## Gravity Newton to Einstein

The motions of the celestial bodies may be understood in terms of two laws articulated by Newton. His second law states that a vector force F1 applied to a mass m 1 brings about a vector acceleration a of m 1 according to

if the observer is in an inertial (non-accelerating) frame of reference. His law of gravitation gives the force due to gravity,

Gm1m 2

Fgravity =--2-r (Newton's law of gravitation) (4.2)

The force on point mass m1 in the presence of m2 is proportional to the product of the masses and the gravitational constant G = 6.67 x 10-11 N m2 kg-2, and is inversely proportional to the square of the distance between the masses. The direction of the force is along the line joining the two masses and the force is attractive. This is indicated with the unit radial vector r and the minus sign.

The acceleration of m1 can be obtained by substituting the force (2) into (1). If m2 is much more massive than m1, and if we ignore the effects of other bodies such as planets, the resultant differential equation represents the classical planetary problem which applies quite well to the earth-sun system. The sun is about 330 000 times heavier than the earth. The solution of the differential equation for the earth's motion indicates that the earth should move in an elliptical orbit about the sun, as indeed it does. The orbit is nearly circular, but not quite; it is slightly elliptical. This results in a changing angular size of the sun as viewed from the earth. It is about 3% smaller on July 6 than it is six months later in January.

Gravity and Newton's second law are also responsible for the spinning of the earth, the orbital motion of the planets about the sun, and the formation of the planets from clumps of gas and dust left over from the formation of the sun. This material coalesced under the influence of the self gravity; each part of a clump finds itself attracted to all other parts. In this process, gravitational potential energy is converted into heat (infrared radiation), kinetic energy of outflowing material, and rotational kinetic energy of the resultant planet.

The spin of the earth has not been constant since its formation; it is gradually slowing due to gravitational tidal forces. The moon and the oceans are mutually attracted to one another by gravity giving rise to oceanic tides. Friction between the oceans and the earth results in a long-term average increase in the length of the day of —0.7 ms per century. Other irregular effects (ocean currents, atmospheric motions, etc.) contribute comparable and unpredictable changes to this slowdown rate, for a long-term average of —1.4 ms per century.

Gravity similarly drives the formation of stars like the sun. It also is the force that causes stars to burn atomic nuclei to provide the radiation pressure against further collapse. Without the inward pull of gravity and the conversion of gravitational potential energy to kinetic energy, the centers of stars would not have the pressure and temperature necessary to sustain nuclear burning. When the nuclear fuel is expended at the center of a star, it will collapse to a white dwarf, neutron star, or black hole, again under the influence of gravity. The resultant supernova explosion (in the collapse of a stellar core to a neutron star) thus is powered by gravity.

On larger scales, the motions of stars and gaseous clouds in galaxies are clearly dominated by gravity, as are the motions of individual galaxies in clusters of galaxies. In fact, there appears to be an excess of gravitational force on these large scales; the current view is that an unknown type of dark matter is the source of this excess force.

On the largest scale of all, gravity runs the universe. The universe itself is expanding, but as it does, it is subject to self gravity which would tend to slow the expansion. In contrast to this expectation, recent work suggests an opposing repulsive force that accelerates the expansion, due to some sort of dark energy.

Much current research is directed toward determining the rate of expansion of the universe. Will it expand forever, or will it slow down eventually coming to rest (much like a rising ball), and then begin to fall inward toward a Big Crunch? The presence of gravitational matter on such large scales can not be described with Newtonian geometry nor with Einstein's special theory of relativity. Einstein's general theory of relativity, general relativity or GR, is required to properly describe the universe insofar as we know it; it pulls together the concepts of gravity and space-time into a unified whole. In fact, it contains a cosmological constant that results in a long-range repulsive force such as that attributed just above to dark energy.

Thus we find that gravity does indeed underlie the entire field of astrophysics.

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