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Author Topic: Lambert's Cosine Law  (Read 51535 times)

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #225 on: 13/02/2015 23:29:27 »
If we rewrite v = SQRT(Gm/r) as s/t = SQRT(Gm/r) then we can eliminate the square root (s/t)^2 = Gm/r. This can be restated as (s/t)(s/t) = Gm/r. If we move one of our distances (s) to the other side it becomes (1/t)(s/t) = Gm/r * 1/s. The value of 1/s then relates to the tangent to the orbit that moves us through an angle to a higher radial position. So we have s/t^2 = Gm/r * 1/s. By multiplying both sides by 1/t we then have an equation for jerk s/t^3 = Gm/r * 1/s * 1/t. Of course we really need to be looking at ds, d^3t and dt in these in this case.
 

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #226 on: 14/02/2015 00:07:28 »
Of course if we did not want to maintain our perfectly circular orbit we could simply remove the accelerating force and allow an elliptical orbit to develop. This leads to the thought that maybe electron orbitals have a lot in common with elliptical orbits.
 

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #227 on: 14/02/2015 01:23:20 »
If we consider we have an acceleration (a) and a jerk (j) our final equation forms become:

a = Gm/dr * 1/ds

j = Gm/dr * 1/ds * 1/dt

With our final velocity after deceleration :

v = SQRT(Gm/r)

We also have an adjusted mass equation:

m = kQ/(Er^2) * G/(gr^2)

This differs from what is usually understood as mass as the electric field and charge components have been added in. So this is a composite mass equation. An investigation of substituting this into the velocity equation would be interesting.
 

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #228 on: 14/02/2015 02:01:52 »
If we consider a large central mass with a smaller mass in orbit and both perfect spheres with a unifomr mass density we can say something about our velocity equation. Perpendicular to the field of the larger central mass we have a gradient of velocities that describe concentric circular orbit. Throughout the volume of the smaller mass these orbital profiles have differing effects from the nearest to the furthest away. At the point in the smaller mass that is nearest to the central body the velocity is at its peak speed and dies away at the surface furthest away. If the centre of gravity of this smaller mass passes exactly along the orbit at the correct velocity then those portions of the mass further away are traveling at a speed that would take a smaller mass out of the orbit. At the nearest point of the smaller mass to the larger body the speed is lower than needed to maintain an orbit at that altidude and a smaller mass at that point would tend to fall out of orbit towards the larger body. Therefore an induced angular momentum of rotation could be induced by such an imbalance. The point nearest the larger mass would be retarded in motion whilst the furthest point will advance in its motion. This is without taking into account any time dilation. Even in elliptical orbits this would be true.
 

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #229 on: 14/02/2015 02:26:47 »
It is interesting to note that there are already equilibrium points known as the Lagrange points. At these points in the 3 body situation objects will maintain an orbit and even a small deviation from equilibrium will be corrected to bring a mass back to equilibrium.

Details can be found here:

http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/lagpt.html


EDIT: Equivalent points should be present throughout the universe due to the interaction of the gravitational fields of galaxies.
« Last Edit: 14/02/2015 02:30:23 by jeffreyH »
 

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #230 on: 14/02/2015 22:25:21 »
In moving away from a stable circular obit we need an acceleration. However, acceleration tells us nothing about the mass that is moving. It tells us nothing about the force that needs to be applied to accelerate the mass. Different sized masses can not be moved away from equilibrium at exactly the same rate of acceleration by applying the same force. Different forces are needed. This says something about the propagation of force through mass. When considering free falling objects of differing masses in the same gravitational field this is not the case. It is always the same force. This indicates an action at the particle level right through a mass and acting simultaneously.
 

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #231 on: 17/02/2015 01:00:15 »
New thought.

x = [kQ/(Er^2) * G/(gr^2)] - [kQ/(Er^2) + G/(gr^2)]

Don't know where this leads yet. I need to look at some particle data. Not sure what x will actually mean.

 

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #232 on: 18/02/2015 01:26:19 »
The two fundamental equations for the de Broglie wave are:

c6a6eb61fd9c6c913da73b3642ca147d.gif= h/SQRT(2*Ke*m)

5470b9993b5d776db89f25ac7cfff3a1.gif= Ke/h

In both equations the energy is kinetic represented here by Ke. This is energy added to the system. Kinetic energy itself has the formula Ke = (1/2)mv^2. We saw earlier that momentum p = mv and that c6a6eb61fd9c6c913da73b3642ca147d.gif= h/p. The momentum is the motion of mass though space and the kinetic energy is a result of this motion. If we consider time dilation and its effects upon motion over time we should be able to derive equations that describe the changes in wavelength, frequency and kinetic energy in any gravitational field. What we also need is the effect on the particles electromagnetic field.

