Chapter 34, Other Planetary Systems Video Solutions, Pathways to Astronomy

Problem 1

Suppose a Jupiter-size exoplanet (radius $71,500 \mathrm{km}$ ) passed in front of a Sun-size star (radius $696,000 \mathrm{km}$ ). What percentage of the star's light would be blocked by the exoplanet?

Laszlo Zalavari

Numerade Educator

03:23

Problem 2

The Doppler method for finding planets can be used to see how fast the Sun moves in response to a planet orbiting it.
a.Using Kepler's third law $\left(P^{2}=a^{3}, \text { for } P\right.$ in years and a in AU), find the period of the orbit and its circumference if the semimajor axis is $0.05 \mathrm{AU}, 0.5 \mathrm{AU}$, or $5 \mathrm{AU}$.
b. Calculate how fast a planet would be orbiting the Sun in meters per second at each of these three distances.
c. The Sun also "orbits" the point of balance between the Sun and the planet (although this point may be inside the Sun). According to Newton's third law, the Sun's speed around this point must be smaller than an orbiting planet's in proportion to the ratio of their masses. What is the Sun's speed in each case above if the orbiting planet has Jupiter's mass? the Earth's mass?

Rajesh Singh

Numerade Educator

00:59

Problem 3

When planets orbit a star, both orbit their common "center of mass" (Unit 17), which lies at a point between them closer to the center of the more massive object in proportion to the ratio of their masses. (If the planet is $1 / 10^{\text {th }}$ as massive as the star, the point will be $10 \times$ farther from the planet than from the star.)
a. By how many AU does the Sun shift in each of the cases in the previous problem?
b. If viewed from 30 light-years away, how large will its angular shift be? (Refer to the angular size formula in Unit $10 .$ )

Abhishek Jana

Numerade Educator

02:00

Problem 4

Gliese 581 is a star with a mass $31 \%$ of the Sun's mass, and it generates only about $1.3 \%$ as much light as the Sun.
a. Use Newton's version of Kepler's third law to show that the nearest planet, which takes 5.37 days to orbit the star, must orbit at a distance of $0.041 \mathrm{AU}$
b. Compare the brightness of light received from Gliese 581 at the distance of $0.041 \mathrm{AU}$ to the brightness of light received at Earth from the Sun. (Use the inverse-square law-Unit 21.2.)

Dominador Tan

Numerade Educator

03:23

Problem 5

Astronomers observe planets orbiting stars with a variety of masses. If the mass of the star is smaller, its force on a planet will be smaller, so the planet orbits more slowly. On the other hand, the star will be farther from the center of mass because the ratio of masses is more nearly equal, so its orbital speed will be faster. It is not immediately apparent which will have the greater effect. The goal of this problem is to show that it is easier to detect the same mass planet around a less massive star using the Doppler method.
a. Use the technique of problem 1 to find the speed of a 1 solarmass star in response to a Jupiter-mass planet in orbit at $1 \mathrm{AU}$
b. Use Newton's version of Kepler's third law (Unit 17.2 ) to find the period of a Jupiter-mass planet orbiting at $1 \mathrm{AU}$ around a 0.25 -solar-mass star. (Note that the planet's mass is still very small by comparison to the star.
c. What is the speed of the planet and 0.5 -solar-mass star above?

Rajesh Singh

Numerade Educator