Translation from "Allgemeine geographische Ephemeriden, Bd I, 1799

Proof that the attractive power of a heavenly object can be so large that light cannot be emitted from it

P. S. LAPLACE

 

(1) The speed v, with which a distance s is traversed during a time t, is well known to be v=s/t.

(2) If the motion is not uniform, the speed v is computed at a certain instant as the quotient of the distance ds and the time interval dt, since the speed during an infinitessimally short interval dt is constant, and the motion can therefore be considered as uniform during this interval .

(3) A constant force acting tends to change the speed, and the change of speed is therefore the most natural measure of the force. However, every force causes twice as large an influence during twice the time, and the change of speed dv must be divided by the time dt during which the force acts. This gives the following general expression for the force P

P = dv/dt = d(ds/dt)/dt

If dt is constant, this yields

d(ds/dt) = d(ds/dt)/dt

or

P = dds/dt2

(4) Consider that a body has a mass M; another body, eg a light particle, is at a distance r. The mass M then influences the light particle with a force -M/rr, where the minus sign indicates that the attractive power from M acts against the direction of motion of the light.

(5) According to (3) this force is also equal to ddr/dt2, so

M/rr = ddr/dt2 = -Mr-2

Multiplication with dr gives

drdr/dt2 = -Mdrr-2

and integration

(1/2)dr2/dt2 = C + Mr-1 where C is a constant, or

(dr/dt)2 = 2C + 2Hr-1

where v is the speed of the light particle at a distance r.

(6) For the determination of the constant C, we denote R as the radius of the attracting body and a as the speed of light at a separation R. that is, at the surface of the body. According to (5) then a2=2C+2M/R, ie, 2C=a2-2M/R. Substituting this into the equation above yields

v2 = a2 - 2M/R + 2M/r

(7) Assume now that another body has a radius R' and mass iM, and that the speed of light at a distance from this body is v'. According to the equation in (6), one has

v'2 = a2 - 2iM/R' + 2iM/r

(8) If we make r infinitely large, the last term in the previous equation vanishes and we obtain

v'2 = a2 - 2iM/R'
The distance to the fixed stars is so large that this assumption is justified.

(9) The force of attraction of the other body shall be considered to be so large that light cannot be emitted from it. Analytically this can be expressed as follows: the speed of light v' is zero. Setting v' equal to zero in equation (8) yields an equation by which the value of iM can be determined. One therefore has

0 = a2 - 2iM/R' or a2 = 2iM/R'

(10) In order to determine a, we assume that the first-named attracting body is the sun, so that a is the speed of the sunlight at the surface of the sun. The solar attraction is, however, so small in comparison with the speed of light, that this speed can be considered as constant. From the phenomenon of aberration it is concluded that the earth travels 20"1/4 in its orbit while light travels from the sun to the earth. If V denotes the average speed of the earth in its orbit, one has a:V=1:tan(20"1/4).

(11) In Exposition du systems du monde, part II, page 305, I took R'=250R. Since mass is volume multiplied by density and volume is proportional to the cube of the radius, the mass is proportional to the cube of the radius multiplied by the density. Setting the density of the sun equal to 1 and that of the other body equal to p, one therefore has

M :iM = lR3 : p R'3

or 1 : i = 1 : p (250)3

or i = (250)3 p

= 1 R3 : p 2503 R3

(12) Substituting the values of i and R' in the equation a2=2iM/R', one finds

a2 = 2(250)3pM/250R = 2(250)2pM/R
or
p = a2R/[2(250)2M

(13) In order for p to be determined b' must be known. The force from the sun at a distance D is M/D2 ; and if D is the average distance to the earth V the speed of the earth, then this force can be expressed as V2/D (see Lande's Astronomic III, paragraph 3539). Thus M/D2= V2/D or M=V2D which, substituted into equation (12) gives for p

p = a2R/(250)2V2D = (8/(1000)2)(a/V)2(R/D)

a/V = [speed of light]/[earth's speed] = 1/tan(20"1/4)

according to (10)

R/D = [radius of the sun]/[avg. distance to the sun] = tan of sun's radiatable radius

This implies

p = 8 tan(16'2")/(1000 tan 20"1/4) 2

This value for p is approximately 4, the same as the density of the earth.

Similar proposals were made by John Michell, Phil. Trans. Roy. Soc. (London) 74, 35 (1783).