As I continue this thread I will be introducing relativistic mass.
« Last Edit: 18/02/2015 02:12:33 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #233 on: 19/02/2015 01:19:10 »
A short digression before I progress this thread. Hawking radiation is due to pairs of particles, a particle and it's anti particle. One of these falls through the horizon whilst the other escapes. In the case of an anti particle falling through the horizon it should, if possible, be able to annihilate with a particle in the interior. If the particle falls in, however, then the universe loses mass. The particle will simply become part of the singularity. This may not be true, I don't know. However, whether a black hole grows or shrinks has profound consequences. I will be following up on this next.
 

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Re: Lambert's Cosine Law
« Reply #234 on: 19/02/2015 01:30:35 »
When plotting change of wavelength of the photon that escapes just outside a black hole horizon there may be a linear relationship between frequency and the speed of the photon as viewed from infinity. The graph of this situation is attached.

With the frequency held constant and the speed of the photon plotted as viewed from infinity we have a linear plot. If we were to add a gamma factor, which I believe is incorrect, the photon will lose all its energy at infinity.

This plot brings to mind the Hubble data. If we consider that the black holes at the centres of galaxies are constantly changing, either shrinking or growing, then the shift that they cause in photon wavelength may also be changing in as yet undetected way. This may be regular across all such black holes and may follow a law of its own. In which case the expansion of the universe may be biased away from its true rate.

It all depends upon how these black holes behave. The black hole sag A* may be able to help answer such questions. In the end the red shift may be only very marginally affected by such black holes. Only time will tell.
« Last Edit: 19/02/2015 01:33:10 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #235 on: 19/02/2015 01:38:15 »
For comparison I have attached the plot with the gamma factor.
 

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Re: Lambert's Cosine Law
« Reply #236 on: 21/02/2015 01:15:50 »
If we re-examine the factors kQ/(Er^2) + G/(gr^2) these can be re-arranged as:

kQg/(Egr^2) + GE/(Egr^2)

This is an interesting set of relationships. The charge is paired with gravitational acceleration and the Electric field is related to the gravitational constant. Since the charge is involved in the attraction and repulsion forces and g is a consequence of the attractive force of gravitation this is worth further investigation. It is also of note that the denominator has the electric field and gravitational acceleration as factors. The units still need to be sorted for validity and this is what shall be undertaken next. This is not a mass equation by any means. It is something entirely different. If this is valid the value should be determined by unit cancellation. What we have left will guide the rest of the investigation.
 

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Re: Lambert's Cosine Law
« Reply #237 on: 21/02/2015 04:01:33 »
In the equation x = [kQ/(Er^2) * G/(gr^2)] - [kQ/(Er^2) + G/(gr^2)] when we remove the gravitational and electric components we may be revealing the magnitude of the quark energy hidden by confinement. This should be taken as pure speculation because it is in no way validated. The equation E = mc^2 defines the energy we can detect. It is the confined energy that is hidden from immediate view mathematically. The jury hasn't even deliberated on this one.
 

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #238 on: 21/02/2015 17:03:20 »
If we consider an individual particle interacting with a perfect sphere we can then say that at the surface of the sphere the particle will experience a force represented by Sg. For a perfect sphere this will apply at any point on its surface. We can then say that the total potential around that surface is Sg*4*pi*r^2. We could then calculate the surface area the sphere would have if contained within its Schwarzschild radius rs. We can then define this as As. The surface of then sphere at normal density is A = 4*pi*r^2. If we consider this as a ratio we can say As/A is a starting point with A/As representing the magnitude of A with respect to As. Can we then say that Sg*A/As represents the true increase in force?
« Last Edit: 21/02/2015 17:12:59 by jeffreyH »
 

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #239 on: 23/02/2015 02:18:54 »
Looking again at kQg/(Egr^2) + GE/(Egr^2) what does the numerator kQg + GE tell us? We have Coulomb's constant incorporating the speed of light, magnetic permeability and electric permittivity. This produces a product with the charge and the gravitational acceleration. Added to this is the electrostatic field strength as a product with the gravitational constant. For a particle such as the proton, what values do we get? Do these balance in any way? That is the next test.
 

