gravitational light deflection: could relativity be ... deflection serret millennium aout...

August 5th, 2018 O. SERRET @ Millennium Relativity 1/18

Gravitational Light Deflection:

could Relativity be Invalidated by GAIA?

By: [email protected]

Abstract:

In this article, we propose an alternative explanation of the relativistic gravitational

deflection of light based on gravitation and refraction. In a first part, it is recalled that

the measurements use the direct comparison coefficient �� made when the light grazes

the Sun. It is also showed the lack of precision of these measurements close to the Sun.

Then we propose an explanation other than that given by the theory of Relativity, by

adding to the Newtonian gravitational deflection of the light the refraction by the solar

corona. In a second part devoted to the measurement of the deflection far from the

Sun, as the satellites allow, we criticize the indirect method used based on the

relativistic parameter PPN ��. Then we propose, thanks to the very high precision of

the GAIA satellite, to use this method of direct comparison with �� to measure the

light rays at a great distance from the Sun. Then the deflection value far from the Sun

could be weaker than expected.

Résumé:

Dans cet article, nous proposons une explication alternative de la déviation

gravitationnelle relativiste de la lumière basée sur la gravitation et la réfraction. Dans

une première partie, il est rappellé que les mesures utilisent le coefficient de comparaison

directe �� réalisé lorsque la lumière rase le soleil. Il est montré également le manque de

précision de ces mesures proches du Soleil. Nous proposons alors une explication autre

que celle donnée par la théorie de la relativité, en ajoutant à la déviation gravitationnelle

newtonienne de la lumière la réfraction par la couronne solaire. Dans une seconde partie

consacrée à la mesure de la déviation loin du Soleil, comme le permettent les satellites,

il est critiqué la méthode indirecte utilisée à partir du paramètre relativiste PPN ��. Nous

proposons ensuite, grâce à la très grande précision du satellite GAIA, d’utiliser cette

méthode de comparaison directe avec �� pour mesurer les rayons lumineux très éloignés

du Soleil. Ainsi la valeur de la déflection loin du Soleil pourrait être plus faible que prévu.

Key words: Gravitational light deflection, bending, light celerity, Hipparcos, Gaia, Against Relativity theory, neo-Newtonian mechanics.

Table of contents I. Introduction ................................................................................................................................... 2

II. The deflection of light near the Sun ............................................................................................. 2

III. The deflection of light far from the Sun ...................................................................................... 4

IV. CONCLUSION .............................................................................................................................. 7

V. References ...................................................................................................................................... 8

VI. Related articles / Neo-Newtonian Mechanics .............................................................................. 9

VII. Appendixes ................................................................................................................................... 10

GRAVITATIONAL LIGHT DEFLECTION: COULD RELATIVITY BE INVALIDATED BY GAIA?


I. Introduction The measurement of Sunlight's deflection by Eddington [1] during the 1919 eclipse was

considered, despite of “only 30% of accuracy” [2], as the first proof of the validity of the theory

of Relativity based on the constancy of the speed of light. Confidence in this theory is such that

today it is no longer measured ‘directly’ the deflection at a distance from the Sun: it is simply

calculated the relativistic parameter PPN γ�. It is according to this method that the

HIPPARCOS satellite readings were analyzed in the 90's, and that one prepares to strip the

readings of the satellite GAIA [3]. Nevertheless, this parameter γ� exists only in the theory of

the Relativity; to take a result of a theory to demonstrate the hypothesis of the same theory is

not rigorous, it looks like a ‘circular reasoning’. Moreover, this phenomenon of gravitational

deflection of light can be explained in another way, as we propose to show it in this article.

