22 Lecture Gravitation
Phys 2101 Gabriela Gonzlez
Kepler's laws: I
Three laws, but all consequences of just one: Newton's law of gravitation! LAW OF ORBITS: All planets move in elliptical orbits, with the Sun at one focus. 2 x2 + 2 =1 2 a a = semimajor axis b = semiminor axis Sun's distance from center =ea e= eccentricity If e=0, orbit is circular.
Animations by by Bill Drennon, Physics Teacher Central Valley Christian High School Visalia, CA USA , http://home.cvc.org/science/kepler.htm
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Solar system orbits: not circular, but close
1 AU = 149 598 000 000 meters
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Kepler's laws: II
LAW OF AREAS: A line that connects the planet to the Sun sweeps out equal areas in the plane of the planet's orbit in equal times. "Areal velocity" dA/dt is constant
dA ( / 2) 1 2 = = dt 2
Angular momentum:
L = = = = 2
dA = dt 2
Conservation of angular momentum!
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Kepler's laws: III
LAW OF PERIODS: The square of the period of any planet is proportional to the cube of the semimajor axis of the orbit.
4 2 3 T2 =
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A very real example
"In spring 2002 S2 was passing with the extraordinary velocity of more than 5000 km/s at a mere 17 light hours distance -- about three times the size of our solar system -- through the perinigricon, the point of closest approach to the black hole. By combining all measurements of the position of S2 made between spring 1992 and summer 2002, we have obtained enough data in order to determine a unique keplerian orbit for this star, presented in Figure 1." "It is highly elliptical (eccentricity 0.87), has a semimajor axis of 5.5 light days, a period of 15.2 years and an inclination of 46 degrees with respect to the plane of the sky. From Kepler's 3rd law we can determine the enclosed mass in a straightforward manner to be 3.71.5 million solar masses. Therefore at least 2.2 million solar masses have to be enclosed in a region with a radius of 17 light hours."
http://www.mpe.mpg.de/www_ir/GC/gc.html Schdel, R. et al. A star in a 15.2-year orbit around supermassive the black hole at the centre of the Milky Way. Nature, 419, 694 - 696, (2002).
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Orbits and Energy
Energy conservation: kinetic + potential = constant! Newton's Gravitational law (circular orbits): U = - GMm/r F=ma GMm/r2=m(v2/r) Then, K = m v2 = GMm/2r = - U/2 E = K+U = - U/2+U= U/2 = - GMm/2r Elliptical orbits: E=- GMm/2a
Total energy does not depend on eccentricity, only on semimajor axis!
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Examples
(a) What is the gravitational acceleration on Earth, on the equator? (Use g = ag- 2R = GM/R2 - 2R) (b) How fast would the Earth had to rotate to produce weightlessness at the Equator? (c)What is the escape speed in Earth? (Use Ve2 = 2GM/R) (d)What would be the escape velocity is the Earth had the same mass concentrated in only a radius of 5mm? (e) How far from the surface will a particle thrown at 112m/s go? E=K+U = (1/2) mv^2 -GMm/r = constant (f) With what speed will an object hit the Earth if it is dropped from the space shuttle orbiting at 400km above the Earth? (g) How long does the space shuttle take to orbit once around the Earth? (law of periods) (h) How high should a satellite be above the Earth to have a circular geosynchronous orbit?
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Example
Neutron stars are extr...