linepioneer_10sketch - Credit NASA

The Pioneer 10
Credit:
NASA Illustration; 
sketch from Pioneer Odyssey





Photographic prints of a couple of images sent to Luca Scantamburlo in the year 2006 by an European insider.They would portray Nibiru, the Tenth Planet, photographed by the space probe Pioneer 10 during a secret fly-by (extended mission). Credit: Photo by Luca Scantamburlo, (C) 2013

Photographic prints of a couple of images sent to Luca Scantamburlo
in the year 2006
by an European insider.
They would portray Nibiru,
the Tenth Planet,
photographed by the space probe Pioneer 10
during a secret fly-by
(extended mission).

Credit:
Photo by
Luca Scantamburlo
© 2013
line








THE SUN'S DARK COMPANION
AND THE PHYSICS OF EQUILIBRIUM




by R.F.

November 10, 2015

Astronomers have known about the perturbations in the orbits of Uranus and Neptune for two centuries now, but have yet to come up with a specific cause or explanation, at least officially. The first of these anomalies was observed soon after the discovery of Uranus in 1781, which many thought the later discovery of Neptune in 1846 would be able to explain, but it was not to be. In fact Neptune was found to have orbital irregularities of its own, and so the search for the unseen "Planet X" responsible for the problem continued. The search carried over to the twentieth century and the discovery of Pluto in 1930, which astronomers of the day had hoped was the missing planet that would resolve the long-standing problem, but its mass turned out to be much too small to affect either of the vastly larger planets significantly. NASA became heavily involved with the problem in the early 70s with many years of computer studies, observations and satellite probes to follow targeted at finding the responsible source. With the view that the object was probably very dim in visible light and more likely to be found in the infrared frequency band, the agency launched the Infrared Astronomical Satellite (IRAS) in January 1983. In December of that year an article appeared on the front page of the prestigious Washington Post announcing that the probe had found a candidate source and imaged it twice at a distance of about 530 AU away in the constellation Orion. A number of articles in the national press followed, the last one appearing in US News and World Report in September 1984. That article was withdrawn the following week, the subject was dropped from national news coverage, and NASA now officially denies it ever existed. 


The article published on the American newspaper Washington Post, on Dec.30, 1983, by Thomas O'Toole. -Photo by L. Scantamburlo, 2015 - Courtesy John DiNardo

The article "Mystery Heavenly Body Discovered", by Thomas O'Toole,
published on the American newspaper
The Washington Post,
on Dec. 30th, 1983.


Photo by L.Scantamburlo, 2015
(Courtesy John DiNardo's archive)



US News World Report, 1984, planet X - Photo by L. Scantamburlo, 2015 - Courtesy John DiNardo

The article "Planet X - Is It really Out There?",
published on the American 
US News & World Report
, on Sept. 10th,1984.

Photo by L.Scantamburlo, 2015
(Courtesy John DiNardo's archive)


The story of Planet X went through a similar media cycle. NASA launched Pioneer 10 and 11 in 1972 with the express purpose of looking for Planet X, although that goal only came out years later. The media announcement in 1983 had primarily dealt with a very remote object, which would have to be fairly large at such a distance to account for the orbital perturbations of the large planets in any case. This left the logical avenues of speculation about Planet X unaffected, since there was no announced observation to contest. That changed in 1992. In November of that year Pioneer 10 began sensing a gravitational pull from something that lasted several weeks, a signal NASA had hoped to receive based on where they estimated the unseen planet might be found. NASA scientists in collaboration with a British astronomer published an in depth analysis of the Pioneer data in two journal articles that appeared in 1995 and 1999, in which they proposed that a planet-sized object most likely exists at about a distance of 56 AU from the Sun in the constellation Taurus. After the second paper's appearance the media discussion stopped once again. The agency now officially denies that Pioneer 10 had detected what they announced and that Planet X exists at all. Even more amazing yet is that both NASA and conventional astronomers are now saying that there really are no unexplained perturbations in the orbits of Uranus and Neptune and there never were.  It was all an unfortunate mistake. The thousands of misguided astronomers over the globe who have been immersed in the problem since 1781 had all gotten it wrong because of small errors in the mass estimates of the two bodies, but with that correction the orbits are now recognized as being perfectly well-behaved, no unseen object is bothering the two planets, everything is just fine. Nothing to be concerned about - move along. A history of this research is found in summary in the beginning of a previous study by the author [1] and in extensive detail in the excellent book and web site [2] of Italian research journalist Luca Scantamburlo. Two other researchers who have also produced a veritable wealth of related data, articles, books and insight on the subject are Andy Lloyd [3] and Barry Warmkessel [4].


