On Saturday, January 8, 2022 at 7:33:34 PM UTC-3, Michael Moroney wrote:
Show us how von Soldner derived his doubled Newton figure with the math/science he used to derive it. Then we can compare and
contrast to Einstein's derivation.
GET THE FIGURE ON THE WEB. MOST OF THE WORK IS TO FIND A FUNCTION THAT VERIFIES THE TRAJECTORY OF STARLIGHT
TOWARD THE EYES OF AN OBSERVER ON EARTH'S SURFACE. THEN HE GENERALIZES (ALWAYS HALF THE TRAJECTORY, DUE TO THIS).Loading Image...
ORIGINAL PAPER TRANSLATED TO ENGLISH (MAYBE BY 1911):
On the deflection of a light ray from its rectilinear motion, by the attraction of a celestial body at which it nearly passes by.
By Johann Georg von Soldner. Berlin, March 1801.
At the current, so much perfected state of practical astronomy, it becomes more necessary to develop from the theory (that is from the
general properties and interactions of matter) all circumstances that can have an influence on a celestial body: to take advantage from
a good observation, as much as it can give.
Although it is true that we can become aware of considerable deviations from a taken rule by observation and by chance: as it was the
case with the aberration of light. Yet deviations can exist which are so small, so that it is hard to decide whether they are true deviations
or observational errors. Also deviations can exist, which are indeed considerable—but if they are combined with quantities whose
determination is not completely finished, they can escape the notice of an experienced observer.
Of the latter kind may also be the deflection of a light ray from the straight line, when it comes near to a celestial body, and therefore
considerably experiences its attraction. Since we can easily see that this deflection is greatest when (as seen at the surface of the
attracted body) the light ray arrives in horizontal direction, and becomes zero in perpendicular direction, then the magnitude of deflection
will be a function of height. However, since also the ray-refraction is a function of height, then these two quantities must be mutually
combined: therefore it might be possible, that the deflection would amount several seconds in its maximum, although it couldn't be
determined by observations so far.
These are nearly the considerations, which drove me to still think about the perturbation of light rays, which as far as I know was not
studied by anyone. Before I start the investigation, I still want to give some general remarks, by which the calculation will be simplified.
Since at the beginning I only want to specify the maximum of such a deflection, I horizontally let pass the light at the location of
observation (at the surface of the attracting body), or I assume that the star from which it comes, is apparently rising. For convenience
of the study we assume: the light ray doesn’t arrive at the place of observation, but emanates from it.
We can easily see, that this is completely irrelevant for the determination of the figure of the trajectory. Furthermore if a light ray arrives
at a point at the surface of the attracting body in horizontal direction, and then again continues its way (at the beginning horizontally
again): then we can easily see, that with this continuation it describes the same curved line, which it has followed until here. If we draw
through the place of observation and the center of the attracting body a straight line, then this line will be the major axis of the curved
one for the trajectory of light; by describing over and under this line two fully congruent sides of the curved line.
C (Figure 1) shall now be the center of the attracting body, A is the location at its surface. From A, a light ray goes into the direction AD or
in the horizontal direction, by a velocity with which it traverses the way v in a second. Yet the light ay, instead of traveling at the straight line
AD, will forced by the celestial body to describe a curved line AMQ, whose nature we will investigate. Upon this curved line after the time
(calculated from the instant of emanation from A), the light ray is located in M, at the distance CM = r from the center of the attracting body.
g be the gravitational acceleration at the surface of the body. Furthermore CP = x, MP = y and the angle MCP = h. The force, by which the
light in M will be attracted by the body into the direction MC, will be 2g/r². This force can be decomposed into two other forces,
2g/r² cos h and 2g/r² sin h , into the directions x and y; and for that we obtain the following two equations (s.Traité de mécanique céleste
par Laplace, Tome I, pag. 21)
d²x/dt² = − (2g cos h)/r² (I)
d²y/dt² = − (2g sin h)/r² (II)
If we multiply the first of these equations by −sin h, the second one by cos h, and sum them up, then we obtain:
(d²y cos h − d²x sin h)/dt² = 0 (III)
Now we multiply the first one by cos h, the second one by sin h and sum them together, then we obtain:
(d²y cos h + d²x sin h)/dt² = −2g/r² (IV)
To reduce in these equations the number of variable quantities, we express x and y by r and h. We easily see that
x = r cos h; y = r sin h
If we differentiate, then we will obtain:
dx = cos h dr − r sin h dh; dy = sin h dr + r cos h dh
And if we differentiate again,
d²x = cos h d dr − 2sin h dh dr − r sin h d²h − r cos h dh² and
d²y = sin h d²r + 2cos h dh dr + r cos h d²h − r sin h dh²
If we substitute these values for d²x and d²y in the previous equations, the we obtain from (III):
(d²y cos h − d²x sin h)/dt² = (2dhdr + rddh)/dt²
Thus we have:
(2dh dr + r d²h)/dt² = 0 (V). And furthermore by (IV),
(d²r − r dh²)/dt² = −2g/r² (VI)
To make Equation (V) a true differential quantity, we multiply it by r dt , thus:
(2r dh dr + r² d²h)/dt = 0
and if we again integrate, we will obtain:
r² dh = C dt, where C is an arbitrary constant magnitude.
