Localization of the Particles within a Thin Spherical Shell

Outside the cluster, the gravitational field is represented by Schwarzschild’s solution which, with our choice of the coördinate system, is given by (1)

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Inside the cluster, the field is determined by (14). Thereby, the function n is to be considered as given. However, n is not completely arbitrary, as the total radius of the cluster is restricted by the lower limit given by (6a).

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Equation (14) represents a complicated relation between the particle density n and the function a representing the gravitational field. The limiting case, however, in which the gravitating particles are concentrated within an infinitely thin spherical shell, between rro A and rro, is comparatively simple.

This case could only be realized if the individual particles had the rest-volume zero, which cannot be the case. This idealization, however, still is of interest as a limiting case for the radial distribution of the particles.

We divide the whole space into three zones for separate consideration, part O to be the part outside the shell, r≥ ro, part I to be the part inside the shell, Tro A, and part S to be the part of the shell ro - Arro.

In O, the gravitational field is represented by (1), in I, it is represented by (2) with constant values of a and b.

It follows that a’ (and a’) have to change within S the faster the smaller A is chosen.

However, as a’ remains finite in S, a itself changes only infinitely little in S.

It is, therefore, permissible in S to neglect a’ compared with a".

We therefore replace (14) within S by

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where a and r are to be treated as constants for integration purposes. We introduce the variable and the “constant”

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and obtain the equation

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z is hereby determined as a function of r within S if n is given as a function of r. When the integration is carried out between ro A and ro we obtain

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where N designates the number of particles in S. It follows from (1) that for

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and from (2) that, because of a and b being constant in I, in I

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It follows from (6a) that

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It turns out that this is just the condition for the numerator of the expression for z,, to be real. (15), for each possible ro, gives the relationship between the sum of the masses of the particles, mN, and the total gravitating mass μ of the cluster.

For large values of ro, with a fixed value of μ, one obtains in the limit:

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The factor </8 is due to the fact that m is measured in grams, 44, however, in gravitational units. (16) therefore simply states that in this limiting case the gravitating mass of the cluster is equal to the sum of the particle masses. The most illuminating way to express this result is the following: Outside the shell (r ≥ro), the gravitational field is given by

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Inside the shell it is given by the same expression, with the difference, however, that r is to be replaced by the constant ro, whereby the inequality must be satisfied.

The number N of particles of the mass m which together form the shell is given by the following consideration: As an abbreviation we introduce

Then we have:

where

☛ can assume values between 0 and 2-3(~27). The quantity is only very little different from zero in this whole region. A few typical values are given in the following table:

This leads to a very interesting consequence: First it is clear that (a)/0 may be replaced by (2)/2 with good approximation and this by (M)/M. This latter quantity is the relative decrease of energy of the cluster when it contracts from an infinite radius to the radius ro. The table shows that this contraction energy has a maximum near σ = 0.15, and for greater values of σ, i.e. smaller values of ro, it decreases again. The physical cause of this effect is that, with decreasing ro, the potential energy of the cluster decreases, but the kinetic energy increases. For sufficiently small values of ro the latter effect surpasses the former.

It is therefore clear that the decrease of the radius with decreasing energy would come to an end for a value of about 0.15, i.e. a radius of about 6.7(/2ro), while the lower limit of the radius as given by the velocity of light is (2+3) (u/2ro). The value of r corresponding to the minimum energy means an upper limit for the particle velocity in the direction of the tangent of about 0.65 times the light velocity.