Exercises

Question 13.328

Kepler’s Third Law states that T2/a3 has the same value for each planetary orbit. Do the data in the following table support this conclusion? Estimate the length of Jupiter’s period, assuming that a = 77.8 × 1010 m.

Planet

Mercury

Venus

Earth

Mars

a (1010 m)

5.79

10.8

15.0

22.8

T (years)

0.241

0.615

1.00

1.88

Question 13.329

Finding the Mass of a Star Using Kepler’s Third Law, show that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is .

Question 13.330

Ganymede, one of Jupiter’s moons discovered by Galileo, has an orbital period of 7.154 days and a semimajor axis of 1.07 × 109 m. Use Exercise 2 to estimate the mass of jupiter.

Question 13.331

An astronomer observes a planet orbiting a star with a period of 9.5 years and a semimajor axis of 3 × 108 km. Find the mass of the star using Exercise 2.

Question 13.332

Mass of the Milky Way The sun revolves around the center of mass of the Milky Way galaxy in an orbit that is approximately circular, of radius a ≈ 2.8 × 1017 km and velocity υ ≈ 250 km/s. Use the result of Exercise 2 to estimate the mass of the portion of the Milky Way inside the sun’s orbit (place all of this mass at the center of the orbit).

Question 13.333

A satellite orbiting above the equator of the earth is geosynchronous if the period is T = 24 hours (in this case, the satellite stays over a fixed point on the equator). Use Kepler’s Third Law to show that in a circular geosynchronous orbit, the distance from the center of the earth is R ≈ 42,246 km. Then compute the altitude h of the orbit above the earth’s surface. The earth has mass M ≈ 5.974 × 1024 kg and radius R ≈ 6371 km.

Question 13.334

Show that a planet in a circular orbit travels at constant speed. Hint: Use that J is constant and that r(t) is orthogonal to r′(t) for a circular orbit.

Question 13.335

Verify that the circular orbit

r(t) = 〈R cos ωt, R sin ωt〉

satisfies the differential equation, Eq. (1), provided that ω2 = kR−3.

Then deduce Kepler’s Third Law for this orbit.

Question 13.336

Prove that if a planetary orbit is circular of radius R, then υT = 2πR, where υ is the planet’s speed (constant by Exercise 7) and T is the period. Then use Kepler’s Third Law to prove that .

Question 13.337

Find the velocity of a satellite in geosynchronous orbit about the earth. Hint: Use Exercises 6 and 9.

Question 13.338

A communications satellite orbiting the earth has initial position r = 〈29,000, 20,000, 0〉 (in km) and initial velocity r′ = 〈1, 1, 1〉 (in km/s), where the origin is the earth’s center. Find the equation of the plane containing the satellite’s orbit. Hint: This plane is orthogonal to J.

Question 13.339

Assume that the earth’s orbit is circular of radius R = 150 × 106 km (it is nearly circular with eccentricity e = 0.017). Find the rate at which the earth’s radial vector sweeps out area in units of km2/s. What is the magnitude of the vector J = r × r′ for the earth (in units of km2 per second)?

Exercises 13–19: The perihelion and aphelion are the points on the orbit closest to and farthest from the sun, respectively (Figure 8). The distance from the sun at the perihelion is denoted rper and the speed at this point is denoted υper. Similarly, we write rap and υap for the distance and speed at the aphelion. The semimajor axis is denoted a.

r and v = r′ are perpendicular at the perihelion and aphelion.

771

Question 13.340

Use the polar equation of an ellipse

to show that rper = a(1 − e) and rap = a(1 + e). Hint: Use the fact that rper + rap = 2a.

Question 13.341

Use the result of Exercise 13 to prove the formulas

Question 13.342

Use the fact that J = r × r′ is constant to prove

υper(1 − e) = υap(1 + e)

Hint: r is perpendicular to r′ at the perihelion and aphelion.

Question 13.343

Compute rper and rap for the orbit of Mercury, which has eccentricity e = 0.244 (see the table in Exercise 1 for the semimajor axis).

Question 13.344

Conservation of Energy The total mechanical energy (kinetic energy plus potential energy) of a planet of mass m orbiting a sun of mass M with position r and speed υ = ∥r′∥ is

  • Prove the equations

  • Then use Newton’s Law F = ma and Eq. (1) to prove that energy is conserved—that is, .

Question 13.345

Show that the total energy [Eq. (8)] of a planet in a circular orbit of radius R is . Hint: Use Exercise 9.

Question 13.346

Prove that as follows:

  • Use Conservation of Energy (Exercise 17) to show that

  • Show that using Exercise 13.

  • Show that using Exercise 15. Then solve for υper using (a) and (b).

Question 13.347

Show that a planet in an elliptical orbit has total mechanical energy , where a is the semimajor axis. Hint: Use Exercise 19 to compute the total energy at the perihelion.

Question 13.348

Prove that at any point on an elliptical orbit, where r = ∥r∥, υ is the velocity, and a is the semimajor axis of the orbit.

Question 13.349

Two space shuttles A and B orbit the earth along the solid trajectory in Figure 9. Hoping to catch up to B, the pilot of A applies a forward thrust to increase her shuttle’s kinetic energy. Use Exercise 20 to show that shuttle A will move off into a larger orbit as shown in the figure. Then use Kepler’s Third Law to show that A’s orbital period T will increase (and she will fall farther and farther behind B)!