To specify C, we note that r²dh (= rrdh) is equal to: the double area of the small triangle which described the radius vector r in the time dt.
The double area of the triangle that is described in the first second of time, is however: = AC v ; thus we have C = AC v.
And if we assume the radius AC of the attracting body as unity, what we will always do in the following, then C = v. If we substitute this
value for C into the previous equations, then: r² dh = v dt,
Thus we have
dh = v dt/r² (VII)
If this value for dh is substituted into Equations (VI), we obtain:
d²r/dt² − v²/r³ = −2g/r²
If we multiply this equations by 2dr , then:
(2dr d²r )/dt² − 2v²dr/r³ = 4gdr/r²
and if we integrate again,
dr²/dt² + v²/r³ = 4g/r + D
where D is a constant magnitude, that depends on the constant magnitudes which are contained in the equation. From this equation
that is found now, the time can be eliminated, hence:
dt = dr/√(D + 4g/r − v²/r²)
If we substitute this value for dt into Equation (VII), then we obtain:
dh = v dr/r² √[D + 4g/r − v²/r²]
To integrate this equations, we bring it into the form:
dh = v dr/r² √[D + 4g²/v² − (v/r − 2g/v)²]
Now we put
v/r − 2g/v = z
then we have vdr/r² = −dz
If this and z is substituted into the equation for dh, the we will have:
dh = −dz/√(D + 4g²/v² − z²)
From that the integral is now: h = arccos [z/√( D + 4g²/v²)] + α
where α is a constant magnitude. By well-known properties it is furthermore:
cos (h −α ) = z/√(D + 4g²/v²)
and if we also substitute instead of z its value:
cos (h −α) = (v² − 2g/r)/r √(v²D + 4g²)
h −α would be the angle that r forms with the major axis of the curved line that has to be specified. Since furthermore h is the angle
which r forms with the line AF (the axis of the coordinates x and y), then α must be the angle that forms the major axis with the line AF.
However, since AF goes through the observation place and the center of the attracting body, then by the preceding, AF must be the major
axis; also α 0, and thus:
cos h = (v² − 2gr)/r √(v²D + 4g²)
For h = 0 it must be r = AC = 1, and we obtain from this equation:
√(v²D + 4g²) = v² − 2g
If we substitute this in the previous equation, then the unknown D and also the square-root sign vanish; and we obtain:
cos h = (v² − 2gr )/r √(v² − 2g)
furthermore by that
r + [(v² − 2g )/2g]r cos h = v²/2g (VIII)
From this finite equation between r and h, the curved line can be specified. To achieve this more conveniently, we again want to
reduce the equation to coordinates. Let (Figure 1) AP = x and MP = y , then we have:
x =1− r cos h; y = r sin h , and
r = √[(1− x)² + y²]
If we substitute this into equation (VIII), then we find:
y² = [v² (v² − 4g )/4g²] (1− x)² − [v² (v² − 2g )/2g²] (1− x) + v²/4g²
and if we properly develop everything,
y² = v²x/g + [v² (v² − 4g )/4g²] x²/4g² (IX)
Since this equation is of second degree, then the curved line is a conic section, that can be studied more closely now.
If p is the parameter and a the semi-major axis, then (if we calculate the abscissa with its start at the vertex) the general equation
for all conic sections is:
y² = px + px²/2a
This equation contains the properties of the parabola, when the coefficient of x² is zero; that of the ellipse when it is negative; and that
of the hyperbola when it is positive. The latter is evidently the case in our equation (IX). Since for all our known celestial bodies 4g is
smaller than v², then the coefficient of x² must be positive.
If thus a light ray passes a celestial body, then it will be forced by the attraction of the body to describe a hyperbola whose concave side
is directed against the attracting body, instead of progressing in a straight direction.
The conditions, under which the light ray would describe another conic section, can now easily be specified. It would describe a parabola
when 4g = v², an ellipse when 4g were greater than v², and a circle when 2g = v². Since we don’t know any celestial body whose mass is
so great that it can generate such an acceleration at its surface, then the light ray always describes a hyperbola in our known world.
Now, it only remains to investigate, to what extend the light ray will be deflected from its straight line; or how great is the perturbation
angle (which is the way I want to call it).
