Download Astronomy 291 Problem Set 2 Due Monday 1

Astronomy 291 Problem Set 2
Due Monday 1 October 2012
1. (2 points) What is the escape velocity from the surface of Phobos (R = 10 km, ρ = 3 g cm−3 )?
2. (2 points) Jupiter has a sidereal rotation period of 0.4135 solar days and a sidereal orbital period of 11.8618 years.
What is the difference, in seconds, between the solar and sidereal days for an inhabitant of Jupiter?
3. (2 points) While reading this problem, your body is being assaulted by the gravitational forces of two objects: the
3.0 × 106 M⊙ black hole in the center of the Galaxy (27,000 light-years distant), and the President of Penn State, working
in his office. Which of the two objects generates more force on you?
4. (3 points) What are the perihelion and aphelion speeds of Mercury (orbital eccentricity of 0.2056)? What are the
perihelion and aphelion distances of this planet? Compute the product vr (speed times distance) at each of these two
points and interpret this result.
5. (2 points) A television satellite is in a circular orbit about the Earth, with a sidereal period equal to the Earth’s
sidereal rotation period (23h 56m ). What is the distance from the Earth’s center to the satellite? If the satellite appears
stationary to an earthbound observer, what is the orientation of the satellite’s orbit?
6. (4 points) A planet has an orbit with a = 5 AU and b = 4 AU about a star whose mass is four times that of the sun.
• What is the orbital period of the planet?
• What is the orbital eccentricity?
• What is the planet’s closest approach to the star?
• What is the planet’s distance from the star when the true anomaly is 90◦ ?
7. (4 points) Assume that the moon moves in a perfect circle and at a constant uniform speed, has a synodic period
of 29.531 days, and that the sun is 74.23615148 times more distant than the moon. What is the difference in the time
intervals between new and first quarter and first quarter and full moon?
8. (4 points) Star A has celestial coordinates α = 06:00:00 and δ = +45◦ 00′ 00′′ and star B has celestial coordinates
α = 15:00:00 and δ = −30◦ 00′ 00′′ . What is the angular distance between the stars? How far is star A from the ecliptic?
On what day of the year does star B transit twice? From what fraction of the Earth’s surface is star A never above the
horizon?
9. (4 points) You wish to send a 500 kg spacecraft from the Earth to Jupiter. What is the Jupiter-Sun-Earth angle at
launch? Assume that the Earth is massless.
10. (3 points) As a student in Astro 291 at Cambridge in 1714, you have the good fortune to have as a teacher Sir Isaac
Newton, whose course lectures will not be exceeded in physical insight, clarity, or wit for nearly three centuries. You are
fascinated by the law of gravity, and wish to unite it with your other passion, astrology. After a few months you make
a startling discovery, which you propose as the fourth law of planetary motion: a new quantity of planetary orbits, the
zodiceleration, is conserved: Z = r × N a, where r and a are the radius and acceleration vectors, and N is an integer
that represents the sign of the zodiac where the planet is located (all quantities are those as viewed from the sun, not
from the Earth, and N = 1 for Aries, N = 2 for Taurus, etc).
You present your calculations to Sir Isaac, who ponders the question over the weekend and finally comes to a
conclusion. Does he report that the zodiceleration is indeed a constant? If so, have you found a new fundamental law of
celestial mechanics? If not, what observational evidence (available at the time) is there that contradicts your assertion?
11. (3 points) Space: the Final Frontier. These are the voyages of the starship Nittany Lion; its eight-month mission is
to free the Galaxy of the enemies of truth, justice, and the American way. The ship is currently in Earth orbit, undergoing
preparations for its voyage. From an observer on Earth, the Nittany Lion moves from west to east against the fixed stars,
and every 19.34 hours it is directly overhead. What is the ship’s sidereal orbital period?
12. (3 points) A comet has an orbit about the sun with an eccentricity of 0.999 and a semimajor orbital axis of 500 AU.
What is the orbital period? What is the perihelion distance? What is the comet’s velocity when it passes the orbit of
Neptune?
13. (3 points) A new asteroid has been discovered with an orbital period of 23.2 years. Astronomers were startled
to discover that while at perihelion the asteroid grazed the surface of the sun. What are the eccentricity and perihelion
velocity of the asteroid’s orbit?
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14. (4 points) If the law of gravity was F = − G0 M
, where G0 is the value of the gravitational constant in
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appropriate units, derive Kepler’s third law for a small object moving in a circular orbit around an infinitely massive
primary (assuming that the other laws of physics are unchanged).
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15. (4 points) While on a secret mission for Star Fleet, you are captured by the cruel crew of the Tom Osborne. You
shudder at your expected fate of having to take the University of Nebraska’s Astro 291 exam, but are relieved to hear that
clemency has been granted and you are cast adrift in space without food or water. After a day, you land on a spherical
asteroid of pure ice. Your suit contains a hidden bottle and writing material. You can throw the bottle at a velocity
of 20 m s−1 . What is the maximum radius that the planet can have in order for your message to have any hope of reaching
Star Fleet Command?
