Astronomy Olympiad

Problems on Gravity & Celestial Mechanics

Newton’s Definition of Force:

Here m is the mass of the body being acted upon by the force and a is the acceleration induced in the body by the force.

Newton’s Law of Gravity:

M & m are the masses of the two interacting bodies and R, the distance between their centers. G is the universal constant of gravitation with the value ~ 6.67 x 10^-11 N-m²/kg²

Newton’s First Theorem – Body inside a spherical distribution of mass:

A body inside a shell of matter experiences no net force from that shell. Imagine the body to be arbitrarily away from the center. Draw two cones with the same solid angle and opposite to each other from the body. The mass contained in either cone may be different, but the distances are correspondingly different too. Thus by Newton’s definition of gravity, the force due to either cone on the body is the same. This can be generalized to a sphere as well as taking the cones in all possible directions from the body, you prove the theorem.

Newton’s Second Theorem (Keplarian situation) – Body outside a closed spherical shell of matter:

The gravitational force on a body that lies outside a closed spherical shell of matter is the same as it would be if all the shell’s matter were concentrated at one point at the center. A rigorous proof can be found in advanced mechanics textbooks. However the simple reasoning seems to be as follows: The vector describing the force exerted on the body by any small section of the shell which is away from the center can be decomposed into a component towards the center of the shell and another tangential to the shell. The tangential components of all vectors drawn due to all sections of the shell will cancel off due to symmetry (only when such symmetry exists) and the radial components will only contribute to gravity on the body.

Acceleration due to Gravity:

Newton’s definition of force when applied to gravitational force, gives the acceleration due to gravity at distance R from the center (above the surface) of the body (planet or star….) as above. This works for bodies near the surface. For bodies at a significant height h, use the later formula where R_pl is the planet’s average radius.

Weight & Weightlessness:

An object feels its weight due to the reaction given by the surface on which it stands (The first equation above). However, when immersed to a volume V inside a fluid with density r, its feels a reduced weight due to the buoyancy of the fluid (upthrust), denoted by the bracket in the second equation. An object on a moving platform will feel the reaction of the platform subdued due to its own acceleration. Thus if the platform is moving up while the object feels a gravity downwards, a_object ~ -g, the object feels a greater reaction force, hence greater weight. Now if the platform is falling with the same acceleration as that due to gravity, i.e. free fall, the object would get no reaction at all, and hence feel weightless. (e.g. astronauts in spacecraft accelerated towards the Earth due to centripetal force, by an amount equal to that of gravity)

Keplar’s Law of Ellipses:

Bodies attracted by gravity towards each other follow an elliptical path in space. The ellipse may be perturbed slightly due to other nearby objects or due to general relativistic effects.

Keplar’s Law of Areas:

Speed increases at perihelion and decreases at aphelion for the planet. Velocity of a body in orbit can hence be given as

Keplar’s Law of Periods:

M is the mass of the central body and R, the distance of the orbiting body from the central body. The mass of the orbiting body has no effect. If T is taken in years & R in AU, then 4p² ~ GM₀. Hence for the solar system calculations T² = R³.

Critical Velocity in Orbit:

As gravity provides the centripetal force for the orbiting body, v_crit depends upon the central body (mass M) and the radius of the orbit R. It is measured tangential to the orbit at the instantaneous position of the orbiting body.

Energy of Orbit:

K & U are the kinetic and gravitational potential energies respectively.  The total energy E depends upon the gravitational interaction between the two bodies, since it is the gravity which binds the bodies together in a system. The negative sign implies that the orbiting body is bound to the central (massive) body and would require that much energy to be provided for its release.

Escape Velocity:

When the body is provided energy enough to overcome the binding energy, i.e. the gravitational potential, the body can be released from the planet’s grip. The required energy translates as giving a normal (along the radius) velocity v_esc to the body.

Virial Theorem:

Where K & W are the kinetic and potential energies respectively and P & V are the pressure and volume of the system. Virial system is an important statement about the energy balance of a system. If the system is isolated and closed, it cannot do any work on the surroundings and hence 2K + W = 0. Consider a star being formed by gathering material from infinity to a sphere. Initially K = 0, P = 0 and volume and consequently potential energy is high. As the star begins to accrete K increases and W decreases until equilibrium is reached. At this point, i.e. half the potential energy of the material is radiated away in forming the star.

Jean’s Criterion:

A cloud of N particles each with mass mp will collapse when its potential energy will exceed its kinetic energy and hence the Jean’s mass/density condition for formation of a star. A 10,000 solar mass cloud at 30oK will have a density of 10,000 molecules/m³ at collapse. A solar mass cloud will require 100 times more density to collapse. This indicates that star formation must be a multi-step process. Larger clouds condense to form smaller blobs and then proceed to make stars.

Tidal Force:

Tidal force due to a body (mass M) on a smaller body (mass m) is the difference in force on two ends of an elongated body with radius r.

