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4. The Solar System generate, create ansi/aim code 128 none on .net projects QR Code Spevcification Investigation 4.4. The Newtonian de ection of light We shall derive the de ection by using the principle of equivalence, in much the same style as we derived the gravitational redshift in Investigation 2.2 on page 16. Consider light passing a star.

Since light must travel on a straight line with respect to a local freely-falling observer, and since these observers all fall towards the center of the star, the light must continually bend its direction of travel in order to go on a straight line with respect to each observer it happens to pass. We can estimate the size of the e ect, at least roughly, by the following argument within Newtonian gravity. Let us consider just one freely-falling observer, who is at rest with respect to a star of mass M at the point where the light beam makes its closest approach to the star as it passes it by.

Let this closest distance be d, the impact parameter. (This is not quite how we de ned the impact parameter, but it is close enough for our approximate argument.) The observer s acceleration towards the star is g = GM/d 2 .

Traveling at speed c, the beam of light will experience most of its de ection in a time of order d/c, the time it takes for the light to move signi cantly further away from the star. During this time, the observer has acquired a speed v = gd/c = GM/cd perpendicular to the motion of the light. By the equivalence principle, the light must also have acquired roughly this same speed transverse to its original direction.

Since its speed in the original direction is very little changed, we can calculate the angle of de ection by simple geometry: the tangent of the de ection angle is v/c. For small angles, the tangent of an angle is equal to the angle as measured in radians. This leads to the estimate that the angle of de ection will be = v/c = GM/c 2 d radians.

The total de ection should be double this, since the light will experience the same de ection coming in to the point of nearest approach as going out: 2GM . c2 d (4.13) This turns out to be exactly the answer that a very careful calculation would give.

But we do not need to do that calculation to check Equation 4.13. We only need to use the computer program Orbit with the right initial data and measure the results.

How would one measure this If we look at the position of a star just once, as its light is passing near the Sun, we won t know how much de ection it is su ering, since we don t know its true position. The way to do it is to measure the position of a star when its light is passing nowhere near the Sun. Then, perhaps six months later, when the Sun is near the position of the star, measure the position again.

It is clear from Figure 4.7 that the apparent position of the star moves outwards, away from the Sun. The only di culty is in seeing the star when the Sun is near.

But during an eclipse of the Sun, all the light from the Sun is blocked by the Moon, and so it is possible to see stars very close to the Sun s position. This is how the e ect was eventually measured. The observed result is twice the number given by Equation 4.

13, consistent with general relativity. Newtonian prediction of the de ection angle in radians =. Exercise 4.4.1: Light de ection by other bodies Any gravitating body will de ect light. Estimate, using Equation 4.13 above, the amount of de ection experienced by a light ray just grazing the surface of the following bodies: (a) Jupiter, whose radius is 7.

1 104 km; (b) the Earth; (c) a black hole of any mass; and (d) you.. Figure 4.7. The Code-128 for .

NET de ection of light by a point-like mass as calculated in Newtonian gravity. The mass has the mass of the Sun, but to show the effect, it has been made almost as compact as a black hole. This allows trajectories to experience strong gravity and exhibit large de ections.

. Eddington was on e of the rst true astrophysicists, a scientist who used the theories of physics to understand the nature of astronomical objects. He was an early champion of Einstein s general relativity. Dyson was the British Astronomer Royal from 1910 to 1933.

. pendently by the Code 128 Code Set B for .NET German astronomer Johann G von Soldner (1776 1833) in 1801. They regarded it as a mere curiosity, since measuring the apparent positions of stars to an accuracy of an arcsecond or so was impossible in their time.

Einstein himself independently re-derived the formula using the equivalence principle in 1909, and he pointed out that the expected de ection might well be observable with the telescopes of his day. However, before a suitable opportunity arose for observing the effect, Einstein moved on to devise the theory of general relativity (1915). In this theory, there is an extra effect that causes light to de ect twice as much, so that the new prediction would be 4GM/c2 d.

(We shall calculate this effect in 18.) A de ection of this size was indeed measured in an eclipse expedition in 1919 led by the British astronomers Sir Arthur Eddington (1882 1944) and Frank W Dyson (1868 1939). The accuracy was enough to distinguish between the old Newtonian de ection and the new general relativistic one.

The veri cation of this prediction of general relativity did more than anything else to make Einstein a celebrity, a household name.. Tides and tidal forces:.
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