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# Interstellar Dispersion

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#### 9301-020a

When propagating through a plasma, two closely-space electromagnetic frequency components of an analog or pulse signal at f1 and f2, will suffer a relative dispersion given by:

```         e2.Ne.L
dTaud  = -------- [1/f12 - 1/f22] = 1.41 x 109 [1/f12 - 1/f22]  seconds
2.pi.m.c```

where:

 e = electronic charge (1.6 x 10-19 C), Ne = electron density in interstellar space (107 m-3), m = electron rest mass (9.11 x 10-31 kg), c = velocity of light (3 x 108 ms-1), L = propagation distance (9.461 x 1018 m).

At high carrier frequencies, [1/f12 - 1/f22] may be replaced by [df/f3], where df is the bandwidth (f2 - f1), and f is the mean frequency (Hz). Clearly, as we increase the carrier frequency interstellar dispersion becomes less significant. Let us now calculate the approximate magnitude of the dispersion effect at radio frequencies using the values given above for a distance of 1,000 light years.

#### 9301-020b

For propagation over 1,000 light years, and at microwave frequencies:

f1 = 1 GHz

f2 = 2 GHz

df = f2 - f1 = 1 GHz

dTaud ~ 1 ns

#### 9301-020c

Note that the value of electron density Ne used in these calculations is at the upper bound for interstellar space. For communications within interplanetary space, Ne = 106 to 1010 m-3, while in intergalactic space, Ne < 10 m-3. The sign of the dispersion is such as to cause the lower frequency to be delayed with respect to the higher frequency, i.e., low frequencies move slower.

At microwave frequencies, the interstellar dispersion over 1,000 light years is comparable to the pulse period or inverse bandwidth. At optical frequencies, dispersion does not limit any wideband analog or pulse modulation bandwidth. This is another important reason why lasers are superior for interstellar communications. In practice, for ground-based observatories, the optical dispersion as the photons pass through the last few kilometers of their journey will limit the bandwidth. Atmospheric dispersion is of the order of 100 ps and is non-stationary, i.e., it depends on atmospheric turbulence.

Astrophysical Formulae, Kenneth R. Lang, Springer-Verlag, 1978, p. 52-58.

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