MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Department of Physics

Physics 8.901: Astrophysics I Spring Term 2004

PROBLEM SET 1 (corrected)

NOTE ON CORRECTIONS: Minor typos in the first equation of Problem 2 and in Problem 3b have

been corrected. Also, note that parts (b) and (c) of Problem 2 are somewhat flawed, in that the premise that

the data points will lie on a straight line is not in fact true for samples of different composition. You may

omit parts (b) and (c). As an extra credit question, note the addition of an optional part (d) for Question 2.

Due: Thursday, February 12 in class

Reading: Phillips, The Physics of Stars, Chapters 1.

1. Historical astronomy: Fundamental length scales. Accurately determining distances and sizes in

astrophysics remains a fundamental and challenging problem to this day. However, a number of surpris-

ingly accurate measurements can be made simply with the naked eye. The ancient Greek philosopher

Aristotle (c.384–322 B.C.) was able to deduce that the Earth was spherical from observations like the

shape of the Earth’s shadow during lunar eclipses and the changing view of stars in the sky during

travel from north to south. A number of fundamental length scales in our solar system were also

correctly deduced by the ancient Greeks.

(a) Size of the Earth. Eratosthenes (c.276–196 B.C.) deduced the size of the spherical Earth using

the following facts: (1) On a particular summer day each year, the Sun penetrated to the bottom

of a very deep well (and thus was directly overhead – this point is called the zenith) in the town

of Syene (now Aswan); (2) On the same day in Alexandria, the Sun at mid-day was 7 south of

the zenith; (3) Alexandria was north of Syene by a distance of just under 5000 stadia, where 1

stadium is about 160 meters. Eratosthenes assumed that the Sun is sufficiently distant that its

rays can be treated as parallel. Use these facts to reproduce Eratosthenes’s inference of the radius

of the Earth. (You may use the value of π.) Compare this to the modern value of 6378 km.

(b) Size and distance of the Moon. Aristarchus (c.310–233 B.C.) calculated these using informa-

tion from a lunar eclipse. Use Timothy Ferris’s composite photograph of a lunar eclipse (on the

web at http://web.mit.edu/8.901/images/moon.png) to make a similar calculation. The dia-

gram below indicates the relevant geometry. You should make use of small angle approximations

where appropriate.

1/4o

1/4o

Umbral Shadow Earth

Center of Solar Disk

Center of Solar Disk

Eastern LimbD

Western Limb

Moon

i. Assume that only the darkest part of the Earth’s shadow (the umbra) corresponds to total

eclipse. Estimate the diameter of the circle roughly corresponding to the umbral shadow on

the composite image, and also the diameter of one of the lunar images. Note that the center

of the shadow does not lie on the line connecting the path of Moon’s center — why not?

ii. Compute the radius of the Moon compared to that of the Earth. Be sure to account for the

proper geometry of the umbral shadow at the distance of the Moon; for this purpose, you may

take the angular diameter of both the Sun and the Moon to be 0.5. Estimate the uncertainty

in your answer, given the uncertainty in your estimate of the diameter of the umbral shadow.

Compare to the modern value of 6378/1738=3.67.

iii. Taking the angular diameter of the Moon to be 0.5, calculate the Earth-Moon distance D

in terms of the Earth’s radius.

(c) Distance to the Sun. Aristarchus also estimated this. In the diagram below, the Moon at Q

is at first quarter, so that the angle EQS is 90. (Note that EQ is not perpendicular to ES.)

The interval from new Moon (at position N) to first quarter (at Q) is 35 min shorter than that

from first quarter to full Moon (at F ). Given that the lunar synodic period (the interval between

two identical lunar phases) is 29.53 d, estimate the Earth-Sun distance (ES) in terms of the

Earth-Moon distance.

E

F

Q

S

β

β

N

2. The age of the Sun. How old is the Sun? Two things are certain: the Sun is older than the Solar

System, and the Solar System is as old as the oldest rocks in the Solar System. In practice, tectonic

activity gives rise to rocks with a wide range of ages, but the oldest terrestrial rocks, and also the

oldest rocks from meteors and from the Moon, have ages approaching 4.55 × 109 years. This sets a

lower limit to the age of the Sun.

In this problem, we illustrate how the naturally occuring isotopes of uranium, 235U and 238U, can be

used to determine the age of rocks. Both isotopes decay via a sequence of α-decays and β-decays to

form stable isotopes of lead; the decay chain of 235U ends up with 207Pb, and the decay chain of 238U

ends up with 206Pb. As a result, the number of uranium nuclei in a rock decays exponentially with

time as:

N5(t) = N5(0)e−λ5tandN8(t) = N8(0)e−λ8t,

where the last digit of the isotope mass number has been used as a subscript label. The decay

constants λ5 and λ8 for the two isotopes correspond to half-lives of T5 = ln 2/λ5 = 0.7 × 109 years

and T8 = ln 2/λ8 = 4.5 × 109 years. The magnitudes of these half-lives are ideally suited to the

determination of rock ages which are over a billion years old.

