inch (corresponding to 4 and 8 km diameters), and write your count.
b) The area you were looking at was 1.9 × 105 km2. How long is each side of the picture? Convert your crater count into the crater density: "craters per million km2".
c) Plot your point on the attached graph (using 6 km for your average crater diameter) and estimate the age of the terrain these craters are on. Make sure you understand that this is a logarithmic plot.
3. Escape Velocity (8 pts.) Verify the escape velocity from the surfaces of Earth and Jupiter that can be found in Appendix 4, and calculate the escape velocity from the surface of the Sun for comparison. (SHOW YOUR WORK.) Also calculate the average speed of a hydrogen molecule (H2) and a carbon dioxide molecule (CO2) from these planets using the average surface temperature in the appendix. As a rule of thumb, a gas is held in the atmosphere if the escape velocity is more than 6 times the average speed of the molecule. Do your results agree with our knowledge of the atmospheres (or lack of atmosphere)?
Part 4:
Write your answers CLEARLY and SHOW YOUR WORK. IMPORTANT NOTE: For some of the questions in this set, there is not a formula where you can go and simply plug in numbers. These questions rely on your understanding, and being able to create your own formulas.
1. Star Lifetimes:
a) For most stars on the main sequence, luminosity scales with mass as M3.5 (see Math Tools 16.1). What luminosity does this relationship predict for i) 0.5M stars, ii) 6M stars, and iii) 60M stars? (Put your answers in units of solar luminosity L .)
b) Using the information in Table 16.1, calculate the lifetimes of 0.5, 6, and 60M stars in units of the Sun's lifetime.
c) The main sequence stars Sirius (spectral type A1), Vega (A0), Spica (B1), Fomalhaut (A3), and Regulus (B7) are among the 20 brightest stars in the sky. Explain how you can tell that all these stars are younger than the Sun.
d) The second longest stage in a star's life occurs when it is fusing helium. Helium fusion provides less energy than hydrogen fusion does (about 12 times less when equal amounts of mass burned), and stars are generally more luminous (about 40L ) during this time. Explain how this affects the lifetime of the star in this stage.
2. Giants and Supergiants:
a) Near the end of the Sun's life, the Sun's radius will extend nearly to Earth's orbit. Estimate the volume of the Sun at that time assuming the Sun is a sphere. Using that result, estimate the average density of the giant Sun. How does that density compare with the density of the Sun today and to the density of Earth's atmosphere at sea level (about 10-3 g / cm3 )?
b) The distance of the red supergiant Betelgeuse is approximately 427 light years. If it were to explode as a supernova, it would be one of the brightest stars in the sky. Right now, the brightest star in the sky other than the Sun is Sirius (which has a luminosity of 26LSun and is 26 light years away). How much brighter than Sirius would the Betelgeuse supernova be (from our point of view) if it reached a maximum luminosity of 1010LSun?
c) There have been some claims that when Betelgeuse explodes it will be like having a second Sun in the sky. Compare Betelgeuse's brightness to the Sun's brightness at Earth. Is this likely to be correct?
3. Mass and Light.: Our galaxy has approximately 50,000 stars of average mass (0.5M ) for every main sequence star of 20M , but 20M stars are about 104 times more luminous than the Sun and 0.5M stars are only 0.08 times as luminous as the Sun.
a) How does the luminosity of the single massive star compare to the total luminosity of the 50,000 less massive stars?
b) How much mass is contained in the lower-mass stars compared to the single highmass star?
c) Which stars - lower-mass or higher-mass - contain more mass in the galaxy, and which produce more light? Explain your answers.
4. Low-Mass Stellar Evolution : Do the worksheet on page 519 of the book ("Explorations: Low-Mass Stellar Evolution"). It uses an animation that can be found at astro.unl.edu/naap/hr/animations/hrExplorer.html.