II. The deflection of light near the Sun II.a) Gravitational light deflection

In his book Relativity, Einstein explains that relativistic deflection is exactly double the Newtonian

deflection: "It may be added that, according to the theory, half of this deflection is produced by the

Newtonian field of attraction of the sun, and the other half by the geometrical modification

(“curvature”) of space caused by the sun" [4]. So, half of the deflection would be due to gravity (hence

the name of 'gravitational deflection'), the other half due for another reason. This reminds on the one

hand that the deflection of light is also predicted in the context of Newtonian mechanics. On the other

hand, note the paradox of talking about gravitational deflection in the framework of the theory of General

Relativity that does not recognize the forces of gravitation…

Let us now introduce the correlation coefficient γ� defined as the direct comparison of deflection

between measurement and Newtonian theory

γ� = ��

− 1 (1)

with � value of deflection, in [mas].

and with the property:

� γ� = 0 �� < Newtonian mechanic is valid >γ� = 1 �� < the theory of Relativity is valid > (2a-b)

Near the solar disk, the coefficient γ� = 1. The theory of Relativity was thus accredited by the scientific

community and then by the general public, although the results were not systematically reproduced [5].

More recently, Eddington was even suspected of to have corrected his measures [6] to prove this

revolutionary theory. Nevertheless, other measurements would have confirmed that the deflection values

are in agreement with the theory of Relativity, even if these measures remain difficult accessible for the

general public. Among the measures available (see Fig. 1), like that of 1973 [7], it appears that:

- the deflection value � = 1750 789 (or 1.75 seconds) at the edge of the solar disk is extrapolated

(and not measured). This limit value could also be close to 1200 mas.

- the deviation �:7;89<=;>? is imprecise (of +/- 350 mas), sometimes giving even negative

values!



Figure 1: Uncertainties on light bending measurement (1973):

what is the extrapolated value at R=1?

II.b) Deflection by refraction

That said, there is another factor that should not be overlooked: refraction by the gases surrounding the

Sun. Indeed, the absolute vacuum does not exist. The Sun is surrounded by a gas halo more or less

homogeneous, as can the Earth, the stars and the galaxies. The uncertainty concerns the density of this

gas and therefore its refractive index.

On Earth, the refractive index of ground air is 1.000293. Although of value very close to unity, this low

value is at the source of phenomena such as the glittering stars or the mirages; these are examples of the

deflection of light by the atmosphere, which is amplified by temperature. At sea level, the gas density is

of the order of 1025 molecules/m3. The air density decreases with altitude: at an altitude of 45 km, at the

limit of the Earth's atmosphere (the stratosphere), its density represents less than 1% of that of the

ground. The refractive index of the air is there of the order of 1.000002 [8]. There are several models of

decreasing, for example in @ABCD DA⁄ :1 − F. H?I or in

@ABCD DA⁄ ;B�.D [9]. From the two previous equations, we

simplify the model (which of course can be changed) to:

J − 1 = @ABCD DA⁄ (3)

Around the Sun, the lower corona (closest to the Sun) has a particle density of 1014 to 106 particles/m3

[10], composed of ¾ hydrogen and ¼ helium molecules at very high temperature, and other particles

such as 1014 electrons/m3. Suppose, therefore, that the refractive index at an altimetry of 2 solar radii is

1.000003; this is barely above the refractive index of 1,000002 at the confines of the Earth's atmosphere.

We will assume for the Sun corona an equally exponential decay as a function of altitude. Let Rs be

the radius of the Sun. The refractive index would then be variable according to Table 1:

Table 1: Corona refractive index

Adding the deflection due to the gases of the solar corona to that due to the Newtonian gravity, we obtain

(see appendix A) values comparable to that predicted by the theory of Relativity, especially when the

light does not tangent the solar disk (this is indeed the case during the observations, the tangential value

at R=Rs being an extrapolation), see Fig. 2.



Figure 2: Gravitational light bending

Remarks:

- Rigorously, in the relativistic calculations, one would have to add the refraction due to the gases to the

values calculated by the only Relativity, which would give a maximum value of more than 2000 mas,

not coherent with the measurements.

- The refraction phenomenon, locally fluctuating by thermal agitation of the gas, could explain the

uncertainty of deviation measurements near the Sun, and the negative deflection measurements recorded

on Figure 1.