Now consider a set of rock samples which were formed at the same time, but with different chemical

composition. They differ chemically because different chemical elements are affected differently by

the processes of rock formation. However, rock formation processes do not favor one isotope over

another. For example, on formation, the relative abundances of 235U and 238U should be the same in

every sample. Similarly, the relative abundances of 207Pb and 206Pb should also be the same in every

sample. However, these abundances will change with time as the uranium isotopes decay into lead.

(a) Consider the ratio of the increase in the number of 207Pb nuclei relative to the increase in the

number of 206Pb nuclei. Show that this ratio is the same for all rock samples which were formed

at the same time, and show it is given by

N7(t) −N7(0)

N6(t) −N6(0)=N5(t)

N8(t)

(

exp(λ5t) − 1

exp(λ8t) − 1

)

.

(b) (You may omit this part: see correction note on first page) Consider a graph in which the

measured abundances in the rock samples of 207Pb and 206Pb are plotted, with N7(t) along the

y-axis and N6(t) along the x-axis. Show that a straight line will be obtained if all the samples

were formed at the same time.

(c) (You may omit this part: see correction note on first page) Given that the current ratio of

naturally occurring 235U to 238U is 0.0071, evaluate the gradient of the straight line for the rock

samples of age 109 years, 3 × 109 years, and 5 × 109 years.

(d) Extra credit: As noted in the correction on page 1, the approach described in parts (b) and (c)

above is not completely correct. For extra credit, consult suitable resources in the library or on

the web and summarize briefly the correct explanation of how uranium-lead radioactive dating is

used for determining ages of old rocks.

3. Central pressure in stars. Useful bounds can be set on the pressure at the center of a star without

detailed structure calculations. Consider a star of mass M and radius R. Let P (r) be the the pressure

at a distance r from the center and m(r) be the mass enclosed by a sphere of radius r.

(a) Show that in hydrostatic equilibrium, the function P (r) +Gm2(r)/8πr4 decreases with r. Hence

show that the central pressure satisfies the inequality

Pc >1

6

(

4π

3

)1/3

G〈ρ〉4/3M2/3,

where 〈ρ〉 is the mean density.

(b) If you assume that the density ρ(r) decreases with r (which is reasonable), it is possible to derive

a tighter lower bound and a useful upper bound on the central pressure. Show that

Pc >1

2

(

4π

3

)1/3

G〈ρ〉4/3M2/3.

(c) In addition, show that

Pc <1

2

(

4π

3

)1/3

Gρ4/3

c M2/3,

where ρc is the central density.

4. The Kepler problem: hyperbolic motion. In class, we derived the Kepler equation for elliptical

motion, ωt = ψ− e sinψ, which relates the eccentric anomaly ψ to the time t since pericenter passage,

the eccentricity e ≤ 1, and the orbital angular frequency ω. For hyperbolic motion, the analog of the

eccentric anomaly is the angle ψ′, defined such that the trajectory can be written as r = a(e coshψ′−1),

where e > 1 and a(e − 1) is the distance of closest approach to the focus (the pericenter distance).

Find an analog to the Kepler equation for hyperbolic motion, which specifies the time t since closest

approach as a function of ψ′.

5. Tidal evolution of the Earth-Moon system. In this problem, you will compute the evolution of

the Earth-Moon system by considering the tidal coupling between the Moon’s orbit and the Earth’s

rotation. Angular momentum may be exchanged between these two components but must be conserved

overall. Energy may be lost from the system via the heat generated by tidal friction. You should neglect

any effects due to the rotation of the Moon.

(a) Write down expressions for the total energy E and total angular momentum J of the Earth-

Moon system. Some useful symbols will be the Earth’s angular rotation frequency ω; the Moon’s

(Keplerian) orbital frequency Ω; the masses of the Earth and Moon, Me and Mm; the Earth’s

moment of inertia I; and the mean separation of the Earth and Moon, a.

(b) Use the equation for J to eliminate ω from the energy equation.

(c) Show that the energy equation can be cast into the dimensionless form

ε = −1

s+ α(j − s1/2)2,

where ε is the total energy in units of (Gmemm/2a0), j is the total angular momentum in units

of (µa20Ω0), s = a/a0 is the dimensionless separation, µ is the reduced mass, and the subscript

“0” refers to values at the present epoch in history.

(d) Find numerical values for α and j. Look up the masses of the Earth and Moon, and take the

Earth’s moment of inertia to be (2/5)MeR2

e and a0 = 3.84 × 105 km.

(e) Graph the dimensionless energy equation to find the two values of s for which ε is an extremum.

(f) Find the same two values of s quantitatively by differentiating the energy equation and solving the

resulting nonlinear equation numerically by the Newton-Raphson method or some other scheme.

Show that ω = Ω at these orbital separations. Find the corresponding orbital period of the Moon

and rotation period of the Earth.

(g) Find the difference in energy ∆E between the current epoch and the time in the future when the

Earth’s rotation and the Moon’s orbit will be synchronous.

(h) Estimate the rate of energy dissipation due to tidal friction by assuming that, twice per day, the

top 1-m layer of the oceans is lifted by 1-m and then lowered. Further assume that a few percent

of this mechanical energy is dissipated as heat.

(i) From the energy dissipation rate and total energy ∆E that must be lost in order for the Earth

to come into rotational equilibrium with the Moon’s orbit, estimate the time (from the current

epoch) when this equilibrium configuration will be reached.