Differential Centrifugal Force:

In general a body would spin around the central body and also around itself and thus the net centrifugal force it experiences will be due to both spins.

Roche limit:

When a solid body is moving in the vicinity of a massive central body such as a planet, if it gets within a certain distance of the planet’s center, it disintegrates. This distance is known as the Roche limit. The object disintegrates if the tidal & differential centrifugal forces together overpower the gravitational force (between the end points) binding the object together.

This limit was originally deduced by Roche for a liquid satellite as.

Q1)            A star with mass 10³² kg. has a planet going around it in a circular orbit. If the planet should be at 1 A.U. from the star then what would be its period of revolution. Hence calculate its average velocity in the orbit.

Q2)            Using Newton’s Law of gravity, calculate the gravitational force between the following pairs: Sun-Earth, Sun-Mars, Earth-Moon, Mars-Phobos, Saturn-Titan, Earth-Ceres (an asteroid in the Jovian belt)

Q3)            Compare using Newton’s law of gravity, the force between the Sun and the Earth to the force between you and the Earth. Which is greater?

Q4)            Using Keplar’s third law make a table of distances (r) and hence (T) for all the planets in the solar system.

Q5)            Does the moon follow Keplar’s third law? Check by calculating the period of revolution of the moon around the earth, accurately and compare the same with the actual observed period.

Q6)            A planet is orbiting around a star of mass 3 M_o in an orbit of radius 1 AU. What should be the velocity of revolution of the planet?

Q7)            A satellite has to be launched into an orbit around Mars to observe its climatic changes and hence guide further missions. If the satellite is to have an orbit of 10,000 km from the Martian surface, calculate its velocity.

Q8)            The space shuttle re-entering the Earth’s atmosphere encounters a piece of debris (scrap metal, nuts & bolts, etc.) from some previous spacecraft. Assume that the debris is traveling at a speed of 1 km/s w.r.t. to the shuttle and has a mass of about 1 kg. What will be its energy when it hits the shuttle? Also, calculate the momentum of the piece when it hits.

Q9)            How much energy will we have to supply Phobos to get it to escape from the gravitational pull of Mars?

Q10)        How much energy does a rocket require to lift-off from the lunar surface to go far enough to escape Moon’s gravity?

Q11)        What is the minimum energy required to reach the Moon starting from the Earth’s surface?

Q12)        Calculate the energy that would be required to transfer a satellite from an orbit 1000 km from the Earth’s surface to 10000 km (again from the surface). If this energy is to be supplied as one big rocket thrust (tangential to the orbit, i.e. as addition to v_crit), calculate the acceleration required. Hence or otherwise compute the velocity that the satellite will acquire when in the higher orbit.

Q13)        A planet with mass twice that of the Earth is orbiting the sun at 4 AU. Compare the period of this planet with that of the Earth. (To compare is to divide one quantity by the other and hence comment as to which one is greater and by what factor)

Q14)        A planet has thrice the mass of the Earth and only twice the size (i.e. radius). What is the surface gravity (g) on this planet? Compare it with Earth’s surface gravitational acceleration.

Q15)        If the Earth were to orbit a star 10 times the Sun’s mass and yet were to retain its period of revolution, i.e. T ~ 1 yr, then at what distance form the star should the Earth be?

Q16)        If the Earth were to suddenly reduce in size to half its size today, then what would be the surface gravity? Also imagine a particle fixed on the Earth’s surface. This particle would take 24 hrs to go around the Earth’s center under present conditions. How long will it take to make one revolution around the Earth, after the reduction in size?

Q17)        If Mars collides with a huge asteroid and hence loses one-fourth the mass in the way of large amount of material flowing out into space, what exactly, would happen to its orbit around the Sun?

Q18)        If we assume that 0.01% of the Sun’s mass was spewed out from the Laplacian Nebula when the Sun was being born and that it was trapped by the gravity of the protostar and hence remained in orbit around the Sun forming a cloudy envelope around it. If all this material were to be concentrated into one object orbiting the sun, at a distance of say 500 AU, what would you expect its period to be? (Hint: As the material has mass comparable to the Sun, use Keplar’s law with a (M + m) instead of just the Solar mass in the denominator)

Q19)        What would be the weight felt by a body when immersed in mercury, (density ~ 13.6 gm/cm³) if it has a mass of 10 kg and a volume of 1000 cc?

Q20)        Imagine that you are sitting in a rocket taking off from the Earth. The rocket is moving with an acceleration of 5g, i.e. five times the value of g at the Earth’s surface. What would be your weight if your mass is 80 kg?

Q21)        What would be your weight if you were to stand on Saturn’s surface? (Assume Saturn is solid)

Q22)        If there were life on Pluto, and a plutonian weighing 100N on Pluto’s surface were to come to the Earth, how heavy would he feel?