2.5 Frequency of gravitational waves

The signals for which the best waveform predictions are available have well-defined frequencies. In some cases the frequency is dominated by an existing motion, such as the spin of a pulsar. But in most cases the frequency will be related to the natural frequency for a self-gravitating body, defined as
∘ ----- ∘ ------- ω0 = πG ¯ρ, or f0 = ω0∕2 π = G ¯ρ∕4π, (12 )
where ¯ρ is the mean density of mass-energy in the source. This is of the same order as the binary orbital frequency and the fundamental pulsation frequency of the body. Even though this is a Newtonian formula, it provides a remarkably good order-of-magnitude prediction of natural frequencies, even for highly relativistic systems such as black holes.

The frequency of the emitted gravitational waves need not be the natural frequency, of course, even if the mechanism is an oscillation with that frequency. In many cases, such as binary systems, the radiation comes out at twice the oscillation frequency. But since, at this point, we are not trying to be more accurate than a few factors, we will ignore this distinction here. In later sections, with specific source models, we will get the factors right.

The mean density and hence the frequency are determined by the size R and mass M of the source, taking 3 ¯ρ = 3M ∕4πR. For a neutron star of mass 1.4M ⊙ and radius 10 km, the natural frequency is f0 = 1.9 kHz. For a black hole of mass 10M ⊙ and radius 2M = 30 km, it is f0 = 1 kHz. And for a large black hole of mass 2.5 × 106M ⊙, such as the one at the center of our galaxy, this goes down in inverse proportion to the mass to f0 = 4 mHz. In general, the characteristic frequency of the radiation of a compact object of mass M and radius R is

1 ( 3M )1 ∕2 ( 10M ) f0 = --- ---- ≃ 1kHz ----⊙- . (13 ) 4π R3 M

Figure 2View Image shows the mass-radius diagram for likely sources of gravitational waves. Three lines of constant natural frequency are plotted: 4 f0 = 10 Hz, f0 = 1 Hz, and −4 f0 = 10 Hz. These are interesting frequencies from the point of view of observing techniques: gravitational waves between 1 and 104 Hz are in principle accessible to ground-based detectors, while lower frequencies are observable only from space. Also shown is the line marking the black-hole boundary. This has the equation R = 2M. There are no objects below this line, because they would be smaller than the horizon size for their mass. This line cuts through the ground-based frequency band in such a way as to restrict ground-based instruments to looking at stellar-mass objects. No system with a mass above about 104M ⊙ can produce quadrupole radiation in the ground-based frequency band.

View Image

Figure 2: Mass-radius plot for gravitational wave sources. The horizontal axis is the total mass of a radiating system, and the vertical axis is its size. Typical values from various sources for ground-based and space-based detectors are shown. Lines give order-of-magnitude constraints and relations. Characteristic frequencies are estimated from f ∼ (G ρ∕4π )1∕2. The black-hole and binary lines are described in the text.

A number of typical relativistic objects are placed in the diagram: a neutron star, a pair of neutron stars that spiral together as they orbit, some black holes. Two other interesting lines are drawn. The lower (dashed) line is the 1-year coalescence line, where the orbital shrinking timescale due to gravitational radiation backreaction (cf. Equation (28View Equation)) is less than one year. The upper (solid) line is the 1-year chirp line: if a binary lies below this line, then its orbit will shrink enough to make its orbital frequency increase by a measurable amount in one year. (In a one-year observation one can, in principle, measure changes in frequency of 1 yr–1, or 3 × 10–8 Hz.)

It is clear from the Figure that any binary system that is observed from the ground will coalesce within an observing time of one year. Since pulsar binary statistics suggest that neutron-star–binary coalescences happen less often than once every 105 years in our galaxy, ground-based detectors must be able to register these events in a volume of space containing at least 106 galaxies in order to have a hope of seeing occasional coalescences. That corresponds to a volume of radius roughly 100 Mpc. For comparison, first-generation ground-based interferometric detectors have a reach of around 20 Mpc for such binaries, while advanced interferometers should extend that to about 200 Mpc.


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