Italian articles published on newspapers (La Stampa, by Mario Di Martino, March 07, 2007, Turin) and magazines (Focus, 05/2007, by Andrea Parlangeli)). - Photo by L. Scantamburlo, 2015 - private studio and archive


Italian articles published on newspapers (La Stampa, Turin, March 07, 2007,
"Rallenta la corsa delle Pioneer. Č colpa della materia oscura?", by Mario Di Martino)
and magazines (Focus, 05/2007, "Chi frena i Pioneer?", by Andrea Parlangeli).
Since the 1980s, both space probes (Pioneer 10 and 11) began a small gravitational pull.
But Pioneer 10 space probe, in 1992, began sensing a further gravitational pull from something of unknown, that lasted several weeks.


Photo by L. Scantamburlo,
private studio and archive (2015)


In the previous study it was argued that whether anomalies in the orbits of Uranus and Neptune exist or not, something about the solar system's arrangement is very strange in any case. The mass of the Sun contains over 99.9% of the mass of the entire known solar system and yet less than 4% of its angular momentum. From classical physics we know that angular momentum is always conserved in a "closed system", so even though various processes are known to rob a star of its angular momentum over time, it seems strange nevertheless that such a gross disparity can exist between the two. That disparity along with the solar system's missing angular momentum was explained in that study. The basis of the argument centered on determining an estimate of the lower bound for the solar system's current angular momentum, developed from recent observations and theoretical work regarding the evolution of protostellar disks, similar to the one that gave birth to the solar system. The estimate for our own protostellar disk's angular momentum was found to be about 2000% larger than its current value, which was used to suggest that astronomy's conventional reckoning of the solar system's current angular momentum is considerably low and that a large object should be lurking somewhere in the outer reaches of the Sun's influence that accounts for the difference. The analysis showed that the missing angular momentum could be explained by a binary companion of the Sun, possibly a brown dwarf, with a mass some 2 to10 times greater than Jupiter's and a period of some 5000 years or greater. Although the argument presented was consistent with known models of the solar system based on the most current research, it would still be highly desirable to find supporting rationale based on only currently observable data. Short of an actual observation is there any currently measurable evidence we can use to derive support for that study's claims? Fortunately, there is

Brosche's Rule

Celestial objects tend to form stable formations because of the way they balance energy.  Each member of such a system moving through space under the influence of only the system's collective gravitational field acquires two forms of energy: the kinetic energy of motion and the potential energy of gravitational attraction that binds the member to the collective. The way objects move in the system is determined by both. Bodies in any such system can be in a completely helter skelter, random state of motion, or they can merely look helter skelter but actually be in a highly coordinated state of equilibrium. So, how can we tell the difference? The answer comes from an amazing discovery of nineteenth century physics called the virial theorem, described in 1870 by German physicist Rudolf Clausius, father of the second law of thermodynamics. The virial theorem has several forms that apply to a wide range of systems acting under various kinds of force field, but for the above collection of objects moving under its own gravitational field the theorem's statement is concise: the system of objects is in a state of equilibrium if twice its average kinetic energy plus the average gravitational potential energy binding the bodies to the system is zero (or nearly so). The negative gravitational energy holding the system together is twice the positive energy of motion that can pull it apart, which is why galaxies persist as bound structures for billions of years. The delicate energy balance of such a system is very powerful and yet the theorem's condition for equilibrium is actually very lenient. There is no prohibition against objects in equilibrium from moving chaotically, which they very often do, nor is there a requirement for the system as a whole to be in thermodynamic equilibrium, which it invariably is not. The mechanical energy of the system is the sole governing influence.   
 
