20 Dark Matter and Mercury's Orbit                          Table of Contents     Previous      Next





For a published version of this work see the Paper:


A Quantum Mechanical View of the Precession of Mercury’s Orbit      


The calculation of Mercury’s precession about the Sun presented in this article offers a quantum mechanical alternative to the calculation based on general relativity (ref 1). Further, this alternative provides a plausible explanation of Dark Matter in which gravitons act collectively as a small planet between Mercury and the Sun. This unseen graviton bundle of a planet influences the orbit of Mercury causing its precession.

My Motivation for this work:
In terms of the most astonishing fact about which we know nothing, there is Dark Matter and Dark Energy. We don't know what either of them is. Everything we know and love about the universe and all the laws of physics as they apply, apply to four percent of the universe. That's stunning - Neil deGrasse Tyson (ref 2).

The diagram at left accentuates the elliptical nature of Mercury’s orbit. Also, it should be noted that the precession phenomena is too small to see over a single orbit. It takes hundreds of orbits to even begin to see this precession. At left is a very, very exaggerated view of Mercury’s precession about the Sun.

The diagram at left is a scaled orbit diagram (ref 3), that shows how close to a circle the orbit of Mercury is.

a.  A new definition for the graviton

1.  The graviton is the mediator particle of the gravitational force in the sense that the photon  
   is the mediator particle of the electromagnetic force.

2.  The photon is the smallest quantum of energy at a given wavelength.  The graviton is the
   smallest quantum of mass at a given distance.

3.  When gravitons connect masses that are light years apart, they should be as
   undetectable as photons with wavelengths that are light years long. We may not be able
   to get photons that are light years long, but this is no limit for gravitons.

4.  Photons can act as if they had mass when they push solar sails, and when they are
  reflected between mirrors. (ref 4) This has not turned photons into gravitons, however it
  gives them some of the properties of gravitons.

5.   It seems not too far a stretch that to think a single graviton is reflected to and fro between
  masses. This graviton is not something separate from observable mass, but comes     
  packaged with observable mass.

  1. b. Dark Energy and Dark Matter

Dark Energy is hypothesized as the accumulation of graviton mass that is not rotating about an observable mass. It is observable in the universe at the largest scales where it connects galaxies. See paper: “The Geometry of Dark Energy” (ref 5).

Dark Matter is hypothesized as the accumulation of graviton mass that is rotating about observable mass. It is found mostly in rotating galaxies. The precession of Mercury’s orbit, considered here, is the simplest example of gravitons acting as Dark Matter.

The three terms, Dark Energy, Dark Matter, and Gravitons, have overlapping meanings and are somewhat confusing. Things are a little clearer when we speak of graviton mass accumulations in astronomical settings as being in the configurations of either Dark Energy Gravitons or Dark Matter Gravitons. This paper could be called a prequel to the Prespacetime Journal paper titled “The Geometry of Dark Energy” (ref 5) which concentrates primarily on gravitons acting as Dark Energy.

c. The Insights

This quantum theory of the graviton will be shown to calculate the correct precession of the orbit of Mercury (and Venus and Earth). The math is simple algebra; the physics is introductory, and the astronomy is basic. The insights themselves are not obvious and will raise some questions, which I will anticipate with some Q&A at the end of each insight.

1) The first insight is that the graviton is a quantum of mass.

a.A single photon has the energy E = hf = hc/λ, where f is the frequency of the photon, h is Planck constant, c is the speed of light, and λ is the wavelength. Note that in this paper both the Planck constant h will be used as well as the reduced Planck constant ħ.

b.The equation that converts energy to its mass equivalent is E = mc2, where m is the mass of the graviton. This assumes that the momentum of the mass is negligible. 

c.Equating the two equations in a and b above, we get: E = hf = hc/λ = mc2.

d.Solving for m, we get:  m = h/λc. This is the quantum of mass that is the graviton.

e.A graviton spans the distance of separation d between two observable mass objects. This distance is the wavelength of the graviton. Thus the graviton is defined by the equation m=h/dc. At the astronomical scales of galaxies and the universe as a whole, this mass will take the form of either Dark Matter or Dark Energy. At our solar system scale this dark matter graviton mass causes the precession of Mercury.

      Question 1: If the graviton fits the Planck-Einstein equation (E = hf), then it must be a   
Answer: The graviton is like a photon, but a photon that cannot move freely because it is    
      tethered at both ends. And yes, this is just a postulate that we can test to see if it makes    

        Question 2: What is the graviton tethered between?

