# The Naked Scientists Forum

### Author Topic: Lambert's Cosine Law  (Read 51859 times)

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #175 on: 27/12/2014 02:03:59 »
For a general wave equation we can show the following progression:

m = g*r^2*h*1/(2pi)*(1/c^2)*[1/lP^2*(gamma*t)/L]

P^2 = 2*Ke*g*r^2*h*1/(2pi)*(1/c^2)*[1/lP^2*(gamma*t)/L]

λ = h/SQRT(Ke*g*r^2*h*1/pi*(1/c^2)*[1/lP^2*(gamma*t)/L])

In the final wave equation the velocity v should be considered the only variable. It is a component of both Ke and gamma. Since we have components such as c^2, h and the Planck length squared it would be interesting to review equations containing these combinations to see if this gives us an insight into gravitation, mass and quantum mechanics.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #176 on: 27/12/2014 03:17:29 »
As a starting point for an angular momentum formula we need to review the following.

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qangm.html

Another useful reference is on wikipedia.

http://en.wikipedia.org/wiki/Azimuthal_quantum_number
« Last Edit: 27/12/2014 03:52:56 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #177 on: 28/12/2014 18:15:30 »
To develop our equation of angular momentum we must first review an earlier post.

"Before trying to link the gravitational field to Lambert's Cosine Law I need to take a detour. This starts with the unit sphere and the unit circle. Using the unit sphere and circle shows some interesting relationships and can be scaled up. This can then be used to describe both subatomic and macroscopic domains.

The circumference of the unit circle is 2*pi. To determine the angle of an arc around the circle whose arc length is equal to the radius we can use (1/2*pi)*360 which can be simplified to 7/44*360. This proportionality will become important when viewing interactions at differing scales and relates to wave frequency, length contraction and time dilation effects. The angle we have determined can be converted to radians to use in calculations.

It is interesting to note that the period of sin x is 2*pi. This can be utilized by considering forward motion and angular rotation as it relates to the unit sphere. The relationship between these two properties can describe the evolution of a wave and can be related directly to the gravitational field. When used it can be shown to show the underlying mechanism of the Pauli Exclusion Principle and the difference in energy levels required between electrons.

There are 3 directions of motion under consideration within this model. One motion is forward direction and is considered to be aligned with the poles of the sphere. The two other directions are angular. The first is around the equator and the second follows a longitudinal path intersecting both poles. The maximum unit of motion in unit time in the polar direction is equal to the unit sphere radius. The maximum unit of motion of the angular paths is 7/44*360 as stated above. If viewed at the Planck scale the angular components cannot reach this speed or none of us would be here. Therefore we can deduce that this dampening in angular momentum must be due to gravity which is what the current physical theories state.

If we follow this line of thinking through to its conclusion we can show that when considering the universe as a whole system light might get infinitesimally near to c but will never actually reach it as long as any gravitational field remains. I will demonstrate the reasons for this conclusion as I proceed."

The angle derived above becomes important only when the radius of an object is 1 Planck length. As this will only apply in some string theories we can disregard it. Not only because it relates to string theory but it relates to light speed which angular momentum should never reach. This will be discussed later.
« Last Edit: 28/12/2014 18:27:59 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #178 on: 29/12/2014 15:48:42 »
In examining angular momentum it is not sensible to have Planck units. The best units to select would be nanometres and nanoseconds. Thus the speed of light can be represented by L*1 nanometre divided by 1 nanosecond. We can then apply a factor to L to determine a non-relativistic speed. Now we may have a problem as the angular momentum of a particle is L = r x mv. The two values for L, 1 light second versus L for angular momentum mus NOT be confused. L = r x mv is the cross product mv the linear momentum and r which is the point of rotation. This is best illustrated in the animation on wikipedia.

http://en.wikipedia.org/wiki/Angular_momentum
« Last Edit: 31/12/2014 14:40:53 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #179 on: 01/01/2015 00:12:17 »
I have decided to temporarily suspend posts to this thread until I have reviewed the following.

http://arxiv.org/abs/1412.4515

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #180 on: 02/01/2015 00:14:46 »
What if gravity or any other such force which is considered inverse square is not that at all but appears that way the further from the source and at macroscopic scales. If instead the law was 1/r^(1+1/n) and n started at an as yet undetermined value > 1 and approached 1 but only got there at infinity then we would have an entirely different situation microscopically where gravity starts as a 1/r law and then changes over time to become nearer and nearer to a 1/r^2 law. What if we started the value of n at 50 as an arbitrary assignment?

