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Author Topic: Lambert's Cosine Law  (Read 51967 times)

Online jeffreyH

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Re: Lambert's Cosine Law
« Reply #275 on: 26/04/2015 22:06:01 »
If instead we use L instead of b15a3315a69ce382beea50cbaa779f80.gif and t instead of 5c2423d9bb6800593fddead13d1534d3.gif
we have 57e288a237fb8102bfe96408da639cb7.gif. With t set at 1 second and L at 299792458 m we can simplify as 3f1b2222c04b1f441db845920e3de839.gif. To find m we use 3f2356b370cd1be487d6204412b887cb.gif. The attached graph shows the coordinate acceleration towards the event horizon viewed from infinity for a mass of m. The value of m is 4.03726E+35 kg.

The pink line shows a standard calculation of acceleration and the blue line the coordinate value. The turning point in the blue line shows the point at which the observed object falling into the black hole appears to slow down.

The y axis shows acceleration m/s-1 and the x axis radial distance from the event horizon. [Correction]The y axis shows acceleration m/s-2 and the x axis radial distance from the centre of gravity.
« Last Edit: 30/04/2015 01:19:55 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #276 on: 26/04/2015 22:50:08 »
To really appreciate what this means please read the following.


http://www.google.co.uk/url?url=http://genesismission.jpl.nasa.gov/gm2/mission/pdf/Giantstars.pdf&rct=j&q=&esrc=s&sa=U&ei=u1s9VeKTGcL4atibgMAL&ved=0CBsQFjAB&sig2=FVpc1EFdKKfTZIo8srjslw&usg=AFQjCNHVd9qJvEk-oK6wJRBUlnbXhZ-B2Q


If true then galactic sized black holes are the only ones that can exist and must have been formed during the early stages of the universe and were responsible for the formation of galaxies. The amounts of mass involved can not exist as ordinary stars and must be a consequence of the slowdown of expansion following the inflationary period.

EDIT: A better source is:

http://www.space.com/858-study-stars-size-limit.html
« Last Edit: 26/04/2015 23:00:16 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #277 on: 30/04/2015 01:08:49 »
The derivative of GM/r^2 is 2GM/r^3 which will show the rate at which g is changing at varying radial distances. This derivative is also used in Malcolm S Longair's book Galaxy Formation Second Edition. I am about to buy this book to pursue this further. I have attached a graph of 2GM/r^3 for the proposed minimum mass black hole.
 

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Re: Lambert's Cosine Law
« Reply #278 on: 02/05/2015 05:43:37 »
The compressibility of matter is the crucial point in this investigation. This relates to gravitational collapse.

http://en.wikipedia.org/wiki/Gravitational_collapse
[Gravitational collapse is the inward fall of an astronomical object due to the influence of its own gravity which tends to draw the object toward its center of mass. In any stable body, this gravitational force is counterbalanced by the internal pressure of the body acting in the opposite direction. If the gravitational force is stronger than the forces acting outward, the equilibrium becomes unstable and a collapse occurs until the internal pressure increases sufficiently that equilibrium is once again attained (the exception being a black hole).]

The crucial sentence is "If the gravitational force is stronger than the forces acting outward, the equilibrium becomes unstable and a collapse occurs until the internal pressure increases sufficiently that equilibrium is once again attained".

The key thing is to plot all potential stages of equilibrium for a variety of mass sizes.

Another crucial point is this.

"According to Einstein's theory, for even larger stars, above the Landau-Oppenheimer-Volkoff limit, also known as the Tolman–Oppenheimer–Volkoff limit (roughly double the mass of our Sun) no known form of cold matter can provide the force needed to oppose gravity in a new dynamical equilibrium. Hence, the collapse continues with nothing to stop it."

The Tolman–Oppenheimer–Volkoff limit is then the key to determining if black holes of 3 solar masses can actually form.
 

