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Science and Astronomy Questions - Page 47 - Space Engine

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Watsisname
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09 Oct 2017 17:35

If you think that's gross, the equation describing a rotating black hole is like a summoning spell for Cthulhu. :P  

But anyway yes, you're thinking about it correctly. :)  If you fall into a black hole inside a box, physics in your box behaves just like physics anywhere else (in zero gee), at least as long as the black hole is large enough, and your box is small enough, and until you get too close to the singularity (or the inner event horizon if it's a spinning black hole).  In either case it's a pretty surreal experience if you look outside that box, as you say.



^This video isn't fully accurate in terms of portraying what this would actually look like, but it's still pretty good and better than almost anything else that I'm aware of.  It's also using the equations for an electrically charged black hole rather than a spinning one, because the math is easier, but it has many of the same features for what the interior is like.
 
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Mr. Missed Her
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11 Oct 2017 03:56

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Mr. Missed Her
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11 Oct 2017 16:41

This is more of a math question than an astronomy question, but whatever:
Does anyone know how to calculate the position in an orbit per time for a body in an elliptical orbit? I've been trying to figure this out for a while, and my efforts to find the answer have led to ↑that↑ ginormous equation up there.
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Gnargenox
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11 Oct 2017 18:01

I have no idea how to, but these online calculators look useful.
http://www.1728.org/ellipse.htm

Calculates the ecliptic longitude and latitude, right ascension, celestial declination, and distance of the planets from Earth.
http://keisan.casio.com/exec/system/1224748262
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12 Oct 2017 06:58

I'm not hopeful for those calculators, because neither seem to be in depth on the body's position in space and time. But I might be on to something, because if I can figure out how to make the radius of a circular orbit oscillate in the right way, it'll be elliptical. This is good, because I have a good grip on circular motion.
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12 Oct 2017 15:08

 
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Watsisname
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12 Oct 2017 15:37

They're very neat as mathematical entities, but that's all they are.  Ring singularities do not exist in nature, but are simply consequences of assuming the mass of a rotating black hole exists at a point to begin with.  That assumption leads to predictions that contradicts itself, and the true nature of them requires understanding the details of the collapse.
 
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12 Oct 2017 19:16

, I'm afraid this may be disappointing, but simply using the equation for an ellipse will not help you determine the position as a function of time for an orbit, because the speed is not constant but rather must obey the .  In fact, there is no algebraic expression which will do it.  The solutions to the equation of motion for a particle in a gravitational well are , meaning they cannot be written as a finite combination of elementary functions.  So this is a fairly advanced problem, and a topic of quite some length in celestial mechanics.

To give a hint to the scope of it, I'll briefly overview a standard approach, which is to move to polar coordinates (problems involving radial forces become much easier to analyze in polar coordinates), and solve for the radius as a function of polar angle.  The relation between radius and time for a particle in a gravitational field is given by a differential equation:



where the first (negative) term is the gravitational force, the second term is the centrifugal force, l is the angular momentum and is the   Using the substitution u=1/r and transforming time into polar angle (and skipping all the steps therein), we obtain



which is a differential equation which is much more readily solvable.  Transforming the solution back to r(ф),



I won't prove this, but it turns out that this ε is in fact the eccentricity of an ellipse, or a conic section in general.  So the orbit is a conic section, and we can determine its exact shape in terms of the masses, gravitational constant, eccentricity, and angular momentum.

Finally, to bring time back into the picture, we relate the angle (or "mean anomaly") with respect to perihelion, back to the other parameters and initial conditions, and essentially are solving r(ф(t)).

So, hopefully that gives some intuition to the problem and how to approach it, though my explanation certainly is not thorough enough to be workable.  If this is something you'd like to pursue more deeply or apply to a project, there are some good resources available out there which can help guide you through it. :)
 
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13 Oct 2017 00:47

 
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Mr. Missed Her
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13 Oct 2017 09:57

Space is very spacious.
 
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Mr. Missed Her
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13 Oct 2017 13:17

Yes! Yes! YES! I HAVE FOUND IT!!!! I HAVE IT YES AHAH HAHAHA [maniacal laughing]


I realized that my point for circular rotation was a polar coordinate in disguise. With this point: , input r to r and Θ to t, you've just converted your polar point to a normal point. So I replaced r with that equation at the end of Watsisname's post, and I had an elliptical orbit. I'll have to refine that big equation* up there, but it should be able to create an elliptical orbit with any position, size, and speed. And, as a bonus, any para- and hyperbolic orbits.


*HELPHELP ITS AN EQUATION
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Watsisname
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15 Oct 2017 20:45

 
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15 Oct 2017 23:34

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