EDIT: This may validate the dark matter halo hypothesis.
« Last Edit: 02/01/2015 00:21:21 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #181 on: 03/01/2015 02:42:28 »
In one of the papers by Roberto Onofrio the Schwarzschild radius of the elementary particles becomes boosted by 33 orders of magnitude. As light should still be trapped then the energy of the gravitational field should become boosted by an equivalent factor. So we could replace the value of 50 by 33. If we stay with 50 for now then the attached graph shows the change in gravitation with respect to an earth sized mass. The radius and mass are held constant while the power of the radial term moves from 1/r^(1+1/50) to 1/r^(1+1/1). At the left of the x axis the situation mimics a Planck mass situation where the whole mass can be treated as a single particle. At the right of the x axis we return to our 9.81 value of g where the mass is made of separate particles. The x axis has been graded so that the intervals between successive points are more closely spaced at the source and separate with movement to the right. This is the only way to achieve a sensible scale and illustrates the change from 1/r to 1/r^2 as a gradient.

EDIT: The x axis is derived by starting with a value of 50/50 at the left and ending at 50/1 at the right. This does not mean that this is valid in this scenario. It was a quick experiment with the values.
« Last Edit: 03/01/2015 02:51:14 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #182 on: 03/01/2015 03:17:52 »
This would of course imply a decrease in energy of the gravitational field the further from the source it is. As stated above the intervals on the x axis were derived. If we remove the derivation then we get the attached plot which does show a very different profile altogether.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #183 on: 04/01/2015 23:38:56 »
We can carry out a very rough guestimate on our first value by using the mass of the hydrogen atom and the mass of the earth. We then get a factor of Mh/Me where Mh is the mass of the hydrogen atom and Me is the mass of the earth. Our first value for g was 38913462.45 so we end up with (6.7e-28/5.97219e24)*5.97219e+24. Our result is 4.36557e-45 for g. However this is a 1/r relationship and does not describe an acceleration in this form so the actual acceleration will be much less. If we apply the same proportionality to the radius of the earth as we did to the mass we can use this in our next guestimate.

EDIT: We would need to obtain the square root value of the reduced radius as a correction. If this value does not match a reasonable radius for the hydrogen atom then this invalidates this method entirely. This will need to match the Bohr radius within a certain range of precision.
« Last Edit: 04/01/2015 23:42:55 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #184 on: 05/01/2015 00:07:05 »
What we get is 2.67495e-23 metres which is 31 orders of magnitude larger than the Schwarzschild radius of the proton. Since Roberto Onofrio is suggesting a particle's Schwarzschild radius is boosted by 33 orders of magnitude this is a very surprising result and only 1 order of magnitude out.

CORRECTION: This should state a 2 order of magnitude difference.
« Last Edit: 10/01/2015 18:12:52 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #185 on: 05/01/2015 00:19:30 »
The attached graph is a log/log plot of the evolution of the variation in the profile of the gravitational field in terms of induced acceleration. The values for mass and radius are still those of an earth sized mass and no scaling down has been attempted.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #186 on: 06/01/2015 21:43:37 »
The scale at which the forces would be acting is the attometre. This is 10^-18 metres which is 5 orders of magnitude greater than the boosted proton Schwarzschild radius of the proton. Our scale could be defined from the boosted radius up to the currently accepted proton radius which is in the femtometre range 10^-15. The proton radius is around 0.84–0.87 fm in size. Can an equation be derived that shows a morphing of gravitation between the microscopic and macroscopic domains? This is the key question. It will not be like the previous guestimate.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #187 on: 07/01/2015 03:38:51 »
The next step is the investigation of the gravitational coupling constant. Details can be found on wikipedia.

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

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #188 on: 09/01/2015 02:18:39 »
A short detour is worth taking before continuing. Consider a point P which is at an infinitesimal distance outside the event horizon of a black hole. At point P a photon is emitted in a direction directly away from the gravitational source. Since gravitation will slow light in its field and the the escape velocity is infinitesimally near c at point P then the emitted photon will not have the energy to achieve a velocity equaling c. Since it cannot reach the escape velocity it should therefore lose kinetic energy as it moves away from the source. The question that arise is does this mean that the photon is therefore trapped and will eventually stop and reverse direction. Since, without any other forces acting on the photon, it will only experience a constant velocity it cannot accelerate to escape the gravitational field. This is an important consideration when investigating the ergosphere surrounding a black hole. This is suggestive of a zone outside the horizon that can still potentially trap photons. Since it has been shown experimentally that light can be slowed substantially when moving through certain mediums it may be that the intensity of a gravitational field can mimic such a medium. Infalling photons that are not directed at the source may well fall into near-horizon orbits. Being perpendicular to the direction of the gravitational field may allow photons to reach a velocity infinitesimally close to c. However this may not be the case at all.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #189 on: 09/01/2015 02:41:51 »
It is useful to read the following article.