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Re: Lambert's Cosine Law
« Reply #279 on: 02/05/2015 06:07:32 »
In discussion of the Tolman–Oppenheimer–Volkoff limit it is instructivbe to note this:

http://en.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_limit

[In the case of neutron stars this limit was first worked out by J. Robert Oppenheimer and George Volkoff in 1939, using the work of Richard Chace Tolman. Oppenheimer and Volkoff assumed that the neutrons in a neutron star formed a degenerate cold Fermi gas. They thereby obtained a limiting mass of approximately 0.7 solar masses, [2][3] which was less than the Chandrasekhar limit for white dwarfs. Taking account of the strong nuclear repulsion forces between neutrons, modern work leads to considerably higher estimates, in the range from approximately 1.5 to 3.0 solar masses.[1] The uncertainty in the value reflects the fact that the equations of state for extremely dense matter are not well known. The mass of PSR J0348+0432, 2.01±0.04 solar masses puts a lower bound on TOV limit.]

where "the equations of state for extremely dense matter are not well known".

Discussion of the uncertainty can be found here:

http://en.wikipedia.org/wiki/QCD_matter#Phase_diagram

"The phase diagram of quark matter is not well known, either experimentally or theoretically."

This then opens the debate on lower black hole mass limit.
 

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Re: Lambert's Cosine Law
« Reply #280 on: 03/05/2015 10:11:32 »
After having read a post in another forum I am going to assume something rather odd. That all particles travel at the speed of light. This is a mathematical device only as NO they don't all travel that fast. However what it does do is make all things equal. The speed of light is also considered to be the speed of gravity. So why not just use the photon? Well the photon is massless. If this assumption is also combined with massive particles then we may learn something from a mathematical derivation. It can always be corrected later to show realistic values.
 

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Re: Lambert's Cosine Law
« Reply #281 on: 03/05/2015 12:42:57 »
For the purposes of this investigation relativistic mass has to be ignored. The object generating the gravitational field will be a Planck mass Schwarzschild metric black hole. So that we can disregard angular momentum.
 

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Re: Lambert's Cosine Law
« Reply #282 on: 03/05/2015 12:54:02 »
If we take c=G=1 then a photon orbital becomes 2M which is at the horizon of the black hole. If we were to direct a constant directed light source so that the photons in the orbital increased over time there would come a point where the gravitation generated by the light would start to cancel with the gravitation immediately inside the horizon such that Ve would fall below c. This will become important later.
« Last Edit: 03/05/2015 12:55:45 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #283 on: 13/05/2015 20:58:43 »
One thing that it would be useful to do would be to determine the forces active in the internal cavity of a mass. I don't think this has been attempted experimentally underground. What it would be nice to be able to do is this.

63f08a7b28f21c74535a4fd7a25e987d.gif

However, since these are vectors , at the centre of gravity they are said to cancel so a summation is not possible. A method is needed to determine the action of opposing forces that does not simply assume cancellation. The effect on particles will not be zero. Something must happen due to the outward attractive force that applies in all directions. This is not a trivial exercise nor is it a worthless one. It has implications for the examination of extreme gravitational sources.
 

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Re: Lambert's Cosine Law
« Reply #284 on: 29/05/2015 23:14:29 »
Going back to the coordinate value of G we can first investigate the relationship between surface area and entropy. If we take the equation fb2b4faef787e41d98ab472054a04d6e.gif then we can find a value of g that will give the area of the surface of the event horizon. We can then substitute the value of coordinate G that would apply at the horizon. In calculating coordinate G it was found that G does not tend to zero as the radius approaches rs. This indicates that the size of a black hole indicated by the range of the apparent horizon when viewed by a remote observer will be smaller than expected. This could be one explanation of why Sag A* did not consume the G2 gas cloud. If the black hole is actually larger than we think but compressed into a smaller area due to spacetime compression then the tidal forces will be lower than expected.

