## APOD: The Doubly Warped World of Binary... (2021 Apr 16)

alter-ego
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### Re: APOD: The Doubly Warped World of Binary... (2021 Apr 16)

johnnydeep wrote:
Fri Apr 16, 2021 4:55 pm
There are at least two statements here I don't quite understand:

1. "...material orbiting smaller black holes experiences stronger gravitational effects that produce higher temperatures."

Is this due to the tidal effects being greater for smaller black holes? Meaning that the gravity gradient is steeper around a smaller black hole and thereby tears at orbiting matter more greatly?
Yes, although I did not see direct reference to "tidal force". Heating is caused by gas migrating from outer radii to inner radii (loss of angular momentum), and the released energy over a small change in orbital radius (dr) is expressed as as the derivative of gravitational potential energy wrt radius. Only considering black-body accretion disk luminosity (i.e. no jets), the temperature proportionality to BH mass, M, can be found. Bottom line: Temperature increases with larger velocity change and larger accretion rate:
Accretion Disk Lecture wrote: Suppose that as each fluid element moves inward that it releases its energy locally, and that its energy is all gravitational. How much energy would an element of mass, m, release in going from a circular orbit at radius r + dr to one at radius r?
Gravitational potential energy is Eg = -GMm/2r, so the energy released is GMmdr/2r2.
...
However, let us now focus on just the radial dependence, writing [release of gravitational potential energy over dr] dEg ∼ GMmdr/r2. That means that the luminosity of this annulus, for an accretion rate [m → dm/dt], is dL ∼ GM[dm/dt]dr/r2
and assuming black-body radiation, the temperature can be related to Luminosity:
Accretion Disk Lecture wrote:For a blackbody, L = σAT4. The area of the annulus is 2πrdr, and since dL ∼ M[dm/dt]dr/r2 we have T4 ∼ M[dm/dt]/r3, or T ∼ {M[dm/dt]/r3}1/4
Rewriting the parameters in terms of the BH mass, M, and assuming a stead-state accretion rate (dm/dt =constant)then T ∝ M−1/4.
Accretion Disk Lecture wrote: This shows that as black holes get bigger, emission from their accretion disks get cooler, all else being equal. For example, a stellar-mass black hole accreting at nearly the Eddington rate has an inner disk temperature near 107 K, but a supermassive 108 M☉ black hole accreting near Eddington has only a 105 K temperature.
FYI: Given the conditions of no jets, Eddington accretion rate yields the Eddington luminosity = Maximum luminosity a body (such as a star) can achieve when there is balance between the force of radiation acting outward and the gravitational force acting inward. When there are jets, the BH luminosity can exceed the Eddington limit because that radiation does not encounter the accretion disk, therefore it does not contribute to hydrostatic equilibrium.
johnnydeep wrote:
Fri Apr 16, 2021 4:55 pm
2. "...relativity causes the black holes to appear smaller and brighter as they approach the camera and larger and fainter as they recede."

This I don't get at all. Does it matter which BH is closer to the camera and/or whether they are eclipsing each other or not? And either way, I still don't get it
Though I don't necessarily question Art's description about brightness, I don't think it clearly addresses your question.
First, the question is about relativistic aberration, and second, the visual aspects mentioned can apply to single, moving BH. Relativistic Aberration not only acts differentially on the accretion disk, but also acts on the entire black hole (the photon ring and the accretion disk). In this visualization, there are two BHs orbiting each other which leads to a periodic, relativistic aberration. To see the largest aberration, the observer needs view the orbital plane edge on, and when the BHs are at maximum separation. There, one recedes when the other advances directly toward the observer; the velocity difference is maximum so the apparent size differences are a maximum. The noted aberration does not occur when they are eclipsing, or when viewed from above. For the latter two cases, the orbital velocities are perpendicular to the line of sight.

Referring the first APOD link and the visualization:
Light rays from accretion disks wrote:The visualization also shows a more subtle phenomenon called relativistic aberration. The black holes appear smaller as they approach the viewer and larger when moving away.
Relativistic Aberration - Orbiting Black Holes.jpg
This effect doesn't rely on gravity bending light. It's behavior is rooted in both Newtonian physics (stellar aberration, velocity << c), and accurately described in Special Relativity. For a moving source, it comes down to the changing propagating cone angles - narrowing toward the observer, widening away from the observer. The following SR visualizations demonstrate this for a moving observer. Keep in mind the moving observer's narrowing field of view demonstrates the same behavior as a narrowing light-propagation cone-angle of relativistic star moving towards a stationary observer. I.e. the observers FoV doesn't change, but the apparent size of the star is brighter and smaller.

The first is a 6-min video demonstrating all SR doppler and aberration effects. The brightening and narrowing sky FoV is shown around 2.5 minutes into the visualization.
Click to play embedded YouTube video.

The second shows how a 360°FoV collapses (quantifying the view vs velocity).
Click to play embedded YouTube video.

This one is for an observer falling into a black hole. When his visible aberrated world blinks out, the screen goes black.
I'll admit, I find this a bit unsettling.
Click to play embedded YouTube video.

Hope all this helps.
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johnnydeep
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### Re: APOD: The Doubly Warped World of Binary... (2021 Apr 16)

Thanks for that, alter-ego! I'll have to read it a few more times before I can claim to understand it.
"To Boldly Go......Beyond The Fields We Know."