Unit 2 Two Body Problem

MAE 4263 Spaceflight Mechanics Dr. Brian Kaplinger

UNIT 2 : The 2-Body Problem

Historical Background We owe much to Sir Isaac Newton

Born Christmas Day, 1642.

Co-invented calculus, discovered inverse square law for gravity, proved many disputed scientific ideas of the day.

Demonstrated that the Earth should be oblate the evidence that later proved it is why mathematics won out over philosophy as the domain for physics. Take that Descartes!

What was Newtons greatest accomplishment?

Historical Background Tycho Brahe was an aristocrat, meticulous,

mechanical genius.

Gathered detailed, accurate recordings of positions for several objects including comets and very detailed data on Mars.

Met Johannes Kepler, his assistant, 18 months before he died.

Brahe was jealous with his data, but impressed by Keplers mathematical skill. Kepler retained the data after Brahe died.

Historical Background Kepler was a poor, sickly man. He was trained as

a pure mathematician.

He was patient but unskilled in gathering experimental observations. Likely would not have been as successful without Brahes data.

Following astronomical ideas published by Copernicus and using Brahes data on Mars position, published 3 laws of orbital motion of the planets.

Laws are descriptions based on observation only.

Keplers Laws 1: Orbit of each planet is an

ellipse, with the Sun located at one focus (1609).

Images from Hyperphysics



2: The line joining a planet to the Sun sweeps out equal area in equal units of time (1609).




2: The line joining a planet to the Sun sweeps out equal area in equal units of time (1609).

3: The square of a planets orbital period is proportional to the cube of its mean distance from the Sun (1619).


Theory of Gravitation Newton developed his Theory

of Gravitation, including the inverse square law from Unit 1, in 1666.

Due to discrepancies with his calculations of the Moons orbit, he did not publish for 20 years.

One day, over drinks, Edmund Halley and Hooke discussed possibility of gravity acting like magnetism made a bet that it would prove Keplers Laws.

Theory of Gravitation Halley asked Newton to weigh

in, who claimed to already have a proof.

Halley convinced Newton that this might be important and that he should publish it.

Newton published Principia in 1687.

Contained Newtons 3 Laws, Theory of Gravitation, hydrostatics, compressible fluids, harmonic motion.

Rocket Launches and Staging Covered in detail in another

course.

Staging motivated by not taking with you what you do not have to.

STS-82 in 1997 Space Shuttle Discovery launched to repair Hubble circular orbit at 580 km altitude.

Orbital mass was 106,000kg or 6% of launch mass

83% of thrust generated by SRBs (69% mass) jettisoned within 2 minutes.

What is important for satellite motion? Source Acceleration (gs)

Earth Gravity 0.89

Oblateness of Earth 10^-3

Solar Gravity 6 x 10^-4

Lunar Gravity 3 x 10^-6

Planets < 3 x 10^-8

What is important for satellite motion? Source Acceleration (gs)

Earth Gravity 0.89

Oblateness of Earth 10^-3

Solar Gravity 6 x 10^-4

Lunar Gravity 3 x 10^-6

Planets < 3 x 10^-8

Provides an empirical argument for study of the 2BP, since a single force results in most of the observed motion by several orders of magnitude.

Important Classes of Orbits Low Earth Orbit (LEO)

between 150 km and 2000 km altitude. Above atmosphere but well below Van Allen Belts (2400 km).

Synchronous Orbit (SSO-special case) has a resonant period with that of the Earths surface. Will pass over locations at the same points in the day (GPS).

Geostationary Equatorial Orbit (GEO) remains fixed over a point on the surface.

FIGURE 2.17 The view from GEO.

Important Classes of Orbits

Where is GEO? Satellite in GEO must have an

orbital period that rotates once for every revolution of the Earth compared to the fixed stars in the sky.

A synodic day is the amount of time it takes the Sun to return to the same spot in the sky (defined to be 24 hrs).

A sidereal day is the amount of time it takes the Earth to make one rotation in inertial space.

Why are these not the same?

Satellites in GEO (Wikipedia).

The 2BP

FIGURE 2.1 (a) Two masses located in an inertial frame.

(b) Free-Body diagrams.

Recap Derivation 2BP Equation of Motion

Nonlinear Differential Equation



Conservation of Energy (Consequence of EOM)




Conservation of Angular Momentum (EOM)




Conservation of Angular Momentum (EOM)

Solution is the Orbit Equation (aka Trajectory Equation)

Orbit Equation

Velocity Components

Velocity Components

Velocity Components Summary:

Velocity Components Overall Orbital Velocity:

Know from energy equation that it should be large as we are close to body and small as we are far away.



Matches our experience with non-orbital ballistic trajectories.

Also needed to match Keplers 2nd Law.



Matches our experience with non-orbital ballistic trajectories.

Also needed to match Keplers 2nd Law.

Velocity Component calculations are consistent with this expectation:

Unit 2 Two Body Problem

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