        Answer:  For Dark Energy gravitons (where masses move in straight lines with respect
        to each other) the tether points are Planck masses defined using the Planck constant h
        (Pm = (hc/G)
0.5.  This result was worked out in a previous paper in the Prespacetime 
        Journal (
ref 2).  For Dark Matter gravitons (where masses revolve about each other) the
        tether points are Planck masses defined using the reduced Planck constant
h. This is the   
        agreed upon definition of a Planck mass  (Pm = (
hc/G) 0.5 (ref 6). This tethering will be
        considered bellow in the second insight. Note that: h=2π
h, where h is the Planck constant
h is the reduced Planck constant.

       Question 3: Why are there two types of gravitons?
Answer: It seems that rotary motion forces the gravitons to spread out. See diagram

2) The second insight: Planets that orbit the Sun are connected to the Sun by a lot of   
      gravitons (a graviton bundle) the number of which (N) can be calculated.

          The kinetic energy of a planet that is rotating about the Sun.

1.This energy is usually thought of as ½ mv2. However, it can also be thought of as a force times a distance as diagramed left. Note that the wavelength of the graviton in this situation is 2πd, the circumference of the orbit. The force is toward the Sun and the motion is along the circumference.

2.The energy in the orbit of the planet (mass m) is 2πd(GMm/d2). This energy in the orbiting planet can be equated to the energy of each graviton (hc/d as explained above) times the number of gravitons N.

force           x    distance  = N (hc/d)

[GMm/d2]    x     [2πd]      = N (hc/d)

[GMm/d] [2π]  = N (hc/d)Solving for N we get:

N = [2πGMm/d] [d/hc]

Simplifying we get:  N = 2πGMm/hc or

N = GMm2π/hc or N = Mm/(hc/2πG)
Since h/2π = h, we get N = Mm/(hc/G) and since Pm2 = hc/G, we get N = Mm/Pm2

The result N = Mm/Pm2 can be written as N = [M/Pm][m/Pm]

In words: the number of gravitons connecting the big mass M and the small mass m is the mass M divided by the Planck mass times the mass m divided by the Planck mass.

This is a very interesting result. It indicates that the Planck mass is the minimum mass that can support a Dark Matter graviton. It also indicates that dark matter gravitons are tethered between Planck masses. The Dark Energy graviton is very similar to the Dark Matter energy. Dark Matter is associated with the reduced Planck constant h, Dark Energy is associated with the Planck constant h.

Question: Does this mean that there is no gravitational mass below the Planck mass?
Answer: This exploration points to that conclusion. However, this does not eliminate inertial mass from existing below the Planck mass.

Question: Is there a way of detecting gravitons in the lab?

Answer: I expect that if we had several physical weights and put them together we would measure a gravitational mass that is greater than the sum of the individual weights when they are separated. I think we can test this idea with something like a modified torsional balance of the type Cavendish used to measure the gravitational constant G (ref 7).

On second thought after looking at some numbers this does not seem feasible. However, The experiment outlined in section 21, which uses centrifuges to produce extreme forces of gravity, may have a chance to give us a “feel” for the reality of gravitons.

Question: Why should the Planck Mass be such a fundamental quantity?
Answer: The Planck mass approximates the mass of a small eyelash. This is much larger than any atom or molecule. Why should this ordinary observable mass be special? Answer: I am tempted to speculate, but I really don’t know.

     Graviton Summary 

     a. The number of gravitons N connecting two masses is: N = M1M2/PM2      
     b. Each Planck mass in M
1 connects to every Planck mass in M2.
     c. Each Planck mass in M
2 connects to every Planck mass in M1.
     d. Each graviton produces a gravitational force of hc/(2πd

     e. Each graviton connecting M1 to M2 has a mass of m=h/dc

     f. The total graviton mass connecting M1 to M2 is: N (h/dc) = (M1M2/PM2)(h/dc)

3) The third insight.

This third insight was that the Dark Matter gravitons between Mercury and the Sun could  produce an angular momentum that would add to Mercury’s angular momentum, causing a slight overshoot in the orbit beyond the preceding 360-degree orbit. The ratio describing this angular momentum change over an orbit would be:

Where ΔDeg is the precession in degrees (eventually to be converted to arc-seconds). Angular Momentum is defined as the mass of an object times its tangential velocity; and Angular Velocity is defined as the rotational speed. The graviton bundles that connect to a planet have the same rotational speed as the planet itself.

This equation is a little tricky because the Dark Matter gravitons have changing mass over the orbit (m = h/dc) because the orbit is not a perfect circle.  We can handle this with two methods. The first method is to average the aphelion and the perihelion distance of Mercury (called the semi-major axis) and use it as a constant radius over an orbit. The second method is to divide the 360-degree orbit into smaller segments (each with its own distance) and calculate the precession of each segment. The total precession would be the sum of all the precession segments.