Celestial systems tend to edge their way towards equilibrium automatically, given enough time, no accretion or other perturbing processes going on and no sufficiently large objects careening through the system disrupting things. This is the second law of thermodynamics in action. One such system that achieved such a balanced state is our own planetary system, which satisfies the virial theorem very precisely as Table 1 below clearly shows [5].  Not only is the theorem satisfied for the average motion of all the planets collectively, but each planet / Sun pair satisfies it individually as well. This may seem like a fortunate state of affairs, but in fact it leads to problems. 


 Table 1.0 Energy and Angular Momentum of the Planets


Beginning in the early 60s German astronomer Peter Brosche at the University of Bonn wrote several ground-breaking papers about an amazing property of celestial systems in equilibrium [6, 7, 8]. Canadian physicist Paul Wesson picked up the work in the mid 80s and produced another set of papers and more decades of research that fully supported and advanced Brosche's original findings [9, 10, 11, 12], which is now an accepted effect of conventional astronomy, although virtually unheard of outside the field. By examining a large collection of celestial systems, Brosche showed empirically that if a gravitational system is in equilibrium per the virial theorem and its average density decreases roughly as the inverse square of the distance from the system's center of gravity, then the system's orbital angular momentum is approximately proportional to the square of its total mass. His analysis showed that the rule (hereafter referred to as "Brosche's rule", my term) is extremely general and has been shown to work for virtually all celestial systems including galaxies and galaxy clusters, star clusters, planetary systems, binary and higher multiple star systems, which together span over 30 orders of magnitude. It even works reasonably well for systems with different fall-off rates for density than the inverse square profile. The closer a given system is to the stated conditions of the theorem, the closer the fit to the angular momentum approximation the rule stipulates, or so one would expect. The rule is now very well attested and works at all scales of celestial systems throughout the universe, but there is one important system it doesn't work for at all - our own solar system. In fact, it's not even close - it's off by over 1000% - Something is very wrong...

Before deciphering the cause of what seems to be happening, we should examine Brosche's rule in a bit more detail to learn what's going on with that and what makes it work. The rule says that gravitational systems in equilibrium with a typical density profile have a strongly preferred value of angular momentum that they don't deviate from significantly. This is an interesting empirical result, but it seems too convenient somehow, almost mysterious. Is there any basis for it in physics? Indeed there is, but it's subtle. The full explanation is mathematical and involved, although the essential thread is fairly easy to understand. To that end the following explanation is by no means a proof but sort of an "executive summary". Several papers with the gory details are listed under Brosche, Wesson and Gribbin in the notes section at the end for the interested reader.     

Let's assume that a typical system of celestial objects in equilibrium can be approximated by a collection of point masses moving under only the influence of the system's collective gravitational field. Assume further that each object n of the collective has mass mn, linear velocity vn and radius of curvature rn. The total angular momentum of such a system is therefore a sum of terms of the form mn rn v. From this we can say more casually in the parlance of physics that the total angular momentum J of the system is on the order of M R v, where M is the total system mass and R and v are representative values of radii and velocities. In a similar vein given that the system is in equilibrium by assumption, we have from the virial theorem that M is on the order of v2 R for the system. This is really the key that makes Brosche's rule work. Combining those two ideas with quasi algebra, we have that J is then on the order of M3/2 R1/2We're almost there. For an object whose density varies as the inverse of Rthe mass of the object varies as R, which applies throughout the system [13]. Equivalently, M1/varies as R1/2. Inserting this equivalence into our running model, we see that the total angular momentum is indeed on the order of M2. The squared mass relationship is a direct result of the system's equilibrium, which imposes an approximate constraint on angular momentum by way of the system's balance of energy described by the virial theorem. The inverse square density profile found throughout the universe fits the final piece of the puzzle in place. 
   