EDIT: Of course as well as the coordinate value of G we also need to substitute the coordinate value of g.
« Last Edit: 29/05/2015 23:21:40 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #285 on: 13/06/2015 15:17:33 »
Going back to the idea of all masses moving at light speed we then have an equation for kinetic energy of the form d80c0869605cdd5f19dd6464a729cce0.gif. This is obviously neglecting relativistic mass. How is this useful? That is the next step in the analysis.
« Last Edit: 13/06/2015 15:19:06 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #286 on: 28/06/2015 11:27:42 »
During the time that the electromagnetic field was being quantized Planck came up with 43d79f478ff1ce0ca452190448293a79.gif where h is Planck's constant and 5470b9993b5d776db89f25ac7cfff3a1.gif is wave frequency. This was verfied experimentally via Compton scattering and the equation 0d3c0ae7cb5f27f74ddf111fe479f64d.gif. Here c6a6eb61fd9c6c913da73b3642ca147d.gif is the incident wavelength 96ca2e3e3e7dabc48ee7a1d0d06b2067.gif is the scattered wavelength and 2554a2bb846cffd697389e5dc8912759.gif is the scattering angle. Now 6e4315b8b985bb1cebf8fb992dd0e4dd.gif where m is the particle mass and c is the speed of light. What was shown above was (1/2)mc^2. How does this relate? That will be answered next.

EDIT: 9776ff4be626811b4e2dad37d08e929a.gif is the Compton wavelength of the target particle.
« Last Edit: 28/06/2015 11:47:17 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #287 on: 28/06/2015 12:45:55 »
Quote from: jeffreyH
During the time that the electromagnetic field was being quantized Planck came up with 43d79f478ff1ce0ca452190448293a79.gif where h is Planck's constant and 5470b9993b5d776db89f25ac7cfff3a1.gif is wave frequency.
You have to be careful with this. It wasn't Planck who quantized the EM field. That happened when quantum field theory was created much later on. f43d56f4d269389e3158d69322b9d577.gif was postulated by Planck as the quantization condition for harmonic oscillators in a black body. It was Einstein who quantized the electromagnetic wave, i.e. light
Quote from: jeffreyH
This was verfied experimentally via Compton scattering ...
Compton scattering confirmed that light was made of particles.
 

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Re: Lambert's Cosine Law
« Reply #288 on: 28/06/2015 15:17:11 »
Quote from: jeffreyH
During the time that the electromagnetic field was being quantized Planck came up with 43d79f478ff1ce0ca452190448293a79.gif where h is Planck's constant and 5470b9993b5d776db89f25ac7cfff3a1.gif is wave frequency.
You have to be careful with this. It wasn't Planck who quantized the EM field. That happened when quantum field theory was created much later on. f43d56f4d269389e3158d69322b9d577.gif was postulated by Planck as the quantization condition for harmonic oscillators in a black body. It was Einstein who quantized the electromagnetic wave, i.e. light
Quote from: jeffreyH
This was verfied experimentally via Compton scattering ...
Compton scattering confirmed that light was made of particles.

Thanks for that clarification Pete. It makes things clearer for the reader.
 

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Re: Lambert's Cosine Law
« Reply #289 on: 25/07/2015 12:28:41 »
I was going to progress this thread on the theme of light speed particles. However I wish to post an equation without any derivation. This has come from various conclusions which I will get to later. The equation is this.

3a0b9329966dbdf30e82b10a2aa0cc24.gif

Here F is force, mP is the Planck mass, me is the mass of the electron, rs is the Schwarzschild radius of the Plank mass and r is the radial distance of the electron from the event horizon.
 

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Re: Lambert's Cosine Law
« Reply #290 on: 25/07/2015 12:46:56 »
Please note: Before anyone does dimensional analysis and complains that this is not a force equation, there is a parameter missing.
 

