Solution

The solution to this exercise allows us to calculate the angular momentum Um and energy measured at infinity Elm from initial satellite velocity measured in shell coordinates.

Figure 10 distinguishes between the angle of launch 80 Shen as measured in shell coordinates and the azimuthal angle $ that tracks the satellite on its orbit

The satellite is launched with initial speed shellanci at an angle 0O shell with respect to the outward direction, both as measured by the shell observer. The satellite's tangential component of velocity, v^ is

shell

In order to compute the angular momentum L (equation [2]), we want to put the proper time dx in the denominator of the derivative in equation [33], replacing dtsheii. To find dx, think of the separation between two flashes emitted by the satellite The proper time dx between these two emissions is measured on the clock carried by the satellite. The relation between dfSheii and dx is just the special relativity expression for time stretching-

The value of the angular momentum of the satellite comes from the preceding two equations:

This last expression for angular momentum has a simple interpretation. Angular momentum has its usual vector cross-product form L = rxp, whose magnitude is L = r p sin 0O She|i Here p is the expression from special relativity, p = y mv. In the present case, simply use on-shell values of shell quantities 0, y, and v.

To calculate energy we need the value of dt/dx, the ratio of far-away time to wristwatch time of the satellite The relation between shell time f^i and far-away time f comes from equation [C] in Selected Formulas at the end of the book:

dt dt, shell

Find the expression for energy from these equations and the definition of energy [3]

'o shell shell dx

Yo shell

Equations [35] and [37] allow us to determine the satellite's angular momentum L and energy E from initial speed and angle of motion measured by a shell observer at radius r0 Then energy and angular momentum specify the shape of the entire orbit from its start onward, thanks to equations [21 ] and [22] that tell us how angle <|> and radial coordinate r tick ahead.

Figure 11 Computer plots Predicting trajectories at a glance The plot of effective potential V/m—shown here for a single value of angular momentum L/m—plus the value of total energy E/m, allow us to make a quick prediction about the trajectory of a particle that orbits or is captured by a black hole. Four different energies are numbered on the central plot of this single effective potential, the corresponding trajectories for these energies appear in the four outer corners of the figure.

Figure 11 Computer plots Predicting trajectories at a glance The plot of effective potential V/m—shown here for a single value of angular momentum L/m—plus the value of total energy E/m, allow us to make a quick prediction about the trajectory of a particle that orbits or is captured by a black hole. Four different energies are numbered on the central plot of this single effective potential, the corresponding trajectories for these energies appear in the four outer corners of the figure.

Figure 12 Computer plot. Case 2 of Figure 11 in more detail Effective potential for black hole and corresponding effective potential for the Newtonian case (with unity added to the Newtonian figure to include the rest energy Elm = 1 for a particle at rest at large radius) Both curves are for angular momentum Um = 4.0 M The radial excursion in the Newtonian case leads to an elliptic orbit. In contrast, general relativity predicts that r-values extend to smaller radii The orbiting satellite spends more time at smaller radii than the Newtonian model would predict—a fact not directly obvious from this effective potential diagram During the additional "dwell time" near the inner edge of the orbit—and elsewhere on the orbit—the angle <|> keeps changing according to the conservation of angular momentum In consequence, the orbit swings out to maximum radius at a changed angle instead of at the same angle predicted by Newtonian mechanics The result is described as an elliptic orbit whose major axis rotates, or "advances" (case 2 in Figure 11) For the planet Mercury, this effect—though very much smaller in magnitude than for the case shown here—results in the advance of the major axis of the orbit by nearly 43 seconds of arc (0 0119 degrees) per century (Project C, following this chapter)

Figure 12 Computer plot. Case 2 of Figure 11 in more detail Effective potential for black hole and corresponding effective potential for the Newtonian case (with unity added to the Newtonian figure to include the rest energy Elm = 1 for a particle at rest at large radius) Both curves are for angular momentum Um = 4.0 M The radial excursion in the Newtonian case leads to an elliptic orbit. In contrast, general relativity predicts that r-values extend to smaller radii The orbiting satellite spends more time at smaller radii than the Newtonian model would predict—a fact not directly obvious from this effective potential diagram During the additional "dwell time" near the inner edge of the orbit—and elsewhere on the orbit—the angle <|> keeps changing according to the conservation of angular momentum In consequence, the orbit swings out to maximum radius at a changed angle instead of at the same angle predicted by Newtonian mechanics The result is described as an elliptic orbit whose major axis rotates, or "advances" (case 2 in Figure 11) For the planet Mercury, this effect—though very much smaller in magnitude than for the case shown here—results in the advance of the major axis of the orbit by nearly 43 seconds of arc (0 0119 degrees) per century (Project C, following this chapter)

Figure 13 Computer plot: Radii (circled numbers 1, 2, and 3) of different circular orbits, each of which lies at the radius of the effective potential minimum The stable circular orbit of smallest radius lies atr = 6M. For details, see the exercises of this chapter. With rockets blasting, you can still explore from r = 6M down to r = 2M, but you cannot be in a stable circular orbit in this region

Figure 13 Computer plot: Radii (circled numbers 1, 2, and 3) of different circular orbits, each of which lies at the radius of the effective potential minimum The stable circular orbit of smallest radius lies atr = 6M. For details, see the exercises of this chapter. With rockets blasting, you can still explore from r = 6M down to r = 2M, but you cannot be in a stable circular orbit in this region

I frame no hypothesis

Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the sun and planets, without suffering the least diminution of its force; that operates not according to the quantity of the surfaces of the particles upon which it acts (as mechanical causes used to do), but according to the quantity of the solid matter which they contain, and propagates its virtue on all sides to immense distances, decreasing always as the inverse square of the distances. Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed; and in receding from the sun decreases accurately as the inverse square of the distances as far as the orbit of Saturn, as evidently appears from the quiescence of the aphelion of the planets; nay, and even to the remotest aphelion of the comets, if those aphelions are also quiescent. But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypothesis; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction. Thus it was that the impenetrability, the mobility, and the impulsive force of bodies, and the laws of motion and of gravitation, were discovered. And to us it is enough that gravity does really exist, and acts according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.

—Isaac Newton about 1686

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