Besides satisfying the virial theorem, the solar system also has a density profile that follows the inverse square relation preferred by Brosche's rule. The Sun has over 99.9% of the mass of the entire system, so beyond about 0.02 AU from the Sun's center, the system's relative mass increase is virtually negligible relative to the whole. Well over 99.99% of the solar system's mass is contained in a circular disk of planets, asteroids, comets, dust and other cosmic debris about 50 AU in radius and about .05 AU thick, if we ignore Pluto which adds almost no mass anyway. If R is the radius of the disk, u is the thickness, and M is the mass, then the density by definition is simply M / (u p R2). This relationship is accurate to within 0.1% whether or not we include any mass other than the Sun's, so clearly the density profile does follow in inverse square relation.  

With the two conditions of equilibrium and inverse square density profile being closely satisfied by the solar system, we can now use Brosche's rule to see how the system stacks up against the larger community of celestial systems. In the following equation the mass of the Sun M is the mass of the solar system to three decimal places, which is to say the mass of the Sun, and J is the total orbital angular momentum derived from Brosche's rule


       J = p M 2                                                                                                         (1)

       = ( 8 x 10-17 kg-1 m2 / sec ) ( 1.99 x 1030 kg )2 = 3.18 x 1044 kg-m2 / sec    (2)

       = 0.225 S-AU 2 / yr.



The standard accepted value for the solar system's total angular momentum is

    JT = 3.21 x 1043 kg-m2/sec                                                                           (3)

         = 0.0227 S-AU 2 / yr.



As can be seen Brosche's rule overestimates the solar system's angular momentum by a factor of 10 or about 1000% above the commonly accepted value. This result is consistent with the results of the author's earlier study [1] although somewhat lower, but in any case Brosche's value is dramatically and inexplicably higher than conventional astronomy maintains.



The Balance of Mass and Energy


As argued in the previous study and as seen above, substantial angular moment appears to be missing from the solar system, and we already know that a good place to look for it is with a binary companion orbiting the Sun. Of the stars nearest our Sun over half have been shown to be in binary or higher multiple systems, and one recent estimate even places the frequency of multiply related stars in the Milky Way as high as 85%.  If our Sun has no stellar companion then it's the statistical exception. But if one does exist, however, why hasn't every astronomer in the world seen it by now? Based on the observation in 1983, the object is apparently small, dim and far away. NASA's astronomer's of the day suggested it might be brown dwarf, which is visible primarily in the infrared band which is exactly the range IRAS was targeted to observe.  A great deal of infrared radiation is blocked by the atmosphere, so finding an object visible primarily in the infrared is much easier from a satellite.          

We don't have the whole story and perhaps never will, but we do have some very interesting data at hand that we can use to assess the situation. The following analysis assumes that the Sun does have a binary companion and we would like to find out how big it has to be to close the angular momentum gap exposed by Brosche's rule.       

Non accreting binary systems of mass M, in our case the Sun and planets, the dark star's mass m, an assumed instantaneous separation distance and relative orbital velocity of r and v, respectively, and G being Newton's gravitational constant ( 6.67384 x 10-11 m3 kg-1 s-2 ), then we have the following total mechanical energy:

   ˝ v2 M m / (M + m) - G M m / r = - ˝ G M m / a                        (4)


The first term on the left is the total kinetic energy of the two; the second term is the total potential energy due to gravitation. The term on the right indicates that the binary system's total energy is constant throughout the orbit;  energy of the system is conserved. The net energy is negative because the two objects are bound to the system gravitationally, which requires energy to unbind.  Equation (4) is actually one form of a standard equation in astrophysics describing the energy of binary systems known as the ?vis viva? equation.  The equation has an equivalent form more useful for our purposes, however, expressed in terms of the total angular momentum J of the binary system: 


    ˝ J w - G M m / r = - ˝ G M m / a                                            (5)


The total angular momentum of the pair is constant due to conservation of angular momentum, and w is the instantaneous rate of rotation of the dark star relative to inertial space. If the Sun's binary system satisfies the virial theorem, and it should because all the planets conform to it so well implying that nothing seems to have perturbed them very recently, then equation (5) can be used to derive its implications. According to the virial theorem the average of twice the first term plus the average of the second term should be identically zero.   

