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Chapter 10: Problem 15

Astronomers have detected a solar system consisting of three planets orbiting the star Upsilon Andromedae. The planets have been named \(\mathrm{b}, \mathrm{c}\), and \(\mathrm{d}\). Planet b's average distance from the star is \(0.059\) A.U., and planet c's average distance is \(0.83 \mathrm{A.U.}\), where an astronomical unit or A.U. is defined as the distance from the Earth to the sun. For technical reasons, it is possible to determine the ratios of the planets' masses, but their masses cannot presently be determined in absolute units. Planet c's mass is \(3.0\) times that of planet b. Compare the star's average gravitational force on planet c with its average force on planet b. [Based on a problem by Arnold Arons.

Short Answer

Expert verified
The gravitational force on planet c is approximately 0.0152 times that on planet b.

Step by step solution

01

Understand the Gravitational Force Formula

The gravitational force between two objects, such as a star and a planet, can be calculated using Newton's Law of Universal Gravitation: \( F = \frac{G \, m_1 \, m_2}{r^2} \). Here, \( F \) is the gravitational force, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two objects, and \( r \) is the distance between the centers of the two objects.
02

Express the Forces on Each Planet

For planet b, the force between Upsilon Andromedae and the planet can be expressed as \( F_b = \frac{G \, M \, m_b}{(0.059)^2} \) and for planet c, it is \( F_c = \frac{G \, M \, m_c}{(0.83)^2} \), where \( M \) is the mass of the star and \( m_b \) and \( m_c \) are the masses of planets b and c respectively.
03

Use the Mass Ratio Given

According to the problem, planet c's mass is \(3.0\) times that of planet b. Therefore, \( m_c = 3 \, m_b \). Substitute this relation into the equation for the force on planet c: \( F_c = \frac{G \, M \, (3 \, m_b)}{(0.83)^2} \).
04

Calculate the Ratio of Forces

To find the ratio of the gravitational forces \( \frac{F_c}{F_b} \), substitute the force expressions from Step 2 and simplify: \( \frac{F_c}{F_b} = \frac{\frac{G \, M \, (3 \, m_b)}{(0.83)^2}}{\frac{G \, M \, m_b}{(0.059)^2}} = \frac{3}{(0.83 / 0.059)^2} \).
05

Simplify and Compute the Ratio

Calculate \( 0.83 / 0.059 \), which is approximately \(14.068 \). Then calculate \( (14.068)^2 \) which is approximately \(197.91\). Thus, \( \frac{F_c}{F_b} = \frac{3}{197.91} \approx 0.0152 \).
06

Interpret the Result

The average gravitational force the star exerts on planet c is approximately \(0.0152\) times the force it exerts on planet b.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton's Law of Universal Gravitation
At the heart of understanding gravitational force is Newton's Law of Universal Gravitation. This fundamental principle describes how masses interact through gravity. According to the law, the force (\( F \)) between two objects is dictated by the formula:\[F = \frac{G \, m_1 \, m_2}{r^2}\]Where:- \( F \) is the gravitational force,- \( G \) is the gravitational constant, a universal value,- \( m_1 \) and \( m_2 \) are the masses of the two interacting bodies, and- \( r \) is the distance between the centers of these two masses.
The beauty of this law lies in its simplicity, demonstrating that the gravitational force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between them.
This means that if you increase the mass of either object, the gravitational attraction will increase, and if you double the distance between them, the force will be reduced to a quarter of its original value. In our solar system scenario with Upsilon Andromedae's planets, this law helps us compare how the star pulls on planet b versus planet c.
Astronomical Unit (A.U.)
The concept of an Astronomical Unit (A.U.) is essential in space sciences, serving as a convenient measure of distance within our solar system. Defined as the average distance from the Earth to the Sun, this unit helps us visualize and compare celestial distances without delving into cumbersome large numbers.
One A.U. is approximately 149.6 million kilometers (about 93 million miles). This distance becomes a yardstick for astronomers to express and calculate orbital distances in our cosmic neighborhood.
  • Planet b in our given scenario orbits its star at 0.059 A.U.
  • Planet c, on the other hand, orbits at 0.83 A.U. from its star.
These distances can be directly used in formulas like Newton's gravitational law, where they would replace \( r \) to find out how far the planets are from their star.
Using A.U. makes these calculations more palatable and understandable, especially when comparing planets within a solar system.
Mass Ratio
In astronomical terms, knowing the actual masses of celestial bodies can be challenging, especially half a light year away! Instead, astronomers often rely on mass ratios to understand the relative masses of planets or stars when precise mass isn't available.
In our example, the problem provides the mass of planet c as \(3.0\) times that of planet b, indicating a mass ratio of \(3:1\). This ratio is crucial because it allows us to compare the gravitational forces exerted by the star on each planet without knowing their absolute masses.
Since gravitational force is directly proportional to the mass of the objects involved, using the mass ratio allows us to replace one mass with a multiple of the other in our equations, simplifying complex astronomical evaluations.
  • For instance, if planet c is three times more massive than planet b, the gravitational force on planet c by the star will be generally stronger, assuming similar distances.
In conclusion, mass ratios provide a simplified yet powerful means to understand gravitational physics in stars and planets, especially when absolute measurements are out of reach.