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Re: Lambert's Cosine Law
« Reply #291 on: 25/07/2015 13:02:36 »
With the missing parameter (tP) Planck time we can derive the following definite integral.

d6377222ab9950956994b1858279146e.gif
 

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Re: Lambert's Cosine Law
« Reply #292 on: 26/07/2015 20:44:58 »
The equation for F2 is not the same as for gravitational potential energy which is of the form:

ea246cdf364c4a24cec3d37f96b2ee4e.gif

The reason for this is the derivation used to arrive at the F2 form. The major difference is in the power of the radius r.
 

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Re: Lambert's Cosine Law
« Reply #293 on: 26/07/2015 23:57:13 »
If we take the equation for a circular orbit

0a721ee760f19669ad7d291ba33b637d.gif

we can derive the following

3b67b29fe9088c057e67c9e6528364f9.gif

multiplying both sides by m gives

8838ab89e97af24d174a15c1f84dfa4c.gif

dividing both sides by 2

fb6dbae0376611367e7493ea274fdba4.gif

simplifying gives

ed0711ab9d21f917ceb6c37d80e421da.gif

This results in the total combined kinetic energy of the orbit. Under the right conditions and with the right modifications could this be a way to quantize the gravitational field?
 

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Re: Lambert's Cosine Law
« Reply #294 on: 29/07/2015 22:30:28 »
Some of the subtleties and difficulties involved in the quantization process can be seen on the following page.

http://edition-open-access.de/sources/5/25/
 

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Re: Lambert's Cosine Law
« Reply #295 on: 03/08/2015 21:59:26 »
In partial relativistic terms we can start with be905d71a1af157cb287b23035db23de.gif. However we need to modify this to be a3d796b85aea56846659103ebf9ea6f4.gif where M is much larger than m.
 

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Re: Lambert's Cosine Law
« Reply #296 on: 03/08/2015 22:10:15 »
From there we can reformulate first as d03777f2d6071b7c62c8aebb7d6b845f.gif and then as b048880e921439d40230f57a54d10081.gif.

NOTE: This only applies to the Schwarzschild metric and circular orbits.
« Last Edit: 03/08/2015 22:13:10 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #297 on: 04/08/2015 00:04:26 »
When the orbit is coincident with the horizon of the black hole v can take on only two values. Zero and infinite. Whilst infinite is the expected value, zero should also be given due consideration. If we look at the situation where maximum kinetic energy is given by 0686df72ff4b90a1aa0b96ce6d246774.gif and then take v to be half light speed in the orbital equation this then equates to a momentum of 742568d55e925f46f19870b53bc9a11f.gif. If considering the Planck mass a velocity of 1/2c would give as twice the Schwarzschild radius. Therefore in a distance of 2 Planck lengths the velocity has to either change to infinite or zero. In my personal opinion zero is a better bet. Since light speed is 1 Planck length in 1 Planck time. This gives a particles 2 Planck lengths in which to come to a stop. Since it would have to be traveling 1/2 a Planck length in 1 Planck time this seems more reasonable than achieving an infinite number of Planck lengths.
 

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Re: Lambert's Cosine Law
« Reply #298 on: 04/08/2015 00:12:18 »
The crucial point to consider is the transition between r at 2rs and r at 1.5rs.
 

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Re: Lambert's Cosine Law
« Reply #299 on: 08/08/2015 01:15:08 »
There are two types of kinetic energy that can be considered in this model. The energy required to maintain a circular orbit at a particular distance and the energy required to achieve escape velocity at a particular distance. Looking at escape velocity it is interesting to look at the magnitude of change in the field perpendicular to the surface but working towards the surface from a remote radial distance. This way discrete differences can be shown by the relationship of (r+1)^2/r^2 where the units are in integer increments of the objects radius. The graph of this function, which applies to any size of mass, is attached. It starts at 20 radial distances and ends at the objects surface.

 

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Re: Lambert's Cosine Law
« Reply #299 on: 08/08/2015 01:15:08 »

 

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