The theory of relativity predicts a deflection twice as large as that predicted by Newtonian mechanics

alone. This difference between the observation and the only gravitational prediction can be greatly

reduced by taking into account the refraction due to the gas of the solar corona. But what happens when

the deviation is far from the Sun?

III. The deflection of light far from the Sun III.a) Current measures:

Thanks to the satellites and the precision of their measurement, it is now possible to get rid of solar

eclipses by measuring the deviation of light far from the Sun. This is what was achieved for example

with the Hipparcos satellite in the 90s. The method is as follows: γ� is first calculated by adjusting on

the stellar data; it is the method used to manage the correlation terms involved in the process, with

γ� = 1 + :LM:N?/LN?PQ:N?M:N?RS:LM:N?/LN?P (4)

as defined in the article at the bottom of this pagea

There we see that the relativistic parameter γ� is consistent with the predictions of Relativity, with

precision close to [(σ:γ�?]. Then we estimate the deflection errors in this relativistic framework

[σ:δVWXYZ[\[Z]?]. These errors do not only include errors on γ�, but on many parameters: satellite position,

measurement accuracy, etc. That is to say that we proceed in three steps [11].

- First step: computing the γ�with its root mean square error (for example γ�=0.997)

- Second step: getting the observational error expected (for example ^:_�)=+/-0.003)

aEquation 27 of “PPN Parameter Gamma and Solar System Constraints of Massive Brans-Dicke Theories” at

arXiv:0911.3401v3 [gr-qc] 19 jan 2010



- Third step: getting the deflection error expected (for example ^:�H;`8abcbad?=+/-0.005 mas)

The problem with this method is that the deflection of light has not been measured "directly" [12], as I

will try to show it below.

Figure 3: Comparison of Coefficient γ� and PPN Parameter γ�

Please distinguish well (see Fig. 3) :



- γ� (Correlation coefficient): direct comparison of deflection between measurement and

Newtonian theory.

- γ� (PPN parameter): amount of space curvature produced by unit rest mass. From the stellar

data, a relativistic parameter γ� has been calculated (with a priori values [13] and constant

hypothesis on cosmological, scalar potential, initial conditions, etc. [14]). It is a parameter

deduced from the theory of Relativity giving a value close to unity.

There is in principle no reason to assimilate γ� to γ�. From a methodological point of view, it would be

necessary to directly calculate the comparison coefficient γ�, as it is moreover done during the

measurement reading close to the Sun. It is the use of this single coefficient γ� that we have for the rest

of the article. γ� is also coherent with the Neo-Newtonian Mechanics (more information in Appendix

B1 and in Related Articles).

III.c) Proposal of experimental measures

Since 2013, the GAIA satellite, placed at the Lagrange 2 point, has been scanning the position of the

stars. The scanning law is optimized to explore the same area, with an angle of observation λ of 45° and

with a complete turn of its astronomical glasses in 63 days.

The (scanned) raw observations are disturbed by a number of phenomena that are for example the global

parallax shift, eccentricity of the orbit, Lissajous orbit, relation with the velocity and aberration

correction, instrumental effect as temperature, precession per orbit, other planets of the ecliptic plane,

etc. and probably refraction. Reprocessing of all these raw data can only be supported by researchers

specialized in Space.

In order to directly calculate the comparison coefficient γ�, we would need to obtain some specific

reprocessed data. It is a question of knowing precisely the two angles of observation of the same star

taken in the plane of the ecliptic, measurement recorded twice according to the same line of sight (in

order to be free from the phenomenon of parallax generating a slight deviation): see Fig.4. It is important

to specify that this obviously supposes that there is no approximation in the determination of the angle

Δ of rotation of the satellite on its orbit which will have to be defined to within 1 mas (the precision to

0.01 mas near Gaia's instruments should allow it). See details of the calculation in Appendix B

Figure 4: Two positions of Gaia

Thus, moving from position [1] to position [2], the increase in the deviation should be close to:



�[2] − �[1] = 4 789 (5)

And according to the Theory of Relativity, this increase of deflection should be double, i.e. 8 mas.