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Most popular questions from this chapter

(a) Suppose a rotating spherical body such as a planet has a radius \(r\) and a uniform density \(\rho\), and the time required for one rotation is \(T\). At the surface of the planet, the apparent acceleration of a falling object is reduced by the acceleration of the ground out from under it. Derive an equation for the apparent acceleration of gravity, \(g\), at the equator in terms of \(r, \rho, T\), and \(G .\) (b) Applying your equation from a, by what fraction is your apparent weight reduced at the equator compared to the poles, due to the Earth's rotation? (c) Using your equation from a, derive an equation giving the value of \(T\) for which the apparent acceleration of gravity becomes zero, i.e., objects can spontaneously drift off the surface of the planet. Show that \(T\) only depends on \(\rho\), and not on \(r\). (d) Applying your equation from \(\mathrm{c}\), how long would a day have to be in order to reduce the apparent weight of objects at the equator of the Earth to zero? [Answer: \(1.4\) hours \(]\) (e) Astronomers have discovered objects they called pulsars, which emit bursts of radiation at regular intervals of less than a second. If a pulsar is to be interpreted as a rotating sphere beaming out a natural "searchlight" that sweeps past the earth with each rotation, use your equation from \(\mathrm{c}\) to show that its density would have to be much greater than that of ordinary matter. (f) Astrophysicists predicted decades ago that certain stars that used up their sources of energy could collapse, forming a ball of neutrons with the fantastic density of \(\sim 10^{17} \mathrm{~kg} / \mathrm{m}^{3} .\) If this is what pulsars really are, use your equation from \(c\) to explain why no pulsar has ever been observed that flashes with a period of less than \(1 \mathrm{~ms}\) or so.

The International Space Station orbits at an average altitude of about \(370 \mathrm{~km}\) above sea level. Compute the value of \(g\) at that altitude.

The figure shows an image from the Galileo space probe taken during its August 1993 flyby of the asteroid Ida. Astronomers were surprised when Galileo detected a smaller object orbiting Ida. This smaller object, the only known satellite of an asteroid in our solar system, was christened Dactyl, after the mythical creatures who lived on Mount Ida, and who protected the infant Zeus. For scale, Ida is about the size and shape of Orange County, and Dactyl the size of a college campus. Galileo was unfortunately unable to measure the time, \(T\), required for Dactyl to orbit Ida. If it had, astronomers would have been able to make the first accurate determination of the mass and density of an asteroid. Find an equation for the density, \(\rho\), of Ida in terms of Ida's known volume, \(V\), the known radius, \(r\), of Dactyl's orbit, and the lamentably unknown variable \(T\). (This is the same technique that was used successfully for determining the masses and densities of the planets that have moons.)

(a) A certain vile alien gangster lives on the surface of an asteroid, where his weight is \(0.20 \mathrm{~N}\). He decides he needs to lose weight without reducing his consumption of princesses, so he's going to move to a different asteroid where his weight will be \(0.10 \mathrm{~N}\). The real estate agent's database has asteroids listed by mass, however, not by surface gravity. Assuming that all asteroids are spherical and have the same density, how should the mass of his new asteroid compare with that of his old one? (b) Jupiter's mass is 318 times the Earth's, and its gravity is about twice Earth's. Is this consistent with the results of part a? If not, how do you explain the discrepancy?

On an airless body such as the moon, there is no atmospheric friction, so it should be possible for a satellite to orbit at a very low altitude, just high enough to keep from hitting the mountains. (a) Suppose that such a body is a smooth sphere of uniform density \(\rho\) and radius \(r .\) Find the velocity required for a ground-skimming orbit. (b) A typical asteroid has a density of about \(2 \mathrm{~g} / \mathrm{cm}^{3}\), i.e., twice that of water. (This is a lot lower than the density of the earth's crust, probably indicating that the low gravity is not enough to compact the material very tightly, leaving lots of empty space inside.) Suppose that it is possible for an astronaut in a spacesuit to jump at \(2 \mathrm{~m} / \mathrm{s}\). Find the radius of the largest asteroid on which it would be possible to jump into a ground-skimming orbit.
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