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Re: Lambert's Cosine Law
« Reply #240 on: 27/02/2015 15:49:25 »
Enough outrageous speculation. Let's get down to business. If we consider the wave as circular rather than sinusoidal we can describe the total velocity by (2*pi*1/4*w*f)/t. Here w represents wavelength and f represents frequency. This is an arbitrary choice of variables. So now we have an angular velocity over time. However if we look at the Lagrangian bd5aabbac9bc3e5a0bc6cd4eae2fe473.gif this is unsuitable as the velocity is not angular. As the wave moves through space it is stretched and so there needs to be a combination of both angular and forward velocity to produce a combined velocity whose momentum can be used in a modified Lagrangian. Tying the two velocities together in this manner and with the frequency held constant we can show how the system will evolve when viewed by a remote observer. This also ties the time dilation directly to the internal actions of the particle. We can thus show the evolution of of the change in the rate of change of a system moving into a gravitational field.

NOTE: This is why a flat spacetime is crucial to the development of these equations.
« Last Edit: 27/02/2015 15:52:19 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #241 on: 27/02/2015 16:29:46 »
It should not be assumed that the forward velocity and angular velocity change in unison. Maybe for the photon this can be true but this exercise is not concerned with photons. We cannot examine the effects of gravitation if these velocities are exactly proportional.
 

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Re: Lambert's Cosine Law
« Reply #242 on: 27/02/2015 16:52:21 »
Jeff! What in the world are you doing in this thread? It's as if you're having a very long drawn out conversation with yourself. What gives?
 

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Re: Lambert's Cosine Law
« Reply #243 on: 27/02/2015 18:37:33 »
I am just using it to put ideas down. Don't worry I haven't gone bonkers.
 

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Re: Lambert's Cosine Law
« Reply #244 on: 28/02/2015 00:39:01 »
If z is the axis of the path of the particle we can derive the equation:

514b00fedf907ba3ea467f653965f452.gif
« Last Edit: 28/02/2015 01:17:36 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #245 on: 28/02/2015 01:26:23 »
Okay. You remind me or yor_on when you do this.  [^]
 

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Re: Lambert's Cosine Law
« Reply #246 on: 28/02/2015 01:37:52 »
So I'm in good company then  ;)
 

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Re: Lambert's Cosine Law
« Reply #247 on: 28/02/2015 04:45:02 »
In the equation x = [kQ/(Er^2) * G/(gr^2)] - [kQ/(Er^2) + G/(gr^2)] when we remove the gravitational and electric components we may be revealing the magnitude of the quark energy hidden by confinement. This should be taken as pure speculation because it is in no way validated. The equation E = mc^2 defines the energy we can detect. It is the confined energy that is hidden from immediate view mathematically. The jury hasn't even deliberated on this one.

Interestingly a reduced mass formulation would be better for this equation. Especially when considering the effects of gravitation.

http://en.wikipedia.org/wiki/Reduced_mass

The equation then becomes x = [kQ/(Er^2) * G/(gr^2)] / [kQ/(Er^2) + G/(gr^2)]
« Last Edit: 28/02/2015 04:46:51 by jeffreyH »
 

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Re: Lambert's Cosine Law
« Reply #248 on: 28/02/2015 20:02:31 »
There is something interesting we can do with the gravitational acceleration equation. This equation is:

g = Gm/r^2

This is like using a reduced mass term as we have neglected the second mass. If we rearrange this equation we can arrive at:

r^2/G = m/g

We know that force is F = ma and as g is an acceleration we can form F = mg. If we rearrange to relect this we get:

r^2G = mg

But r^2G does not equal mg as the form r^2/G = m/g represented a proportionality.

This can be illustrated via

1/2 = 2/4

1*2 <> 2*4

1*2*x = 2*4

In this case it is easy to determine that the missing factor is 2^2

However if we investigate the units of both sides of the gravitational acceleration equation something interesting can be found.

G units = m^3 kg^-1 s^-2 cubic metres per kilogram second squared

left side units = m^5 kg^-1 s^-2

mg units = m kg s^2 metres per second squared

missing units kg^2 m^-4 is surface density squared

So we can make the equations equal with the right value for surface density squared as:

r^2Gd^2 = mg

Where d is the surface density squared. The value of d here is undetermined. However, calculating its value is very important. This indicates that it is only the surface density and not the mass as a whole that generates the force of gravity. This also explains the event horizon surface area relationship to entropy. The remaining gravitational energy is internal to the mass.
« Last Edit: 28/02/2015 20:06:05 by jeffreyH »
 

Offline jeffreyH

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Re: Lambert's Cosine Law
« Reply #249 on: 01/03/2015 17:06:49 »
So if the force is related to surface density, then as we increase the radius away from the surface of the gravitating object what density are we then relating to? It must still be the surface density of the gravitating mass but as it dies away in an inverse-square manner can this tell us anything about the density of the gravitational field at points away from the source? In other words, can we quantize gravitation?
 

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Re: Lambert's Cosine Law
« Reply #249 on: 01/03/2015 17:06:49 »

 

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