http://news.harvard.edu/gazette/1999/02.18/light.html

The extremely low temperature of the medium will effectively transfer kinetic energy in the form of heat. This implies that the ergosphere around a black hole is acting like a bose-einstein condensate. This can come about via infalling matter achieving this state with low vibrational energy. This implies a uniform change in acceleration on the particulate matter within the ergosphere so that no heat is generated and no force is felt. Since kinetic energy is equivalent to a potential heat transfer the field, or something, must be taking up this heat. Either that or x-rays with enough energy are effectively escaping the ergosphere and taking this heat away. However, what if there is no matter close enough to the black hole to be falling into the ergosphere? In this case there will be no medium through which light can be effectively slowed down. In the case of the encounter of gas cloud G2 with sag A* it may simply be the case that the black hole was starved of matter with which to generate a sufficiently dense medium with which the gas would react.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #190 on: 10/01/2015 18:25:25 »
Consider the whole surface of the horizon of a rotating black hole. Because of this rotation frame dragging will carry matter around the equatorial disk. Velocities will be high. However at the poles this velocity is absent and therefore makes it easier for matter to enter the black hole's event horizon at those regions. This could result in energy release in the form of photons in all directions. Some will fall into the black hole at the pole. Some will initially form eccentric orbits across the horizon while others will be propelled away from the poles. The jets away from the poles being intense will carry some matter with them in the form of maybe gas molecules or nuclei and separate electrons.

NOTE: Due to entanglement a proportion of the photons expelled via the jets could provide information on entangled photons falling into the black hole.
« Last Edit: 10/01/2015 18:47:58 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #191 on: 10/01/2015 19:11:40 »
If we investigate path profiles around a rotating sphere it could be possible to construct a model of this scenario. This can then be related to smaller scales. Ultimately it can be applied to the particle itself. The following is a good point in the history of particle physics to start.

http://en.wikipedia.org/wiki/Spin_(physics)#History

"Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Wolfgang Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can share the same quantum state at the same time.

The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig, one of Landé's assistants, suggested in early 1925 that it was produced by the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the speed of light in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the theory of relativity. Largely due to Pauli's criticism, Kronig decided not to publish his idea.

In the autumn of 1925, the same thought came to two Dutch physicists, George Uhlenbeck and Samuel Goudsmit at Leiden University. Under the advice of Paul Ehrenfest, they published their results. It met a favorable response, especially after Llewellyn Thomas managed to resolve a factor-of-two discrepancy between experimental results and Uhlenbeck and Goudsmit's calculations (and Kronig's unpublished ones). This discrepancy was due to the orientation of the electron's tangent frame, in addition to its position.

Mathematically speaking, a fiber bundle description is needed. The tangent bundle effect is additive and relativistic; that is, it vanishes if c goes to infinity. It is one half of the value obtained without regard for the tangent space orientation, but with opposite sign. Thus the combined effect differs from the latter by a factor two (Thomas precession).

Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics invented by Schrödinger and Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators, and introduced a two-component spinor wave-function."

Pauli's objections were overcome but should they have been? This will be investigated on the way to our wave interaction equation.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #192 on: 11/01/2015 00:32:02 »
Another useful piece of history to review is this:

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

"Llewellyn Hilleth Thomas (21 October 1903 – 20 April 1992) was a British physicist and applied mathematician.[1] He is best known for his contributions to atomic physics, in particular:

Thomas precession, a correction to the spin-orbit interaction in quantum mechanics, which takes into account the relativistic time dilation between the electron and the nucleus of an atom.

The Thomas–Fermi model, a statistical model of the atom subsequently developed by Dirac and Weizsäcker, which later formed the basis of density functional theory.

Thomas collapse - effect in few-body physics, which corresponds to infinite value of the three body binding energy for zero-range potentials."

Particularly this:

"While on a Traveling Fellowship for the academic year 1925–1926 at Bohr's Institute in Copenhagen, he proposed Thomas precession in 1926, to explain the difference between predictions made by spin-orbit coupling theory and experimental observations."

A section on the Thomas interaction energy can be found here:

http://en.wikipedia.org/wiki/Spin–orbit_interaction

This includes a Lorentz factor for a moving particle. This is an important consideration for interacting wave equations, especially with regard to changes in a gravitational field. Time dilation can be related directly to this factor.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #193 on: 11/01/2015 23:33:48 »
If we return to the poles of the black hole it should be true that a higher proportion of Hawking radiation will be generated here than elsewhere along the surface of the horizon. The interaction of particle wave ensembles away from the poles should be a good area to investigate for particle interactions with gravitation within an intense field. If entanglement is prominent here we may also be able to determine some aspects of the behavior of particles having fallen through the horizon in these polar regions. A study of the profiles of many polar jets should give valuable data in this regard.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #194 on: 12/01/2015 06:01:45 »
Now if we want a balanced wave equation we first restore the factors of 2.