These predictions have been done without taking account of a possible refraction, which should be very

weak at this distance: no more than 2 mas (see Table B5 of Appendix B). Rigorously a very slight value

of refraction should be added to Relativity prediction too. So:

� �[2] − �[1] ≤ j klm aℎ;J < oℎ;p=d pq H;`8abcbad rp<`> s; bJc8`b>8a;> > �[2] − �[1] ≥ u klm aℎ;J < v;p − v;rapJb8J w;xℎ8Jbx9 rp<`> s; bJc8`b>8a;> > (6.a-b)

If it is not possible with GAIA satellite, these measurements could also be done with the Hubble Space

Telescope or tomorrow with the JWST. Realizing this simple 'direct' comparative measure (with ′γ�′) would give an additional confirmation of the theory of Relativity, or maybe not…

IV. CONCLUSION

The Neo-Newtonian Mechanics (NNM) offers another explanation for the

gravitational deflection of light. Thus, near the Sun, the refraction due to the gas

of the solar corona, which is not considered by the theory of Relativity, could

explain a deflection superior to the single Newtonian gravitational deflection. And

at a far distance from the Sun, the Newtonian gravitation would intervene

essentially in the deflection.

To ensure this, thanks to the precision that the GAIA satellite allows, another

analysis of the readings could be easily envisaged: it would be necessary to favor

the direct comparison coefficient of the measurements γ� instead of the indirect

parameter PPN γ�. For a star taken in the plan of the ecliptic, the GAIA satellite

would have to make two readings at 93 ° of divergence on the same line of sight

to measure the difference of deflection. If it is higher than 8 mas, this direct

measure would then confirm the theory of the Relativity; or on the contrary if it

is lower than 6 mas, it would open the way to other theories, such as the neo-

Newtonian mechanics.



V. References 1 Dyson, Eddington & Davidson (1919), A Determination of the Deflection of Light by the

Sun’s Gravitational Field, from Observations made at the Total Eclipse of May 29, 1919,

Royal Society, file:///G:/Light%20deflection/Eddington%201919.pdf

2 Clifford M.W.(2006), The Confrontation between General Relativity and Experiment,

Springer, https://link.springer.com/article/10.12942/lrr-2006-3#Sec2 §3.4.1

3 Gaia website, ESA, https://www.cosmos.esa.int/web/gaia/scanning-law

4 Einstein A (1920)., Relativity the Special and General theory, pages 152-155,

https://ibiblio.org/ebooks/Einstein/Einstein_Relativity.pdf

5 Mikhailov A., The deflection of light by the gravitational field of the Sun, SAO/ NASA ADS

(1959), http://adsbit.harvard.edu//full/1959MNRAS.119..593M/0000598.000.html

6 Bonnet-Bidaud J-M. (2008), Relativité, les preuves étaient fausses, Ciel et Espace,

http://bonnetbidaud.free.fr/ce/relativite2008/pdf_CE/RELATIVITE-052-456.pdf

7 Jones B. (1976), Gravitational deflection of light: solar eclipse of 30 June 1973, The

Astronomical Journal, http://rsta.royalsocietypublishing.org/content/roypta/220/571-

581/291.full.pdf

8 Pérard A., (1925), Indice de réfraction de l’air dans le spectre visible entre 0° et 100°C., J.

Phys. Radium, 6 (7), pp.217-227., https://hal.archives-ouvertes.fr/jpa-

00205209/document

9 Bénard P., (2006), La réfraction atmosphérique en montagne, http://www.saint-

barthelemy.pyreneus.fr/refract/calculs/calculs.html

10 Proisy P. (1949), L'indice de réfraction de l'atmosphère solaire et la trajectoire des rayons

lumineux au bord extrême du disque solaire, Annales d'Astrophysique, Vol. 12, p.123,

http://adsabs.harvard.edu/full/1949AnAp...12..123P

11 Vecchiato A. & Al (2003), Testing general relativity by micro-arcsecond global astrometry,

Astronomy and Astrophysics, v.399, p.337-342, https://arxiv.org/abs/astro-ph/0301323

