# Why does LIGO have an arm length of few kilometers? Is the distance dependent on Gravitational Wave wavelength?

Antennas for capturing radio waves need to have $\frac{\lambda}{2}$ length for optimum reception of signal. Does it imply LIGO arm length is $\frac{\lambda}{2}$ of Gravitational Wave it is trying to capture?

Pretty close. The effective LIGO arm length is 1600km (the light beam is reflected forth and back 400 times). LIGO is most sensitive at approx. 150Hz (advancedligo.mit.edu/summary.html), which would be a wavelength of 2000km... so the LIGO arms are approx. $\lambda /2$. The noise minimum depends on the noise spectrum, of course, so the sensitivity max. won't be exactly where one would expect it.

• SNR is high inorder for LIGO to work. But so many random events happened which emitted Gravitational Waves how come we have such clear signals. Is it because of the way LIGO is aligned? I should have asked this in a new question. – user43794 Feb 20 '16 at 11:50
• @user43794: Nothing on Earth (or even in the solar system) can emit enough gravitational waves to be detectable by LIGO. The noise in LIGO is seismic noise and quantum noise from the laser system. – CuriousOne Feb 20 '16 at 11:55
• Well what i meant was so many Galaxies have so many events which emit Gravitational Waves. Will not they get mixed up? I believe LIGO is just looking at one frequency not an event. That is why there is such a clear signal. – user43794 Feb 20 '16 at 12:02
• The LIGO arms are etalons and the figure of 400 trips is a measure of the Q of the etalon i.e. how many times the light reflects before getting too faint to be useful. This doesn't mean the effective length is 400 x 4km. The optical length of the arms is just 4km. – John Rennie Feb 20 '16 at 12:04
• @user43794: Black hole or neutron star mergers are expected to happen at a frequency of one every 10,000-100,000 years per galaxy. That's not enough to cause the noise background. – CuriousOne Feb 20 '16 at 12:07

The reason aerials are made with a particular length is because the interaction with the radio wave is a resonant process. Whether the transmission is AM or FM there is a central frequency. The radio aerial length is chosen so that this central frequency makes the electron density in the aerial resonate, which enhances the signal.

However gravity wave detection is not a resonant process so there is no reason why the arm length should bear any relation to the wavelength of the gravity wave. The arm length is chosen to maximise the signal and minimise noise.

A signal like the one detected causes a strain of around $10^{-21}$ so for example if the arms were one metre long they would change length by $10^{-21}$m, which is too small to be detected. make the arms a kilometre long and the length change becomes $1000 \times 10^{-21} = 10^{-18}$m, which is just detectable and indeed was detected.

But the longer you make the arms the harder (or more the the point the more expensive) it is to make them to the required precision. The length was chosen to give the best sensitivity for the funds available.

• One can make wide-band RF antennas, they just don't look like a simple dipole. Their lower frequency limit is still given by their physical dimension. LIGO doesn't have to be resonant to achieve its best signal to noise ratio at its effective optical length. For longer wavelengths it loses sensitivity because it doesn't cover much of the wave. It can still detect the wave, it just can't get good SNR. – CuriousOne Feb 20 '16 at 12:19
• @CuriousOne: yes aerials don't have to be resonant, but the reason for making them a set fraction of a wavelength is to make them resonant. This does not apply the LIGO. The upper and lower frequency limits are set by environmental noise not any supposed relationship to the gravitational wave wavelength. – John Rennie Feb 20 '16 at 12:24
• The reason why a simple stick is resonant is because it's a simple stick. If you pattern an antenna correctly, it receives the exact same amount of power but not just at one frequency but in a wide frequency band. LIGO is, if you will, a wide band antenna. The lower limit is still set, for both, by an effective size. – CuriousOne Feb 20 '16 at 12:33
• But i thought superimposition of waves with different wavelengths will cancel out each other. So it means Gravitational waves are narrow bandwidth waves. Am i correct? – user43794 Feb 25 '16 at 20:58