General Astronomy, 29:61
Fall, 2004
Fourth Homework Set...September 18, 2004.
Due September 24, 2004

Show calculations and give reasons for your answers. Don't go around confused and despondent; if you do not know how to get started, ask us for help. The purpose of problem sets is to promote thinking and lead to understanding, not produce a confiteor of revealed truth. Note: Data for some of these problems can be found in the appendix of the textbook.

(1) How close behind the asteroid Vesta would one have to go to see a total, umbral eclipse of the Sun? In answering this problem, you also need to take into account the fact that the distance of Vesta from the Sun is different than for the Earth.

(2) A certain planet is similar to the Earth, and has a moon similar to ours, which is also in an inclined orbit. On this planet, the synodic period of the moon is 24.230 days. The nodal period is 14.538 days. An eclipse is observed. How long after would an observer see another, similar eclipse?

(3) Imagine that our Moon were in a circular orbit, with a radius equal to its mean distance from the Earth (the so-called semimajor axis). Let's also assume that the Moon's orbit is in the plane of the ecliptic. Calculate the duration of the total solar eclipse that occurs every month, i.e. the time from the start of an eclipse somewhere on the Earth to the end of the eclipse at any other point on the Earth.

(4) Calculate the maximum possible duration of a total solar eclipse. Hint: Think in terms of the angular size of the Moon, the angular size of the Sun, and the angular speed at which the Moon moves relative to the Sun.

(5) Calculate the orbital angular momentum and the rotational angular momentum of the planet Earth (use formulas presented in lecture), and compare the values of the two. Is the orbital or rotational angular momentum of the Earth greater?

(6) A small planet has a mass of $10^{23}$ kg. Its motion is defined in an $(x,y,z)$ coordinate system centered on the Sun. In this coordinate system, the vector $\vec{r}$ giving the position of the planet is

$$\vec{r} = 3.0 \hat{e}_x + 2.0 \hat{e}_y + 0 \hat{e}_z$$ (1)

where the units are in astronomical units. In this notation, $(\hat{e}_x, \hat{e}_y,\mbox{ and } \hat{e}_z$ are the unit vectors in the $(x,y,z)$ directions, so that the x, y, and z components of the position vector are 3.0 astronomical units and 2.0 astronomical units, respectively.

The planet is moving with a velocity vector

$$\vec{v} = 1.0 \hat{e}_x + 4.0 \hat{e}_y + 0 \hat{e}_z$$ (2)

with units in kilometers/sec. Calculate the vector angular momentum of this planet.

(7) A rotating object has an angular momentum of 10 kg-m$^2$-sec$^{-1}$ in the $z$ direction. A constant torque of 1 kg-m$^2$-sec$^{-2}$ is applied in the $x$ direction for 5 seconds. What is the angle of the rotation axis, relative to the $z$ axis, after the torque has been applied?

(8) Let's assume you make very accurate measurements of the position of the North Celestial Pole and the First Point of Aries, and thereby accurately establish the Celestial Coordinate system. You measure the position of a star close to the celestial equator, like Procyon ($\alpha$ Canis Minoris, at RA $\simeq$ 8h40m, Dec $\simeq$ 5$^{\circ}$. You then compare your position for Procyon with that of a star chart made in 1900. The values for Right Ascension and Declination disagree. Why is this so? How big is the discrepancy in Right Ascension?