12 Serret O. (2017), Hipparcos did not measure directly the light bending!, GSJ,

http://www.gsjournal.net/Science-Journals/Research%20Papers-

Mechanics%20/%20Electrodynamics/Download/6998

13 Lebach D.E. & Al. (1995), Measurement of the Solar Gravitational Deflection of Radio

Waves Using Very-Long-Baseline Interferometry, Physical Review Letters 75(8):1439-

1442 ·

14 Pireaux S. (2002), Light deflection experiments as a test of relativistic theories of

gravitation, pages 31 to 84, https://cp3.irmp.ucl.ac.be/upload/theses/phd/spireaux.pdf

Acknowlegment

I’m grateful to Voltaire for improving sentences before translation.



VI. Related articles / Neo-Newtonian Mechanics

EXPERIMENTS

Lorentz derivation How to Demonstrate the Lorentz Factor: Variable Time vs. Variable Inertial Mass

http://file.scirp.org/pdf/JMP_2015022510573131.pdf

Velocity Addition

Velocity Addition Demonstrated from the Conservation of Linear Momenta, an

Alternative Expression http://file.scirp.org/pdf/JMP_2015050609513342.pdf

An improvement of the accuracy of Fizeau’s experiment http://gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/7247

Force in Synchrotron Net Force F = γ3ma at High Velocity


OBSERVATIONS

Mercury perihelion About the ovoid orbits in general, and perihelion precession of Mercury in particular (2)

http://www.mrelativity.net/Papers/51/ Mercury%20Millennium%20Serret%205%20janvier%202018.pdf

Light deflection

Hipparcos did not measure directly the light bending!

http://gsjournal.net/Science-Journals/Research%20Papers-

Mechanics%20/%20Electrodynamics/Download/6998

Gravitational Light Deflection: could Relativity be Invalidated by GAIA?

Present article

Shapiro time delay

Shapiro Time Delay derivates from Refraction

http://www.mrelativity.net/Papers/51/Shapiro%20SERRET%20Millennium%20juillet%202018.pdf

Pioneer Anomaly The Pioneer Anomaly explained by the Processing of the Doppler Effect

http://gsjournal.net/Science-Journals/Research%20Papers-Relativity%20Theory/Download/7330

Dark Matter

The flat rotation curve of our galaxy explained within Newtonian mechanics

https://physicsessays.org/browse-journal-2/product/1240-7-olivier-serret-the-flat-rotation-curve-of-our-

galaxy-explained-within-newtonian-mechanics.html

Dark Energy

Gravity vs. Dark Energy, about the Expansion of the Universe


About Doppler-Fizeau

Next article

Gravitational waves

Gravitational waves or particle radiation?

https://www.physicsessays.org/browse-journal-2/product/1588-12-olivier-serret-gravitational-waves-or-

particle-radiation.html

Gravitational

redshift Next article

Muon lifetime Next article

CRITICISM OF RELATIVITY

Lorentz derivation Reply to “A Simple Derivation of the Lorentz Transformation”




VII. Appendixes APPENDIX A / REFRACTIVE DEFLECTION CLOSE TO THE SUN

A1) HYPOTHESES

Figure A1: Geometry of refraction

Half deflection:

�/2 = z/2 – |S − b} (A1)

Initial condition:

b~ = z/2 + |~/2 (A2)

Descarte’s refraction:

JC 9bJ:�C? = JS 9bJ:bS? (A3)

JS 9bJ:�S? = J} 9bJ:b}? (A4)

Spherical Bouger’s refraction:

JCH~ 9bJ:b~? = JSHC 9bJ:bS? (A5)

JSHC 9bJ:bS? = J}HS 9bJ:b}? (A6)

Geometry:

��@:��?DA

= ��@:�B�A?D�

(A7)

��@:�P?D�

= ��@:�B�P?DP

(A8)

|S + �S = |C + bS (A9)

|C + �C + :z − b~? = z (A10)

Geometry of initial deflection:

HC cos:|C? = H~ − = (A11)

HC sin:|C? = �~ (A12)



= = �~ tan:|~/2? (A13)

Hypothesis on refractive index:

JC − 1 = D�DA

:J~ − 1? (A14)

JS − 1 = D�D�

:J~ − 1? (A15)

J} − 1 = D�DP

:J~ − 1? (A16)

A2) PROCESSING THE HYPOTHESES

By Eq.(A9): |S = |C + bS − �S (A17)

By Eq.(A11): cos :|C? = DAB�D�

(A18)

with Eq.(A13): cos :|C? = DAB�A ZY�:�A/S?D�

(A19)

and with Eq.(A12) : cos :|C? = DABD� �[�:��? ZY�:�A/S?D�

(A20)

cos :|C? + tan:|~/2? sin:|C? = DAD�

(A21)

Because �AS ≪ 1 then cos ��A

S � ≈ 1 and a8J ��AS � ≈ sin :�A

S ? (A22)

cos :|C − |~/2? = DAD�

(A23)

|C = Fxp9 �DAD�

� + �AS (A24)

By Eq.(A3): 9bJ:bS? = @�@P

9bJ:�C? (A25)

By Eq.(A7): 9bJ:bS? = @�@P

DAD�

9bJ:b~? (A26)

By Eq.(A2): 9bJ:bS? = @�@P

DAD�

xp9:|~/2? (A27)

bS = F9bJ: @�@P

DAD�

xp9:|~/2?? (A28)

By Eq.(6): 9bJ:b}? = @P@�

D�DP

9bJ:bS? (A29)

And with Eq.(A27): 9bJ:b}? = @P@�

D�DP

@�@P

DAD�

xp9:|~/2? (A30)

9bJ:b}? = @�@�

DADP

xp9:|~/2? (A31)

b} = F9bJ: @�@�

DADP

xp9:|~/2?? (A32)

By Eq.(A4): 9bJ:�S? = @�@P

9bJ:b}? (A33)

And with Eq.(A31): 9bJ:�S? = @�@P

@�@�

DADP

xp9:|~/2? (A34)

9bJ:�S? = @�@P

DADP

xp9:|~/2? (A35)



�S = F9bJ: @�@P

DADP

xp9:|~/2?? (A36)

By Eq.(A17), (A24), (A28) and (A36):

|S = Fxp9 �DAD�

� + �AS + F9bJ: @�

@PDAD�

xp9:|~/2?? − F9bJ: @�@P

DADP

xp9:|~/2?? (A37)

By Eq.(A1) and (A17),

S = �

S – |C − bS + �S − b} (A38)

With Eq.(A24) and (A28),

S = �

S – Fxp9 �DAD�

� − �AS − F9bJ: @�

@PDAD�

xp9:|~/2?? + �S − b} (A39)

Comment:

If J} ≈ JS then b} ≈ �S (A40)

And so, for refraction alone:

S ≈ �

S – Fxp9 �DAD�

� − �AS − F9bJ: @�

@PDAD�

xp9:|~/2?? (A41)

A3) NUMERICAL APPLICATION

By hypothesis R1>Ro, let us have R1/RS = 7 (A42)

For R0/RS = 3 then R1/Ro =7/3=2.33 (A43)

n0 = 1+6.10-6 (A44)

n1 – 1= (n0 -1)/3 = 2 10-6 (A45)

n2 – 1= (n0 -1)/7 = 0.9 10-6 (A46)

Without gravitational deflection, i0=π/2 (A47)

Then θ0=0 (A48)

then δ/2=5.4 10-7 rad (A49)

or δ=224 mas (A50)



APPENDIX B / GRAVITATIONAL LIGHT DEFLECTION

B1) Can we use the Newtonian equations in the context of Neo-Newtonian Mechanics?

the light celerity in space is conventionally:

x = 299 792.459 �� (B1)

with an uncertainty on the measurement of +/- 1 m/s

Neo-Newtonian Mechanics (NNM) is nothing but Newtonian mechanics without the principle of

equivalence (the inert mass would be different from the gravitational mass at very high speed): see