The precession (ΔDeg) was calculated using both methods. The first method, calculating the distance d as the average of the perihelion and aphelion gave a precession of 43.2 arc-seconds. The second method using 10-degree increments gave a precession of 45 arc-seconds (using a ruler and protractor to measure the distance of Mercury at each orbit position given in the diagrams of Mercury’s orbit (ref 3).

The calculation that follows uses the first method (figure bottom left). This method resulted in 43.2 arc-seconds, which agrees very well with measured precession values. It is also much easier to calculate.

Why should the average of the perihelion and aphelion distances work to compensate for mass changes over an orbit?

Answer: The length of a semi-major axis is often termed the size of the ellipse. It can be shown that the average separation of a planet from the Sun as it goes around its elliptical orbit is equal to the length of the semi-major axis. Thus, by the "radius" of a planet's orbit one usually means the length of the semi-major axis (ref 9).

Following this lead, the Dark Matter bundle of gravitons moving with Mercury can be considered to be a planet. The orbit of this planet is a circle that has a constant radius “d” that is the average of the perihelion and aphelion distances (the Semi-Major Axis) of Mercury.

  1. d.Setting up a ratio to Solve for the precession of Mercury:

     1. The basic premise is that if we know the Dark Matter mass MDM and the mass of
         Mercury M
Mercury we can calculate the precession ΔDeg as:



            The graviton bundle moves like a rigid rod that rotates with Mercury about the Sun. Thus

         the angular velocity of the Dark Matter and Mercury are the same.

       2. The velocity of a rotating object is defined as the angular velocity times the radius.

         The angular momentum is the mass of the object times the velocity of the object. And:

     Where d is the distance to Mercury and d/2  is the distance to the center of MDM.

        3. Since the angular velocity of both MDM and MMercury are equal (see diagram above) we    

        4.  We now have the precession in a simple form:  ΔDeg = 180 [MDM/MMercury]

     5.  However, we still need to get more specific about MDM. MDM is the mass of each
          graviton times the number of gravitons.
MDM = (MMercuryMSun/PM2)(h/dc) .
With some cleanup:  MDM  =  hMSunMMercury (dc PM2)



This d is now the semi-major axis shown as davg in the diagram.

We can put the value of MDM into the equation to get:

ΔDeg = 180 [MDM/MMercury]

  6. Substituting for MDM we get:

              We can now substitute (hc/G) for PM2 and get:

     And since h/h = 2π we get:

      Simplifying we get the precession equation for an orbit of Mercury:

This equation could have been developed in terms of either the Planck mass (PM) or G.

        The gravitational constant G was chosen because it is a more familiar quantity.

  1. e.Solving the equation:           

Note: that this equation for the precession ΔDeg is not specific for Mercury even though we started out analyzing Mercury’s orbit.

       Constants:    Gravitational Constant G: 6.674 × 10-1,  Mass of Sun MSun: 1.989 × 1030 kg,

       Speed of light c: 2.99 × 108 m/s, Speed of light squared c2: 8.94 × 1016


       Variables:    d:  The Semi-Major Axis in meters, a variable depending upon the planet.

                          RDays:  (to be used shortly) Days in an Earth year divided by the days in a  
                planet’s orbit. We need this factor since we measure precession with    
                respect to the Earth.

    1.  ΔDeg =  (1130.97)(G)(MSun) / (dc2)  the starting equation for precession

                 = (1130.97)(6.674 x 10-11)(1.989 × 1030) / [(d)(8.94 × 1016)]

                 = (1/d)(1.679 x 106) degrees for each orbit of the planet.

              We can convert degrees to arc-seconds by multiplying by 3600 to get:

    2.  ΔDeg = (1/d)(1.6749 x 106)(3600) = (1/d)(6.02964x109) arc-seconds per orbit of planet.  

       We need to multiply the above term by RDays to get precession per orbit of Earth.

    3.  ΔDeg = (RDays) (1/d)(6.02964 x 109) arc-second of precession for each orbit of Earth.

               To get the result in arc-seconds per Earth century we need to multiply by 100.

    4.  ΔDeg =  (RDays/d)(6.02964 x 1011) arc-second for each Earth century.  (the goal)

We can now create a table comparing the precession of Mercury, Venus, and Earth.

f. Table of Precessions using   ΔDeg =  (RDays/d)(6.02964 x 1011)

g. Comparing precessions


       The following table compares the precessions obtained via three sources:

               1. The graviton theory presented here.

               2. Measured via astronomy
               3. Obtained from general relativity calculations



h. Graviton Theory vs. General Relativity Theory

Both theories can calculate the precession of Mercury, Venus and the Earth to good accuracy. Is there a link between these two theories and if so what is it? In section 21 I offer a theory and an experiment to determine if the theory has merit.

a. This graviton theory has Newton’s law of gravity as fundamental, even though it needs to be compensated for a Dark Matter graviton effect. This graviton effect makes it look as if a small planet is orbiting within the orbit of Mercury as predicted by French astronomer Urbain Le Verrier.

