The Kepler problem

Kepler's third law

Problem:

Use Kepler's third law to calculate the mass of the sun, assuming that the orbit of the earth around the sun is circular, with radius r = 1.5*108 km.

Solution:

Problem:

Haley's Comet approaches the sun to within 0.570 A.U., and its orbital period is 75.6 years.  (A.U. is the abbreviation for astronomical units, where 1 A.U. = 1.5*1011m is the mean Earth-Sun distance.)  How far from the sun will Haley's comet travel before it starts its return journey?

Solution:

Problem:

The time of revolution of planet Jupiter around the Sun is TJ ~ 12 years.  What is the distance between Jupiter and the Sun if the Earth-Sun distance is 150*106 km?  Assume that the orbits are circular.

Solution:

Problem:

A satellite of mass 200 kg is placed in Earth orbit at a height of 200 km above the surface.  It has a circular orbit.  What is the period of the satellite?
Given:  REarth = 6.4*106 m,  MEarth = 5.98*1024 kg

Solution:

Problem:

Io, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of 4.22*105 km.  From these data, determine the mass of Jupiter.

Solution:

Problem:

The source of the first gravitational wave event observed by the LIGO collaboration in 2015 has been interpreted as the merger of two black holes in a binary system, each with a mass of roughly 35 solar masses (implying a radius for the event horizon of about 100 km for each, if assumed spherical), where a solar mass is 1.989*1030 kg.  A full understanding requires general relativity, but assume Newtonian mechanics and Newtonian gravity as a first approximation for the orbital motion.  At the peak amplitude of the detected gravitational wave, its measured frequency indicated that the two black holes were revolving around the center of mass about 75 times per second. 
What was the approximate separation of the centers for the two black holes at this point in the merger event?

Solution:

Problem:

A particle of mass m is released a distance b from a fixed origin of force that attracts the particle according to the inverse square law F(x) = –k/x2.   Find the time required for the particle to reach the origin.  Use this result to show that, if the Earth were suddenly stopped in its orbit, it would take approximately 65 days for it to collide with the Sun.   Assume that the Sun is as a fixed point mass and Earth’s orbit is circular.

Solution:


Energy and angular momentum conservation

Problem:

A comet of mass m approaches the solar system with a velocity v0, and if it had not been attracted towards the sun, it would have missed the sun by a distance d.  Calculate its minimum distance z from the sun as it passes through the solar system.  Make and state any reasonable simplifying assumptions.

Solution:

Problem:

Assume for this problem Earth is a sphere of radius R and mass M.  An object of mass m enters Earth's atmosphere at distance R' > R from Earth's center with speed v at angle α from the radial direction.  Ignoring any friction or air resistance, what at angle β (from the radial direction) will it hit Earth's surface?

image

Solution:

Problem:

A uniform spherical planet of radius a revolves around the sun in a circular orbit of radius r0 and angular velocity ω0.  It rotates about its axis with angular velocity Ω0 (period T0) normal to the plane of the orbit.  Due to tides raised on the planet by the sun, its angular velocity of rotation is decreasing.  Find an expression which gives the orbital radius r as a function of the angular velocity Ω of rotation and the parameters r0 and T0 at any later or earlier time.

Solution:


Kepler orbits

Problem:

For a satellite orbiting a planet, transfer between coplanar circular orbits can be affected by an elliptic orbit with perigee and apogee distances equal to the radii of the respective circles as shown in the Figure below.  This ellipse is known as Hohmann transfer orbit.
Assume a satellite is orbiting in a circular orbit of radius rp with circular orbit speed vc.  It is to be transferred into a circular orbit with radius ra.

image
(a)  Find EH/Ec, the ratio of the total energies of the satellite in the Hohmann and the initial circular orbit.
(b)  Determine the equation for the ratio of the speeds v/vc as a function of the ratio of the distances r/rp from the focus for the Hohman transfer orbit.  Evaluate v/vc at r = rp.

Solution:

Problem:

This question is about an elliptical “transfer orbit” from an inner circular orbit A to an outer circular orbit B.  The transfer starts at point P and is completed at point Q.  The transfer orbit is an ellipse which is tangent to A at point P and tangent to C at point Q.
(a)  Derive a formula for the relationship between v and r for circular orbits.  Is the speed in orbit C greater or less than the speed in orbit A?
(b)  For the transfer, should the satellite speed up or slow down at point P?
(c)  For the transfer, should the satellite speed up or slow down at point Q?

image

Solution:

Problem:

In outer space, two small balls of equal unknown masses m and charges +q and –q are held at rest a distance d0 apart.  Then the balls are simultaneously launched with equal speeds v0 in opposite directions that are perpendicular to the line connecting the balls.  During the subsequent motion of the balls orbit  each other, and their minimum speed is v.  Find the mass m of each ball.

image

Solution:

Problem:

The escape speed from the surface of a slowly rotating airless spherical planet is vesc. What is the minimum initial speed of a projectile launched from a pole that allows it to land on the equator?

Note:
For Kepler orbits:
p/r = 1 + e cos(φ - φ0),
p = L2/mα,   e = (1 + 2EL2/mα2)½,  L = angular momentum

Solution:

Problem:

A particle of mass m is moving in a central potential of the form V(r) = -α/r.
(a)  What is the total energy of the particle if it is moving in a circular orbit of radius R?  What is its speed?
(b)  What is the energy of the particle if it is moving in an elliptical orbit of semi-major axis R?  What is its speed at r = rmin, the perigee of its orbit?

Solution:

Problem:

Find the maximum time a comet (C) of mass m following a parabolic trajectory around the Sun (S) can spend within the orbit of the Earth (E).  Assume that the Earth's orbit is circular and in the same plane as that of the comet.

Solution: