In consultation with the other members of your group, solve the problems below.
1. At what distance would a person have to hold a nickel (which has a diameter of about 2.0 cm) in order for the nickel to subtend an angle of (a) 1o, (b) 1', (c) 1"?
Use small angle formula: theta = linear size/distance, where theta is in radians.
d = r/theta = 2.0 cm/(1 degree * pi/180 degrees) = 1.1 m
d = r/theta = 2.0 cm/(1 arcmin * (1 degree/60 arcmin) * (pi/180 degrees)) = 69 m
d = r/theta = 2.0 cm/(1 arcsec * (1 arcmin/60 arcsec) * (1 degree/60 arcmin) * (pi/180 degrees)) = 4100 m
2. Suppose you live at a latitude of 40o N. What is the elevation of the sun above the southern horizon at noon at the time of the winter solstice?
On the winter solstice, the sun has a declination of -23.5 degrees.
At a latitude of 40 degrees N, the declination at zenith (90 degrees from the horizon) is 40 degrees.
Thus, the zenith angle (angle from zenith to source) is 40o - (-23.5o) = 63.5o
The elevation above the horizon (altitude) is then 90o - zenith angle = 26.5o
Note: you may find this easier to visualize if you draw a sketch of the angles involved.
3. The synodic period of Mercury is 115.88 days. Calculate its sidereal period in days.
1/S = 1/P - 1/E, for inferior planets (1/S = 1/E - 1/P, for superior planets).
1/115.88 = 1/P - 1/365.25
1/P = 1/115.88 + 1/365.25 = 0.011367
P = 87.970 days
4. An asteroid has a semi-major axis of 4.00 AU and an eccentricity of 0.500. What is its distance from the sun at perihelion and aphelion? What is the velocity of the asteroid at perihelion and aphelion? Be sure to label which one is which.
rperi = a (1 - e) = 4.00 (1 - 0.500) = 2.00 AU
rap = a (1 + e) = 4.00 ( 1 + 0.500) = 6.00 AU
Velocity comes from vis viva equation:
v2 = G(M1 + M2)[2/r - 1/a]
where M1 is the Sun's mass and M2 is the mass of the asteroid (negligible).
At perihelion:
v2peri = 6.67 x 10-11 x 1.99 x 1030( 2/(2.00 x 1.496 x 1011) - 1/(4.00 x 1.496 x 1011) )
vperi = 25.8 km/s
At aphelion:
v2ap = 6.67 x 10-11 x 1.99 x 1030( 2/(6.00 x 1.496 x 1011) - 1/(4.00 x 1.496 x 1011) )
vap = 8.60 km/s
5. Imagine a planet like the Earth orbiting a star with 4 times the mass of the Sun. If the semi-major axis of the planet's orbit is 1 AU, what would be the planet's sidereal period?
Use Newton's form of Kepler's laws:
P2 = 4 pi2 a3/G(M1 + M2)
P2 = 4 pi2 (1.496 x 1011)3/(6.67 x 10-11 x 4 x 1.99 x 1030)
P2 = 2.49 x 1014 s2
P = 1.58 x 107 s = 0.5 years