The long term average of the second term is very easy to determine since the only thing that varies is r whose average is simply a, the orbit's semi major axis. Because of conservation of angular momentum, J is constant and again only one thing in the first term varies, the rate of rotation, whose average value comes indirectly from Kepler's third law [14] that relates the orbital period and semi major axis. The average angular rate in radians per unit time derived from the third law [15] is then 

w = 2 p ( G (M + m ) / ( 4 p2 a3 ) )˝                          (6)

   

The virial theorem for our case can then be expressed as


    J w - G M m / a = 0                                                                  (7)


    J ( G (M + m ) / a3 )˝ - G M m / a = 0                                   (8)


On solving for J and then m and neglecting the m / M term that crops up along the way because its magnitude is negligible relative to the adjacent terms, we have the desired expression for the dark star's mass. Note that the expression has been cast in a form to make dealing with units more convenient.

    J = m ( G M a )˝                                                                    (9)


    m = J / ( 2 p ( G M a / 4 p2 )˝ )                                           (10)    


The semi major axis a of the orbit is computed from the orbit's period T using Kepler's third law:

    a = ( T2 G (M + m ) / 4 p2 )1/3                                           (11)


Equations (9) and (10) are simple and require no knowledge of the orbit's eccentricity, which is usually somewhat hard to get.  If we use the angular momentum in equation (2) calculated using Brosche's rule and various possible values for the orbital period, we can now estimate how large the dark star's mass has to be using (10) for a range of orbits to account for the solar system's missing angular momentum. The following table collects together these mass calculations for various orbits that have been proposed by different researchers and others the author found interesting.


Table 2.0 Mass of the Dark Companion



Bottom Line


The standard reckoning of the solar system's angular momentum is inconsistent with established physics and therefore highly doubtful.  Whatever the reason may be we know it has nothing to do with instabilities in planetary motion because applying the virial theorem to the solar system the motions of the planets are in a strong state of equilibrium. But this very fact is also the clue that something is wrong.  If the solar system is in perfect equilibrium and its density follows an inverse square law, then it matches the conditions of Brosche's rule precisely and should be close to its predictions of the solar system's angular momentum ?  one would think within 25 or 30% or so at the least.  But it clearly is not; it's off by almost 1000%. This implies that something significant isn't accounted for ?  the conventional value of the solar system's angular momentum is simply much too small. The study proposes that a binary companion of the Sun is the cause. The analysis shows that a dark body of about one to two Jupiter masses adds relatively little additional mass to the solar system proportionally and yet given an appropriate orbital period the body can provide enough angular momentum to close the gap to account for the discrepancy uncovered by Brosche's rule. Table 2 lists a number of such estimates for orbital periods proposed by various researchers over the years and others that simply looked interesting to the author. Given that each Sun / planet pair satisfies the virial theorem, there is also every reason to believe the Sun / dark companion pair will as well, which is to say that equilibrium throughout the expanded solar system should be preserved even if such an object exists.

The driving force of the study is clearly Brosche's rule. It was only in the 1960s that astronomer Peter Brosche found that when a celestial system drifts towards equilibrium its orbital angular momentum drifts towards a specific value determined by the system's overall mass. This unexpected behavior turns out to be a direct consequence of the distribution of energy within the system brought about by the equilibrium process itself. Celestial systems of every size and description throughout the cosmos have been shown to conform to this rule in studies stretching over 40 years. There is no known theoretical reason why our own solar system shouldn't follow the rule as well, unless conventional astronomy hasn't accounted for everything. Apparently, it hasn't.


Evidence from the Kuiper Belt


Recent discoveries in deep-field astronomy strongly support the likelihood that somewhere in the Kuiper Belt a large object is orbiting the Sun. A major clue came from the Mt Palomar observatory in 2003 with the discovery of the extremely unusual dwarf planet Sedna (90377).  With an orbital period of an amazing 11,400 years it was supposedly much too long and its comet-like orbit way too elliptical for an object so large, but there it was.  Researchers studied it for some time, agreed that it was an unexplained anomaly of some sort, shrugged their shoulders and moved on. Then about 9 years later, another one showed up. Dwarf planet 2012 VP113 ("Biden") with a period of over 4200 years was discovered at the Cerro Tololo Observatory in northern Chile in the fall of 2012. Biden's orbit is less extreme than Sedna's, but it still has an outsized period by planetary standards, a narrow cometary orbit and a mass over half the size of Sedna's.  Even with the highly elliptical orbits of the two unusual objects, their perihelia are so large, both more than 75 AU, that they always stay well outside the orbit of Pluto.  Why should two ?anomalies? look so much the same?
         