Related articles /Neo-Newtonian Mechanics. Most of his results are comparable to those predicted by

the theory of Relativity, such as the advance of perihelion of Mercury or the limited addition of

velocities. But unlike the theory of relativity, neo-Newtonian mechanics is compatible with Quantum

Mechanics. According to Neo-Newtonian mechanics, the photon is a minute mass particle subjected to

gravity. Its speed is hardly less than the asymptotic speed. Suppose for example that

9 ≅ x:1 + 3. 10B�? = 299 792.459 km/s (B2)

9 remaining within the limit of uncertainty of the speed of light.

Let x� the celerity of light at an infinite distance, without gravitational attraction

and xD the speed of light when it is closest to the Sun, at the distance R

At an infinite distance, the potential energy being hypothetically zero, the energy of the particle is equal

to its kinetic energy:

E:∞? = :γ�:x�? − 1? m�sS (B3)

With γ� the Lorentz factor and m� the gravitational mass of the photon

Note: Be careful not to confuse the Lorentz factor γ� with the comparison coefficient γ� or the PPN

parameter γ�.

Close to the Sun,

E:R? = :γ�:xD? − 1?m�sS − G m� M / R (B4)

By Energy conservation,

E:∞? = E:R? (B5)

γ�:xD? − γ�:x�? = G M / :R sS ? = 2 10B� (B6)

then xD ≈ x� (B7)

so x ≈ �a; (B8)

This shows that in neo-Newtonian mechanics the celerity of light remains practically constant in

vacuum, whether the photon is at an infinite distance or very slightly accelerated near the Sun. And since

the Lorentz factor γ� of the photon is practically constant (close to 2.10-6), we can well reduce ourselves

to the Newtonian equations with a quasi-constant celerity of light.



B2) Detail of deflection computing

Figure B1: Frame F (in orange) and frame O (in white)

Symbol Label Formula

a Semi major axis GM/c²

e Eccentricity (R/a)-1

b Semi minor axis (impact parameter) a (e²-1)^(1/2)

p (or l) Semi-latus rectum b²/a

θ Angle in F frame

ρ(θ) Radius in F frame p / (1 – e cos (θ))

Ro Minimum distance to the Sun ρ(θ) > Ro

X, Y Coordinates in F frame

x, y Coordinates in O frame

x Abscissa in O frame a + (Ro – ρ(θ) cos(θ))

y Ordinate in O frame ρ (θ) sin(θ)

y’ Tangent ordinate (derivative) b²/a² x/y

φ Angle of the tangent Atan y’

δ/2 Instantaneous semi deflection δ/2 = π/2 – φ

Table B1: Formulas



Figure B2: Gravitational light deflection in the ecliptic plane

Astronomical glasses oriented at λ=45° doing a complete turn in 63 days, the same star A has to be

seen again after a rotation angle Δ of (63*1.5/365.25*360° ≈) 93°



Table B2: Results for λA = 45° and R=1.648 1011 m



Table B3: Results for λA = 42° and R=1.648 1011 m



Using the table above, let us calculate the deviation of the photon according to whether it is in position

[1] or [2], and then deduce the deviation variation.

Position [1] [2]

λ¡ 45° 42°

RΔ 1.165 E+11 1.103 E+11

δ/2 from 0° to 45° 1.85 mas

δ/2 from 0° to 48° 2.05 mas

δ/2 from 0° to 90° 2.61 mas 2.76 mas

δ[1]= δfrom -90° to -45° 0.76 mas

δ[2] = δfrom -90° to +48° 4.81 mas

Variation from δ[1] to δ[2] 4.05 mas

Table B4: variation of δ between position [1] and [2]

Distance from the Sun center

[m]

7 E+08 1,1 E+11

Distance from the Sun center

[Rs]

1 157

Refraction index (n-1) 6 E-6 0.04 E-6

Refractive deflection [mas] 306 2

Table B5: refraction values

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