In contrast, general relativity has the curvature of space-time as fundamental and posits that Newton’s law of gravity is an approximation when speeds are slow and masses are small.

b. This graviton theory was developed from quantum mechanical concepts and has the graviton as a quantum of mass that corresponds to the photon’s quantum of energy.

General relativity does not at present have a connection to quantum mechanics.

c. The concept of Dark Matter gravitons developed here; in conjunction with the Dark Energy gravitons developed previously (ref 5), accounts for Dark Energy and Dark Matter.

General relativity at present does not account for either Dark Energy or Dark Matter.

d. This graviton theory cannot as yet (see section 21 for a possible explanation) account for the curvature of space-time effect measured by Sir Arthur Eddington showing that the sun can curve the light from other stars (ref 10).

        General relativity predicts and calculates the bending of starlight by observable masses.

e. This graviton theory has no free space; an all-encompassing massy network connects the stuff of the universe. For an object to move (accelerate) it has to lengthen and shorten its attached graviton bundles. This makes the background of space a fundamental component of motion.

General relativity says that acceleration of a mass in free space is equivalent to gravity.

f. Gravitons connect all the masses in the universe. Colliding masses could produce changing patterns on an existing sea of graviton waves. We can think of this as gravity waves. This is a speculation without experimental verification.

General relativity has space-time that connects all the masses in the universe. Colliding masses are thought to produce propagating waves on this existing pattern of curved space-time. Experiments have been made that support this theory.

i.  Parting Thoughts

French astronomer Urbain Le Verrier predicted that a planet (Vulcan) was causing Mercury’s precession (ref 11). According to Tom Levenson,  “Vulcan is remarkable because the idea of this little body inside the orbit of Mercury makes perfect sense,” (ref 12). The work presented here shows that Le Verrier was on the correct track.  There is a mass between Mercury and the Sun. This mass consists of a bundle of long-wavelength gravitons with a mass value of 05.3 x 1016 kg.

Dark Matter as developed here, is a line of mass (a bundle of gravitons) that connects Mercury and the Sun and has a mass of about a million kg per meter for 5.76 x 1010 meters. And the question is, why can’t we see this huge mass directly and can only detect it via the subtle phenomena of precession? I believe the answer is that this graviton mass is a wave phenomenon and manifests quite differently than observable mass such as a golf ball or planet; and to detect it we need to couple to it. This is similar to radio reception. Even if we had a radio transmitter that could produce 1 megawatt of power at a wavelength of 1000 kilometers, we cannot sense this transmission with an antenna that is only 1 meter long; it is not capable of coupling to the energy. However, if we had an antenna that was 1000 kilometers long, we could easily couple to this energy.

The graviton waves we are considering for Dark Matter and Dark Energy behave in a fashion similar to electro-magnetic waves (photons), even though they are not exactly photon electro-magnetic phenomena. We cannot directly sense the graviton mass connecting Mercury and the Sun because it has a wavelength of 5.76 x 1010 meters and a period of 3.23 minutes. This low frequency is below the ELF band (Extremely Low Frequency) and would be in a humorously postulated Ludicrously Low Frequency (LLF) band. We haven’t given much thought on how to couple to this type of electromagnetic like energy. However, the Mercury-Sun combination has the correct distance for coupling to this immense graviton mass.


1: General relativity calculation of precession:    

2: Neil deGrasse Tyson http://www.azquotes.com/quote/1087535?ref=dark-energy

3: Diagrams of Mercury’s orbit: http://calgary.rasc.ca/orbits.htm

4: Photons acting as mass: http://phys.org/news/2014-03-capturing-condensing-realistic-conditions.html

5: Prespacetime Journal- The Geometry of Dark Energy:

6: Apparatus for measuring the gravitational constant G.

7: Planck mass: https://en.wikipedia.org/wiki/Planck_units#Cosmology

8: Richard Muller explains the Higgs boson: https://www.quora.com/How-can-you-explain-the-Higgs-boson-to-a-layman/answer/Richard-Muller-3

9: Semi-major axis of an ellipse: http://www.astro-tom.com/technical_data/elliptical_orbits.htm

10: Arthur Eddington: https://www.wired.com/2009/05/dayintech_0529/

11: The prediction of the planet Vulcan:

12: Tom Levenson: http://news.nationalgeographic.com/2015/11/151104-newton-einstein-gravity-vulcan-planets-mercury-astronomy-theory-of-relativity-ngbooktalk/

                                                                                               Table of Contents     Previous      Next