Solution

(6.67 X 10~8 dyn cm2/g2)(2 X 1033 g) r (3 X 1010 cm/s)2

As we have said, this derivation should not be considered rigorous - it is more of a dimensional analysis. However, it gives a result that agrees with the full general relativistic calculation for shifts that are not too large. The actual result is

The shift for spectral lines in the Sun is very small. The shift for white dwarfs is measurable. The two best cases so far are Sirius B (3 X 10_4) and 40 Eridani (6 X 10_5).

There is an interesting way to measure the gravitational redshift on Earth. It utilizes a phenomenon known as the Mossbauer effect (Fig. 8.10). This involves the emission of a gamma-ray by a nucleus held firmly in place by a solid crystal. In a free nucleus, the gamma-ray would lose a little

Nucleus

If we use the fact that (1 _ x)1/2 = (1 _ x/2) for x V 1, equation (8.4) gives the same result as equation (8.3).

If we use equation (8.4) and take r2 _ and use the approximation for small shifts, we obtain

Recoiling nucleus

Photon

If we compute the wavelength shift, AA, we find

Recoiling Crystal

Nucleus in atom in crystal

The Mossbauer effect. (a) Emission of a gamma-ray by a free nucleus.To conserve momentum, the nucleus recoils.The recoiling nucleus carries away some energy. Therefore, the energy of the gamma-ray is less than the energy difference between the two levels involved in the particular transition. (b) If the nucleus is part of an atom, which is, in turn, part of a crystal, the whole crystal must recoil. Since the crystal is much more massive than the nucleus, its recoil is negligible.This means that the energy of the gamma-ray is always equal to the difference between the two levels involved in the transition.

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