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Astronomy/importance of Keplers second law

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Question
Hello
how much Kepler's second law is important in space science? how are you using it in launching space crafts and Shuttle?
thanks

Answer
Kepler's 2nd law is extremely important in the discipline of celestial mechanics which is concerned with computing the future positions of planets, for example. Given this, it is also critically important in astrodynamics - by which we compute orbits and-or trajectories whereby spacecraft are sent to other planets, e.g. Mars, Venus.

The law's relevance, then, is in the ability to find the energy of orbits as well as the velocities, say at aphelion or perihelion of a planet's orbit.  Kepler's 2nd law holds at every point of an orbit (equal areas swept out in equal intervals of time) so we also have:

r^2 (2π/T) = h

where 'h' is a constant ('specific relative angular momentum') which is twice the rate of area description (i.e. by the radius vector). Thus, if the radius vector is r1, then h = 2A1, when A1 = π(r1)2. Hence, at aphelion and perihelion only we have:

V = h/r

For the perihelion velocity we have:

V_P = h/ a(1 - e)

where a is the semi-major axis, and e is the eccentricity.

For the velocity at aphelion:

V_A = h/ a (1 + e)

Then the ratio of velocities is:

(V_P/V_A) = (1 + e)/ (1 - e)

Which can be extremely important in planning the trajectory of a spacecraft where its path is contingent on that velocity difference.


The correct energy equation can then be written:

V^2 - u /r = C

where C is an energy integration constant. Here, u = G (m1 + m2)

With further working, one can obtain  the "vis viva equation", one of the most important in celestial mechanics, viz.


V^2 = u (2/r - 1/a)  

Exactly HOW the 2nd law and the preceding are used in launching spacecraft is beyond the scope of this response, but if you are really interested you can find out by obtaining any good text on astrodynamics. (Google 'astrodynamics texts')


Otherwise, you can go to this MIT online course for astrodynmics:

http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-346-astrodynamics-fal

(A video lecture is included)

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Philip Stahl

Expertise

I have more than forty years of experience in Astronomy, specifically solar and space physics. My specialties include the physics of solar flares, sunspots, including their effects on Earth and statistics pertaining to sunspot morphology and flare geo-effectiveness.

Experience

Astronomy: Worked at university observatory in college, doing astrographic measurements. Developed first ever astronomy curriculum for secondary schools in Caribbean. Gave workshops in astrophysics and astronomical measurements at Harry Bayley Observatory, Barbados. M.Phil. degree in Physics/Solar Physics and more than twenty years as researcher with discovery of SID flares. Developed of first ever consistent magnetic arcade model for solar flares incorporating energy dissipation and accumulation. Develop first ever loop solar flare model using double layers and incorporating cavity resonators.

Organizations
American Astronomical Society (Solar Physics and Dynamical Astronomy divisions), American Mathematical Society, American Geophysical Union.

Publications
Solar Physics (journal), The Journal of the Royal Astronomical Society of Canada, The Proceedings of the Meudon Solar Flare Workshop (1986), The Proceedings of the Caribbean Physics Conference (1985). Books: 'Selected Analyses in Solar Flare Plasma Dynamics', 'Physics Notes for Advanced Level'. 'Astronomy and Astrophysics: Notes, Problems and Solutions'.

Education/Credentials
B.A. Astronomy, M. Phil. Physics

Awards and Honors
American Astronomical Society Studentship Award (1984), Barbados Government Award for Solar Research (1980), Barbados Astronomical Society Award for Service as Journal Editor (1977-91)

Past/Present Clients
Caribbean Examinations Council, Barbados Astronomical Society, Trinidad & Tobago Astronomical Society.

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