16. (4 points) Two planets have circular orbits around the sun; the semimajor axes of their orbits are 0.6 AU and
2.2 AU, respectively. They revolve around the sun in opposite directions (i.e., one goes around the sun in a clockwise
direction, the other in a counterclockwise direction). What is the synodic period of the planets?
If the planets revolve in the same direction (for example, both go around the sun in a counterclockwise direction), what
is the period of the “minimum energy” transfer orbit?
17. (4 points) A new planet has been recently discovered; careful observations show that it moves 1600′′ in its orbit
each year. The new planet has a satellite whose orbital period is 24.3 days and semimajor axis is 2.2′′ (when the planet
is in opposition). What is the mass of the newly discovered planet (assume that the satellite is much smaller than the
planet)?
18. (4 points) The USS Nittany Lion goes to maximum warp speed in response to a distress call from Federation
planet Etats Nagihcim, which is being terrorized by the nefarious renegade pirate ship Tom Osborne. The planet has a
solitary moon that has 1% of the mass of the planet and has a sidereal orbital period about Etats Nagihcim of 30 days.
The Nittany Lion arrives only to find that the planet had refused to surrender and was zapped with deadly Ω-radiation
from the Tom Osborne. As a result, 99% of the mass of the planet was instantaneously vaporized and removed from the
system, but the moon was unscathed and remained in a circular orbit about the planet at the original distance. What is
the moon’s new orbital period?
19. (6 points) While performing a survey for new planets, you discover a bright comet located at α = 21:58:15.0,
δ = +54◦ 48′ 33′′ . The following night at the same solar time the comet is at α = 23:02:26.0, δ = +59◦ 53′ 23′′ ; 24 hours
after the second observation you locate the comet at α = 00:14:26.4, δ = +62◦ 19′ 55′′ . If the comet’s apparent angular
velocity varies linearly with time, how far will it move in the 24 hours following the third observation?
20. (5 points) A star has equatorial coordinates of α = 10h 23m 17s and δ = −12◦ 22′ 59′′ . What are its ecliptic
coordinates? (Assume an inclination between the systems of 23.5◦.)
21. (6 points) While visiting Australia (149◦ East longitude, 35◦ South latitude) you go outside on 2 June at 1:00 pm
Greenwich Mean Time (i.e., standard time at 0◦ longitude).
• What is the sidereal time?
• What is the altitude of Polaris?
• What is the zenith angle of a star on the meridian with δ = +12◦ ?
• What is that star’s azimuth at transit?
• The moon is full; what are its approximate α and δ?
• What are the coordinates of circumpolar stars from this point in Australia?
22. (6 points) In 1862 Abraham Lincoln signed the Homestead Act, which allowed settlers in specific territories to
obtain ownership of 160 acres (0.25 square miles) if they farmed the land for a specified period of time. A year later
your great-great grandparents headed west to mark the borders of their homestead with four posts. Post 1 is planted at
longitude 98.0 W and latitude 40.5 N; Post 2 is placed exactly 0.5 miles west of Post 1 (the distance is measured along
the line of constant latitude). An obvious method to mark the boundaries of the plot is to go 0.5 miles straight north
or south of the two posts to find the remaining two corners. To maximize the area of land, should your ancestors plant
Posts 3 and 4 0.5 miles directly north or south of Posts 1 and 2, respectively (or does it matter)? Calculate the surface
area of the largest possible claim.
23. (8 points) A solar observatory has been installed in the Cathedral of St. Donald the Compassionate (latitude +43◦ 17′ ).
The solar entrance is located 17.2 meters above the floor. When the sun transits on the winter and summer soltice: 1) At
what distance along the floor from the wall containing the solar entrance will the images of the sun fall? 2) What will be
the physical diameters of the north-south and east-west diameters of the images of the sun? 3) What will be the physical
velocity along the floor of the images of the sun? (You can ignore refraction in these calculations, and assume that the
sun has an angular diameter of 30.0′ at all times.)
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24. (10 points) As Aristarchus’s younger sibling, you have always chafed under the knowledge that mother liked the
“smarter” child best. While musing on the subject of astronomy one day, you hit upon the same idea that Aristarchus
would the following year: a way to measure the relative sizes of the Earth, Moon, and sun. Over the next six months you
make careful observations, and find that the angular diameters of the sun and moon are 0.75◦ and 0.85◦ , respectively;
the ratio of the distance to the sun to the distance of the moon is 47.6, and the diameter of the Earth’s shadow at the
distance of the moon is 2.47 times the diameter of the moon.
Calculate the sizes of the sun and the moon and their distances from the Earth relative to the size of the Earth.
You submit your astonishing findings for publication, but alas, they are rejected because they are considered “ridiculous”.
Six months later you see your brother hailed as a genius by the Athens Enquirer for duplicating your work, with less
accurate data.
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