Two body problem

Problems in celestial mechanics

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Celestial mechanics One of the most basic approximate model references in. This paper studies the dynamics of two celestial bodies that can be regarded as particles under the gravitational action between them. The two body problem is the first approximate result of the real motion of various celestial bodies, the theoretical basis for studying the precise motion of celestial bodies, and also a basic problem in celestial mechanics. It is the only one that can be thoroughly solved so far Celestial mechanics Therefore, it is of great significance. It has been proved that under the action of gravity Equation of motion It can be solved strictly.

catalog

one definition
two Calculation of two body collision time
three Proof of formula
four mathematical problem

definition

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The motion of two particles meeting the following conditions: ① the gravity of other objects is not considered; ② The interaction force between them is along the line of two points, and the magnitude of the force is a function of the distance between the two points. The two body problem can be reduced to an equivalent single body problem. The motion of double stars in astromechanics, planets and their satellites, stars and planets, and the vibration of diatomic molecules in physics all belong to or approximately belong to two body problems. The mass of the sun is more than 700 times the total mass of other stars in the solar system, so the sun is the central object of the solar system. Each planet approximately forms a two body system with the sun, and the gravitational influence of other planets on the planet is only a small perturbation of its orbit around the sun. Therefore, the research of celestial mechanics is based on the solution of the two body problem.

Calculation of two body collision time

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(It can be solved by using Kepler's third law and introducing the reduced mass when the linear motion is regarded as the degeneration of the ellipse)

T^2=4π^2a^3/GM

Consider the straight line as the degeneration of the ellipse, and the semi major axis is a=1/2 of the distance

The time required for collision is T/2

Proof of formula

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Question: The distance between two relatively stationary objects (which can be regarded as the physics of particles) is r0, and the masses are m1 and m2 respectively. The time required for collision under the action of gravity alone.

answer:

Taking the direction of m2 relative to m1 as the positive direction, these two objects are regarded as a system, because they are not subject to external forces, so these two objects are in motion Momentum conservation , at any time, there are

m1v1+m2v2=0 (1)

From (1)

v1=-m2/m1*v2 (2)

It can be seen from (2) that at any time, the ratio of v1 and v2 is equal, so the two objects are displacement The ratio is equal to the ratio of v1 and v2:

s1/s2=v1/v2 (3)

s1+(-s2)=r0 (4)

The displacement s1 of m1 is obtained from (2), (3) and (4)

s1=m2/(m1+m2)*r0 (5)

Assuming that two physics collide with point P, we can get that at any time, the distance s11 and r between m1 and P meet the following requirements:

s11=m2/(m2+m1)*r (6)

When two objects are infinitely far apart, their Gravitational potential energy Is 0, then the gravitational potential energy is

E=∫(Gm1m2/r^2,+∞,r0)dr=-Gm1m2/r (7)

Therefore, at any moment, the sum of kinetic energy of m1 and m2 Ek meets:

Ek=-Gm1m2/r0-(-Gm1m2/r)=1/2m1v1^2+1/2m2v2^2 (8)

The instantaneous speed m1 at any time is obtained from (1), (6) and (8)

v11=(2Gm2^3/(s11(m1+m2)^2)-2Gm2^2/(r0(m1+m2)))^(1/2) (9)