[math-fun] Flux of the gravitational wave event (was Re: Maps from earthshine?)
Eugene Salamin <gene_salamin@yahoo.com> wrote:
"Keith F. Lynch" <kfl@KeithLynch.net> wrote:
(One final aside on moonlight and gravitational waves: Has anyone else noticed that the peak flux of the gravitational wave event, GW150914, on Earth was seven times the flux of the light of the full Moon? If it was visible light, you could not only have seen it, but could have read by its light. LIGO isn't really very sensitive!)
http://www.aei.mpg.de/~schutz/download/lectures/AzoresCosmology/Schutz.Azore... the flux (power per unit area) carried by a gravitational wave of frequency f and strain h is F = (pi/4) (c^3/G) f^2 h^2
The equation I found has an 8 instead of the 4. I'll try to find it again. Except for the pi/8 or pi/4 constant factor, the equation is easy to deduce from dimensional analysis.
?? = (3.2 mW/m^2) (h / 1e-22)^2 (f / 1 kHz)^2.
At the peak of the recently detected wave, h = 1e-21, f = 100 Hz,
I had heard that the (Earth-received) frequency at the peak flux was about 250 Hz. So I got amout 10 mw/m^2. As a sanity check, I divided the reported peak power output of the event, 3.6E+49 watts, i.e. 200 solar masses per second annihilated, by the area of a sphere 1.3 billion light years in radius. I get about 20 milliwatts per square meter. What accounts for the factor of two discrepancy? Probably polarization. LIGO, if I understand correctly, is sensitive to only one of the two polarizations. Also, red shift. 200 solar masses per second at the event equals about 180 solar masses per second on Earth, as the event is receding from us at about a tenth of the speed of light. Also, the received frequency at the time of peak power, about 250 Hz, was originally about 270 Hz. That's an impressive orbital period (1/270 second) for two 30-solar-mass objects. ("Only" 3 solar masses were annihilated in total, because the event lasted less than a second.)
Here's a tricky point about gravitational radiation. Because the radiation is quadrupole, the radiation frequency is twice the mechanical rotation frequency of the source. Yes, the LIGO detector is sensitive to only one polarization, the one aligned with its arms. The cross polarization affects both arms equally, so doesn't produce an interference fringe shift. For an elliptically polarized wave, only the one polarization component is detected. Because this is a three dimensional situation, there could also be an undetected out-of-plane component to the polarization. With two LIGO detectors, at different orientations, it should be possible to glean some partial information about the polarization. One would expect elliptical polarization corresponding to the orientation of the orbital plane as seen from Earth, but with the black holes having spin, it isn't so simple. -- Gene From: Keith F. Lynch <kfl@KeithLynch.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Monday, February 22, 2016 9:04 PM Subject: [math-fun] Flux of the gravitational wave event (was Re: Maps from earthshine?) Eugene Salamin <gene_salamin@yahoo.com> wrote:
"Keith F. Lynch" <kfl@KeithLynch.net> wrote:
(One final aside on moonlight and gravitational waves: Has anyone else noticed that the peak flux of the gravitational wave event, GW150914, on Earth was seven times the flux of the light of the full Moon? If it was visible light, you could not only have seen it, but could have read by its light. LIGO isn't really very sensitive!)
http://www.aei.mpg.de/~schutz/download/lectures/AzoresCosmology/Schutz.Azore... the flux (power per unit area) carried by a gravitational wave of frequency f and strain h is F = (pi/4) (c^3/G) f^2 h^2
The equation I found has an 8 instead of the 4. I'll try to find it again. Except for the pi/8 or pi/4 constant factor, the equation is easy to deduce from dimensional analysis.
?? = (3.2 mW/m^2) (h / 1e-22)^2 (f / 1 kHz)^2.
At the peak of the recently detected wave, h = 1e-21, f = 100 Hz,
I had heard that the (Earth-received) frequency at the peak flux was about 250 Hz. So I got amout 10 mw/m^2. As a sanity check, I divided the reported peak power output of the event, 3.6E+49 watts, i.e. 200 solar masses per second annihilated, by the area of a sphere 1.3 billion light years in radius. I get about 20 milliwatts per square meter. What accounts for the factor of two discrepancy? Probably polarization. LIGO, if I understand correctly, is sensitive to only one of the two polarizations. Also, red shift. 200 solar masses per second at the event equals about 180 solar masses per second on Earth, as the event is receding from us at about a tenth of the speed of light. Also, the received frequency at the time of peak power, about 250 Hz, was originally about 270 Hz. That's an impressive orbital period (1/270 second) for two 30-solar-mass objects. ("Only" 3 solar masses were annihilated in total, because the event lasted less than a second.)
participants (2)
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Eugene Salamin -
Keith F. Lynch