Since the figure of the trajectory is now specified, we can consider the light ray again as arriving. And because I want to specify only the
maximum perturbation angle, I assume that the light ray comes from an infinitely great distance.
The maximum must take place in this case, because the attracting body longer acts on the light ray when it comes from a greater than
from a smaller distance.
If the light ray comes from an infinite distance, then its initial direction is that of the asymptote BR (Figure 1) of the hyperbola, because in
an infinitely great distant the asymptote falls into the tangent. Yet the light ray comes into the eye of the observer in the direction DA, thus
ADB will be the perturbation angle.
If we call this angle ω, then we have, since the triangle ABD at A is right-angled: tan ω = AB/AD
However, it is known from the nature of the hyperbola, that AB is the semi-major axis, and AD the semi-lateral axis. Thus this magnitudes
must also be specified. When a is the semi-major axis, and b the semi-lateral axis, the parameter is:
p = 2b²/a
If we substitute this value into the general equation of hyperbola
y² = px + px²/2a
then it transforms into:
y² = 2b²x/a + b²x²/a²
If we compare this coefficients of x and x2 with those in (IX), then we obtain the semi-major axis
a = 2g/(v² − 4g ) = AB
the semi-lateral axis
b = v/√(v² − 4g) = AD
If we substitute this values for AB and AD into the expression for tangω, then we have:
tangω = 2g/v √(v² − 4g)
We now want to give an application of this formula on earth, and investigate, to what extend a light ray is deflected from its straight line,
when it passes by at the surface of earth.
Under the presupposition, that light requires 564.8 seconds of time to come from the sun to earth, we find that it traverses 15.562085
earth radii in a second.
Thus v = 15.562085. If we take under the geographical latitude its square of the sine 1/3 (that corresponds to a latitude of 35˚16'), the
earth radius by 6,369,514 meters, and the acceleration of gravity by 3.66394 meters (s. Traité de mécanique céleste par Laplace, Tome I,
pag. 118): then, expressed in earth radii, g = 0.000000575231.
I use this arrangement, to take the most recent and most reliable specifications of the size of earth’s radius and the acceleration of
gravity, without specific reduction from the Traité de mécanique céleste . By that, nothing will be changed in the final result, because it is
only about the relation of the velocity of light to the velocity of a falling body on earth. The earth radius and the acceleration of gravity
must therefore taken under the mentioned degree of latitude, since the earth spheroid (regarding its physical content) is equal to a sphere
which has earth’s radius (or 6,369,514 meters) as its radius.
If we substitute these values for v and g into the equation of tang ω, then we obtain (in sexagesimal seconds) ω = 0".0009798, or in even
number, ω = 0".001.
Since this maximum is totally insignificant, it would be superfluous to go further; or to specify how this value decreases with the height
above the horizon; and by what value it decreases, when the distance of the star from which the light ray comes, is assumed as finite and
equal to a certain size. A specification that would bear no difficulty.
If we want to investigate by the given formula, to what extend a light ray is deflected by the moon when it passes the moon and travels to
earth, then we must (after the relevant magnitudes are substituted and the radius of the moon is taken as unity) double the value that was
found by the formula; because the light ray that passes the moon and falls upon earth, describes two arms of the hyperbola.
But nevertheless the maximum must still be much smaller than that of earth; because the mass of the moon, and thus g, is much smaller.
The inflexion must therefore only stem from cohesion, scattering of light, and the atmosphere of the moon; the general attraction doesn’t
contribute anything significant.
If we substitute into the formula for tang ω the acceleration of gravity on the surface of the sun, and assume the radius of this body as
unity, then we find ω = 0".84. If it were possible to observe the fixed stars very nearly at the sun, then we would have to take this into
consideration. However, as it is well known that this doesn’t happen, then also the perturbation of the sun shall be neglected.
For light rays that come from Venus (which was observed by Vidal only two minutes from the border of the sun, s. Hr. O. L. v. Zachs
monatliche Correspondenz etc. II. Band pag 87.) it amounts much less; because we cannot assume the distances of Venus and Earth
from the sun as infinitely great.
By combination of several bodies, that might be encountered by the light ray on its way, the results would be somewhat greater; but
certainly always imperceptible for our observations.
Thus it is proven: that it is not necessary, at least at the current state of practical astronomy, to consider the perturbation of light rays by
attracting celestial bodies.
Hopefully no one finds it problematic, that I treat a light ray almost as a ponderable body. That light rays possess all absolute properties
of matter, can be seen at the phenomenon of aberration, which is only possible when light rays are really material. And furthermore, we
cannot think of things that exist and act on our senses, without having the properties of matter.
nihil est quod possis dicere ab omni
corpore seiunctum secretumque esse ab inani,
quod quasi tertia sit numero natura reperta.
Lucretius de nat. rer. I, 431