Astronomy 420 – Galaxies and Cosmology
Problem Set # 1
due 26 January 2005 (Wednesday)
1. Hipparcos’ Size of the Solar System: In this problem, we recreate Hipparcos of Samos’
2nd century B.C.E. calculation of the sizes and distances of the Sun and the Moon. Our inputs are
a few basic observations and the radius of the Earth, r E , which had been determined previously by
Eratosthenes in the 4th century B.C.E. from the length of gnomon shadows.
(a) Our starting point is the realization that at half moon, the Sun-Moon-Earth angle is
90◦ . Relate the Moon-Earth-Sun angle, θ, to the Earth-Moon distance, d EM , and the Earth-Sun
distance, dES . The following diagram may help:
(b) During solar eclipses, the Sun and Moon describe the same solid angle on the sky. Relate
the lunar radius, rM , to the solar radius, rS , and θ.
(c) Hipparcos noted that lunar eclipses of maximum duration, during which the Moon lies
squarely on the ecliptic and therefore passes through the very center of the Earth’s shadow, the
period of total obscurity approximately equals the length of time required for the Moon to enter
θ
rE .
the Earth’s shadow. Therefore, in the following diagram, x = 2r M . Show that rM = 1+cos
3
(d) During lunar eclipses of maximal duration, the Moon spends roughly τ E in the Earth’s
shadow, where τ has units of time. Knowing that the lunar cycle is τ M , relate dEM to rM .
(e) Given the Earth’s radius, rE = 6371 km, θ = 89◦ 510 , τE = 7.5 min, and τM = 27.3 days,
find rM , rS , dEM , and dES . Note that θ is a difficult measurement to make; Aristarchos incorrectly
measured θ = 87◦ .
–2–
2. Deep Impact: Compare the energy of the Deep Impact probe slamming into Comet Tempel 1,
scheduled for 2005 July 4, to the energy of the Hiroshima blast. Recall the probe weighs 317 lb and
will be traveling at a relative velocity of 23,000 m.p.h.. Hiroshima had an energy of approximately
13 kilotons, where 1 kiloton = 4.184 × 10 12 Joule.
3. Magnitudes: The globular cluster IAU C0923−545 has an integrated apparent visual magnitude of V = +13.0 and an integrated absolute visual magnitude of M V = −4.15. It is located
9.0 kpc from Earth and is 11.9 kpc from the Galactic center, just 0.5 kpc south of the Galactic
midplane. [Carroll & Ostlie Q22.3]
(a) Estimate the amount of interstellar extinction between Earth and IAU C0923−545.
(b) What is the amount of interstellar extinction per kiloparsec?
4. Parallax and Space Motion: The trigonometric parallax angle of Altair is 0.00 198 ± 0.00 006.
It also exhibits a proper motion of 0.00 66 yr−1 and has a radial velocity of −26 km s−1 . [Carroll &
Ostlie Q22.26]
(a) How far is Altair from Earth?
(b) What angle does Altair’s space motion vector make with its radial velocity vector?
(c) Find Altair’s transverse velocity and space motion relative to the Sun.
5. Galactic Structure: Briefly describe the three major components of the Milky Way Galaxy:
the disk, bulge, and halo. Discuss their relative ages, morphologies (shapes), and kinematics.
6. Galactic Year: Approximately how many times has the Sun circled the center of the Galaxy
since the star’s formation? [Carroll & Ostlie Q22.1]
7. A Keplerian Galaxy: [Carroll & Ostlie Q22.17]
(a) Beginning with Kepler’s third law (Eq. 2.35), derive an expression for Θ(R), assuming that
the Sun travels in a Keplerian orbit about the center of the Galaxy.
(b) From your result in part (a), derive analytic expressions for the Oort constants A and B.
(c) Determine numerical values for A and B in the solar neighborhood, assuming R 0 = 8.0 kpc
and Θ0 = 220 km s−1 . Express your answers in units of km s −1 kpc−1 .
(d) Do your answers in part (c) agree with the measured values for the Milky Way Galaxy?
Why or why not?
8. The Milky Way’s Black Hole: If the accretion rate at the Galactic center is 10 −3 M yr−1
and if it has remained constant over the past 5 billion years, how much mass has fallen into the
center over that period of time? Compare your answer with the estimated mass of a possible
supermassive black hole residing in the center of our Galaxy. [Carroll & Ostlie Q22.28]