This:

λ = h/SQRT(Ke*g*r^2*h*1/pi*(1/c^2)*[1/lP^2*(gamma*t)/L])

Becomes:

λ = h/SQRT(2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L])

Then we need h on the right hand side so:

λ/h = 1/SQRT(2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L])

Inverting we get:

h/λ = SQRT(2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L])

Squaring:

(h/λ)^2 = 2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L]

Multiplying by 2pi:

2pi(h/λ)^2 = 2Ke*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]

Finally:

2pi = (λ/h)^2*2Ke*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]

We can now adjust g, r, wavelength or gamma*t but they must all balance in order to conserve the equality. This involves conservation laws somehow. Which ones, maybe all, I don't know yet. I may easily have made a mistake here so beware. This does not take into account the effects of an external g force but since 2pi is the circumference of the unit circle a multiplication be ge (external g) will result in either an expansion or contraction of radius depending upon where it is less than  or greater than 1. Being based upon metre units this is the correct pivot point.

NOTE: Since an increase in the unit radius would imply a slower spin rate then as the gravitational field intensifies this will correlate with time dilation. In this form the equation says nothing about a change in radius with increasing velocity.
« Last Edit: 12/01/2015 06:30:16 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #195 on: 13/01/2015 22:51:46 »
The reciprocal of momentum 1/p = λ/h. This may relate somehow?? to phonons or even the reciprocal lattice.

See crystal momentum:
http://en.wikipedia.org/wiki/Phonon

http://en.wikipedia.org/wiki/Reciprocal_lattice
"Simple cubic lattice

The simple cubic Bravais lattice, with cubic primitive cell of side a, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side 2pi/a (1/a in the crystallographer's definition). The cubic lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space."

How this could be incorporated is beyond me currently. It is simply noted here.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #196 on: 15/01/2015 21:16:26 »
We have two interesting components to the unit circle equation. Ke = 1/2mv^2 and (λ/h)^2 = 1/p. Since mv = p we can define Ke as 1/2v(mv). That is 1/2v(p). By combining both we arrive at 1/2v(p)1/p which becomes 1/2v.

So we can now reformulate the equation as 2pi = 2*1/2v*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L] from which we then get 2pi = v*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]. So we now have the particle velocity included in our rearranged formula. This eliminates the kinetic energy altogether. We now have velocity over a time dilated interval. Now 4 components can change and are all related directly. These are the velocity of the particle, the change in time over which the velocity occurs (time dilation as viewed externally), the radius and associated g force at the surface defined by the radius. Do these need to balance to maintain a relationship with the 2pi? That I have not determined yet. However the equation is becoming simpler. It does suggest a tie in with black hole entropy with the 1/lP^2 factor.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #197 on: 15/01/2015 23:30:44 »
If we look at the following sequence rearranging for v we can see that velocity is also contained within the gamma factor.

2pi = v*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]

2pi/v = g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]

v/2pi = 1/g*1/r^2*1/h*c^2*[lP^2*L/gamma*t]

v(apparent) = 2pi*([L^3]/[gamma*t^3*g*h]*[lP^2]/[r^2])

Here v(apparent) is not necessarily the same as v in gamma. The values for earth of g and r can be plugged in to see what results are obtained at various velocities inside the gamma factor. This is the next step.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #198 on: 16/01/2015 19:37:51 »
Well I think that I can safely say that the above equation is rubbish.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #199 on: 22/01/2015 20:40:26 »
Having scrapped the rubbish in post 197 I can now investigate whether or not the earlier wave equation is valid. Firstly I want to point out a conclusion I have reached. If we have two perfect spheres identical in every respect; same radius, surface area, volume and mass, then on a direct line between the two centre's of gravity the force can be said to cancel as it is equivalent and opposite in direction. From this midpoint on the line between the centres we can describe a plane perpendicular to the line on which gravity will cancel at any point as the forces betwwen the masses will cancel. However the vectors will mean that any particle not exactly positioned on the adjoining line will tend to describe a straight line path along this plane until it reaches equilibrium again at the point where the plane meets the line. This is equivalent to an object dropped down a shaft through one perfect sphere that goes from one surface, through the centre of gravity and reverses direction at the opposite surface. Both of these scenarios can be said to be operating in a flat spacetime as long as the only forces present are the masses concerned.

The two sphere scenario raises another point. If we consider the sources to be two black hole an infinitesimal distance apart at the event horizons with enough distance to hold a particle between them without touching either horizon then because the gravitational force operates equivalently and in opposite directions there should be extreme length contraction which also implies extreme time dilation. So in the situation at the centre of the earth we should have a maxima of time dilation for the size of mass of the earth. This also indicates a compression of a central particle in all directions indicating that this is a proof that black holes do in fact form at the centre of a mass and work outwards.
« Last Edit: 22/01/2015 20:43:25 by jeffreyH »

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##### Re: Lambert's Cosine Law
« Reply #199 on: 22/01/2015 20:40:26 »