Posted: November 3rd, 2015

Astronomy

Astronomy

The following assignment covers Chapters One to Three. Please answer the following questions and submit your work using the assignment link. Show all your work in a clear and concise fashion, and include each step you take to reach the answer. The way in which you obtain your answer is as important as the final answer, itself, and marks will be given for showing all the relevant steps. Please refer to the appendix in the textbook (pages 392–395) to obtain any needed planetary values, and to Table 2-2 on page 41 for any needed constants.

1. [12 marks total]

(a) [4] Use Kepler’s first law to derive explicit expressions for the perihelion and aphelion distances for a planet in terms of the semi-major axis a, and the eccentricity e, for an elliptical orbit. Calculate these distances for Mercury.

(b) [4] What are the perihelion and aphelion speeds of Mercury?

(c) [4] Calculate the maximum angular separation of Mercury as seen from Earth. (d) [4] What is the angular size of Mercury in arcseconds when it is at this location?

2. [9 marks total] Problem 19 in Chapter Three of the text (page 72):

(a) [4] Consider two particles in circular orbits in Saturn’s rings 108 m from Saturn’s center. One is located 1 m farther from Saturn than the other. By Kepler’s laws, they have different periods and must occasionally pass each other. How fast do they pass by each other? (Hint: You could compute Vcirc for each particle and subtract, but the difference in orbits is only one part in 108 , so you would have to maintain accuracy to at least eight decimal places, which is unrealistic. Instead, use calculus and differentiate Vcirc = (GM) 1/2 (r -1/2 ) with respect to getting dVcirc = difference in velocity dVcirc = 1/2 (GM) 1/2 r -3/2dr where dr is the difference in distance. Note that the text has chosen to ignore the negative sign for this question, which indicates the direction of the change in velocity

(b) [3] What if the particles are about 200 km apart, as in the example of Figure 3-17a? Compare your result with the 60 m/s figure given there (based on a more detailed calculation including gravitational attraction of the particles for each other)

. (c) [2] How fast would a shuttle in a 200-km-high circular orbit approach a space lab in a 201-km-high circular orbit? (Answer: about 0.6 m/s.) 1

3. [16 marks total] Do not solve the following question numerically unless specifically asked to do so. The flux from the Sun’s surface is given by LJ = 4pRJ 2 sTJ 4 , where TJ is the temperature on the Sun’s surface.

(a) [3] What is the incident luminosity from the Sun seen on Earth’s surface?

(b) [4] Only a part of the luminosity incident on Earth’s surface is reflected back. The fraction of radiation reflected from Earth’s surface, or albedo, is given by A. The other part is absorbed and converted into heat, which is then emitted as thermal emission from the planet. Write an expression for the luminosity absorbed by Earth and the luminosity emitted by Earth. Assume Earth has had ample time to come up to thermal equilibrium. Use the symbol Tsurf to represent the surface temperature of Earth.

(c) [3] Find an expression for the surface temperature of Earth, when the Earth is at thermal equilibrium. Note that thermal equilibrium means that the amount of radiation absorbed by Earth is the same as the amount of radiation emitted by Earth.

(d) [3] Calculate the surface temperature for Earth if A = 0.36. If Earth’s observed maximum temperature is 310 K, does the calculated temperature predict an accurate surface temperature? Explain.

(e) [3] Repeat the calculation to find the surface temperature of Venus (A = 0.76). Venus has an observed maximum temperature of 750 K. Is the calculated surface temperature for Venus accurate? Why or why not?

4. [8 marks total] Problem 16 in Chapter Three of the text (page 72):

(a) [4] In the past, when the Moon was 0.25 as far from Earth as it is now, estimate how much more massive Earth’s average tidal bulge was than it is today.

(b) [4] How much stronger was the net accelerating effect of this bulge on the Moon?

5. [5 marks total] It has been proposed that a spaceship might be propelled in the Solar System by radiation pressure, using a large sail made of foil. The reflectivity of foil causes the radiation force to be twice what it is normally. Calculate the area (in square kilometers) that the sail must be for the radiation force to be equal to the Sun’s gravitational attraction. Assume that the mass of the ship and sail is 1500 kg and Q=1.1

6. [8 marks total] Problem 17 in Chapter Three of the text (page 72):

An early measurement of the outer radius of Uranus’s rings was about 5.0 × 107 m from the center of the planet. Compare this to the Roche limit for icy bodies near Uranus, using the expression derived here for touching particles. What is the significance of this result? Would you expect Uranus’s ring particles to aggregate into a satellite?

7. [17 marks total] Consider a small dust grain of mass m = 4.19 × 10-3 kg and radius a = 1 cm, which becomes separated from a parent body in a circular orbit a distance 2.5 AU from the Sun. The dust is subject to a force from the Poynting-Robertson effect given by FP R = a 2 4c 2 L 2 JGMJ r 5 !1/2 where a is the size of the particle and r is the radius of its orbit. We are going to take a series of steps to determine how much its orbital radius changes after one complete orbit around the Sun. Please do not calculate the values explicitly unless asked to do so.

(a) [4] Use calculus to compute the acceleration that the dust undergoes, dv dt , where v is the orbital speed of the dust grain. dv dt is equal to dv dr times dr dt. Compute dv dr and multiply it by unknown quantity dr dt. 1 Adapted from Halliday, D., Resnick, R., & Walker, J. (1997). Question 46P (p. 868). In Fundamentals of Physics Extended (5th ed.). New York: John Wiley & Sons. 2

(b) [4] Newton’s law tells us that mass times acceleration is equal to the force. Write an expression for the force from the Poynting-Robertson effect and solve for dr dt.

(c) [4] dt is the time the dust takes to complete one orbit—otherwise known as its period. Using Kepler’s law, write an expression for dt or the period of the dust’s orbit, making sure you change the elements of the standard equation to the notation given in this problem. You can simplify this expression by noting that m is negligible compared to M.

(d) [3] Use your result for dt in the equation for dr dt and solve for dr (the change in radius over one period).

(e) [2] Calculate how much the dust’s orbital radius changes after one complete orbit around the Sun.

8. [10 marks total] We’ve learned that due to tidal forces, the Moon is slowly spiraling outward, away from the Sun. As this happens, the Moon’s angular diameter will get smaller in the sky. Eventually, solar eclipses will no longer occur, since the Moon’s disk will be too small to cover the disk of the Sun.

(a) [4] Using the Moon’s current orbit, find the Moon’s angular diameter in the sky (in degrees) at its closest approach to Earth. This is its current largest angular diameter.

(b) [2] The smallest angular size of the Sun occurs at a distance of 1.515 × 108 km at the summer solstice. What is the Sun’s angular size at this distance in degrees?

(c) [2] How far away does the Moon have to be to match the Sun’s smallest angular size?

(d) [2] If the Moon is moving away from Earth at a rate of 4 cm/y, how many years will it take to move to the distance calculated in part c)? What is the year of the last total solar eclipse?2

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