general relativity - How would the gravitational strain waveform look like for a planet in orbit with a star?

So far I've read and seen about various type of waveforms for different sources of Gravitational waves such as black holes, neutron stars etc. So the waveforms would be for Compact Binary Coalescence etc. I was wondering would a planet orbiting a star also have a similar waveform assuming one planet orbiting a star. Or would it be something else? If it was what would it be?

Answer

It would be a similar waveform to the early part of the waveform of a couple black holes merging, except it would have constant amplitude. As for the black holes there would be two polarizations.

So, specifically it would be sinusoidally varying strains, and if we try to detect it as in LIGO the waveform would vary sinusoidally over time, with a frequency twice the rotation rate of the planet around the star. So the frequency of the gravitational waveform would be, for the earth-sun system, twice per year, i.e., a frequency of 2 cycles per year. The wavelength would be c/(2/1 year), and since c = 1 ly/yr, that would be a wavelength of half a light year. We'd need a very large interferometer to detect it.

A caveat on what is stated above. The shape of the waveform would be sinusoidal for what is called the far field, like in radio waves, meaning signigficantly further away than the wavelength. We'd have to be at least 10 light years or so away to see it as a sinusoidal wave. Closer it it would be a much more complicated waveform, as we'd be in the near field. It is the gravitational radiation of a quadrupole moment, a more complex shape close in. Even further away, in the far field, the waveform amplitude varies with the angle off the orbit's plane.

You can see the far field waveform in the wiki article at https://en.m.wikipedia.org/wiki/Gravitational_wave. But note that it's pretty far into the article, it has the exact mathematical form in the far field.

The total power the arth sun gravitational wave carries is about 200 watts. Since the system looses that energy the sun-earth orbit gets tighter - but an extremely small amount each year, about the diameter of a proton. To decay into the sun that way it would take about $10^{13}$ times longer than the age of the universe now.

So you see, we know that waveform and what it does, it is just too weak for us to detect it. We do detect the waveforms from more tightly and faster orbiting objects around each other, like black holes, and we expect to detect also neutron stars around each other or around black holes if not too far apart. A big space-based interferometer of a million km leg length is being considered to be launched in the 2020's, assuming the funding gets there. We will more many more astrophysical objects, but not planets around suns with it, just too weak.

But yes, a sinusoidal waveform, at twice the orbital frequency.