Gravity of Extended Bodies

Problems

practice

Almost all stars will die as hot, dense lumps, but some of them decide to get fat and cold for awhile first. In 5 billion years our sun will expand into a red giant (red because it will be much cooler) swallowing up Mercury, Venus, Earth, and (possibly) Mars. A few hundred million years after that, it will begin shrinking until it becomes a white dwarf (white because it will be much, much hotter) about as small as the Earth but still retaining much of it mass.

Pairs of stars in orbit around one another are common in our galaxy — more common than lone stars like our sun. The stars never have the same mass and one will always age faster than the other. The diagram above shows a binary system where one star is a red giant and the other is a white dwarf. If the two stars are close enough, the white dwarf may begin "feeding" on the red giant. The matter torn off the surface of the red giant spirals into the white dwarf forming what is known as an accretion disk. Determine the separation between the two stars shown in this diagram. Let…

r =	separation between the stars
a =	radius of red giant
b =	radius of white dwarf
m_a =	mass of red giant
m_b =	mass of white dwarf

Ignore any centrifugal effects and state your answer using the given symbols only.

Suppose that the Earth was an infinite flat slab of thickness t with the same mean density as the Earth. Calculate t in order that this infinite flat Earth has the same acceleration due to gravity on its surface as is found on the actual spherical Earth.
1. Solve it once the easy way, using Gauss's law for gravity.
  ∯ g · dA = −4πGm
2. Solve it once the hard way, using Newton's law of universal gravitation for continuous distributions of matter.
  
  g(r) = − G ⌠⌠⌠
  ⌡⌡⌡ r̂ dm
  
  r²
3. How does this number compare to the radius of the actual spherical Earth?
Dark matter
1. The orbital speed of the planets decreases with distance from the Sun. Why does this happen? Derive a formula that shows the relationship.
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2. The orbital speed of the stars remains roughly constant with distance from the center of the Milky Way. (This is true for other galaxies as well.) What does this tell us about the distribution of mass in galaxies? Derive a formula that shows the relationship.
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3. Calculate the mass of the Milky Way given an orbital speed of 230 km/s and a radius of 50,000 light years. Give your answer in solar masses (m_☉ = 1.99 × 10³⁰ kg) and compare it to the number of stars in the Milky Way (~10¹¹).
4. Dwarf galaxies, star clusters, and gas clouds beyond the edge of the visible galaxy have nearly the same orbital speed as the stars within visible galaxy. There is evidence that rotational speeds remain roughly constant at 220 km/s out to distances of 300,000 light years or six times the radius of the Milky Way. What is so amazing about this observation and what does it imply?
The word nebula (plural nebulae) means cloud in latin. In astronomy, a nebula is a diffuse collection gas and dust that looks something like a cloud. Nebulae are larger than stars, but smaller than galaxies — on the order of 10-1000 solar systems in diameter. A few representative images are shown below.

Helix Nebula, STScI

Rosette Nebula, source unknown

Horsehead Nebula, NOAO

A simplified model of a nebula is a spherical collection of matter whose density varies linearly from a maximum at its center to zero at its "surface". Determine the following quantities both inside and outside such a simplified nebula in terms of its radius R, the distance from the center r, the density at the center ρ₀, and fundamental constants…

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1. density
2. gravitational field strength
3. gravitational potential energy per unit mass

numerical

Prolate spheroid
The image below shows a typical alignment of the three most popular objects in the solar system: the Sun, Earth, and Moon (not to scale). The letters around the Earth represent various locations on its surface.

A =	point closest to the Moon
B =	point farthest from the Moon
C =	point closest to the Sun
D =	point farthest from the Sun

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Complete the following table.
Use the results of your calculations to explain the effects that the Moon and Sun have on the shape of the Earth.

quantity	Moon	Sun
mass (kg)	7.348 × 10²²	1.989 × 10³⁰
distance from Earth (m)	3.844 × 10⁸	1.496 × 10¹¹
gravitational field (N/kg) from object
on Earth, at Earth's center (g)
on Earth, nearest to object (g_near)
on Earth, farthest from object (g_far)
ratio of difference in field at opposite sides of the Earth to field at center (1:x)

Oblate spheroid
Compare these images of Earth and Jupiter (not to scale).

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Complete the following table.
Use the results of your calculations to explain the effect that rotation has on the shapes of Earth and Jupiter.

quantity	Earth	Jupiter
mass (kg)	5.9742 × 10²⁴	1.8988 × 10²⁷
period of rotation (h)	23.935	9.9250
radius (m)
polar	6,356,750	66,854,000
equatorial	6,378,140	71,492,000
oblateness: ratio of difference in radii to equatorial radius (1:x)
rotational speed (m/s)
at either pole
on the equator
centrifugal acceleration (m/s²)
at either pole
on the equator
gravitational acceleration (m/s²)
at either pole
on the equator
ratio of centrifugal to gravitational accelerations at the equator (1:x)

investigative

Determine the apparent acceleration due to gravity at your current location on Earth. Here is the general procedure for doing this.

Start by determining your latitude and altitude. That information is probably available somewhere on the internet or you could measure it yourself with a GPS enabled device.

The Earth is an oblate spheroid. Use the following fancy formula to calculate your distance (r) from the center of the Earth…

r = h + √	(a² cos φ)² + (b² sin φ)²
	(a cos φ)² + (b sin φ)²

where…

h =	your altitude
φ =	your latitude
a =	earth's equatorial radius (6,378,137 m)
b =	earth's polar radius (6,356,752 m)

Compute the magnitude of the gravitational acceleration at your location.
Compute your distance from the Earth's axis; that is, the component of r parallel to the plane of the equator.
Compute the magnitude of the centrifugal acceleration at your location.
Combine the gravitational and centrifugal components to determine the apparent acceleration due to gravity (g′) at your location. Determine both the…
1. magnitude and
2. direction (relative to r, the vector that points directly toward the center of the Earth)
(Draw yourself a little picture before you try to answer this part.)
Compare your results to the values often found in textbooks and reference tables.
1. 9.8 m/s² (this value with its two significant digits of accuracy should agree with your results)
2. 9.798 m/s² (an average value calculated over the surface of the Earth)
3. 9.80665 m/s² (the defined unit known by the short name of standard gravity)

algebraic

Determine the rotation curve for…
1. a spherical galaxy with a uniform mass distribution
2. a thin, disk-shaped galaxy with a uniform mass distribution

calculus

Given the nebula in practice problem 4, determine the…
1. location
2. value
of the maximum gravitational field in terms of its radius R, the density at the center ρ₀, and fundamental constants.
Determine the gravitational field and gravitational potential, inside and outside the following mass distributions…
1. a sphere of mass M and uniform density ρ.
2. a simplified model of the Earth, whose total mass M is evenly split between the core and the mantle — ½M for each part. (Assume the crust, oceans, and atmosphere make a negligible contribution to Earth's mass.)
3. a simplified model of the Earth consisting of a core with density 2ρ and mantle with density ρ. (Assume the crust, oceans, and atmosphere make a negligible contribution to the Earth's mass.)
4. a galaxy (including dark matter halo) with a density that decreases according to the function ρ = ρ₀/r², where r is the distance from the galactic center.

g(r) = − G	⌠⌠⌠ ⌡⌡⌡	r̂ dm
		r²