In the same paper in the Journal Nature announcing their discovery of 2012 VP113 [16], astronomers Chad Trujillo and Scott Sheppard also analyzed a set of known long-period objects to see if there might be a common thread.  When they compared the orbits of the two dwarf planets with those of 10 well-known asteroids and comets whose orbits reached beyond 150 AU from the Sun, all the objects including the two dwarf planets showed a very interesting trait. The perihelion of each body occurred at very nearly the same point in space that it crossed the plane of the solar system.  For so many orbits to stay bunched up like this on their own after billions of years of evolution the astronomers concluded that something large has to be shepherding objects in its vicinity. Given that idea they estimated that a planet of 2 to 15 Earth masses at a distance of about 250 AU could explain the bunching, but that other possible scenarios would also work just as well, such as a Neptune-sized planet much further out [17].            

Following the article in Nature, other researchers added some interesting twists to the two astronomers' concept.  In 2014 astronomers at the Complutense University of Madrid, Carlos de la Fuente Marcos and his brother Raul, did a detailed analysis of the orbits of the two dwarf planets along with a collection of long period asteroids. They concluded that the estimate of 250 AU for the distance of an unseen planet proposed by Trujillo and Sheppard was credible and that even a second object 200 AU from the Sun might also exist [18, 19, 20, 21]. At about the same time, Italian physicist Lorenzo Iorio working for the Italian government in Bari, Italy, took up the problem by analyzing the orbits of closer planets to refine the estimate of the perturbing object's distance away. Using the known perihelion precessions of Earth, Mars and Saturn, Iorio was able to predict possible bounds for the unseen planet of about 496 - 570 AU if the planet has about two Earth masses and 970 - 1111 AU if it has about 15 Earth masses [22].

The relevance of these studies is less the specific predictions themselves, which differ considerably in any case, than the changed view of conventional astronomy which now accepts that something very large and distant has to exist to be causing the effects observed with the solar system's long-period bodies. Whether one or more objects is responsible, whatever is out there is adding angular momentum to the solar system, bunching up the orbits of dwarf planets and asteroids, and influencing the perihelion precession of planets.  It will take a while before astronomers fully sort out what's happening, but the debate about whether something is really out there is now all but over.  


Conclusion


The solar system has a number of anomalous properties that imply the existence of one or more large, uncatalogued objects orbiting the Sun.  Most of the recent studies that predict this possibility have focused on irregularities in the motions of orbiting bodies observed over the last several decades. This study on the other hand focuses specifically on the solar system's high degree of equilibrium. The bodies of the inner solar system, which contains over 99.99% of the whole, satisfies the virial theorem's energy distribution criterion for equilibrium to almost three decimal places; the same is also true for each Sun/planet pair individually for all 9 planets. The solar system is clearly in a strong state of equilibrium.  If a celestial system is in equilibrium and its density profile follows an inverse square law with distance, which the solar system conforms to on both counts, then the system's orbital angular momentum and mass should be closely related according to Brosche's rule. Using the commonly accepted values for the two parameters, our solar system fails this condition abysmally. Instead the rule estimates a dramatically higher orbital angular momentum than the currently accepted value, which demonstrates that something in the solar system with a very large moment of inertia is missing from the count.  Using the excess angular momentum and the energy balance equation from the virial theorem, the mass of the neglected object was calculated for a range of potential orbital periods and was found to lie between about one and two Jovian masses. 

Supporting evidence from the Kuiper Belt also indicates that the Sun has large distant companion. The discovery of the two dwarf planets Sedna and "Biden" with their extremely long periods and highly elliptical orbits was viewed as so unusual that it seemed to several astronomers that something else might be involved. Long period comets and asteroids arriving in the inner solar system should be coming from all directions, all things being equal, but that isn't what the data shows. Several studies have now proved that a strong bias exists in which part of the sky these objects come from and how their orbits seem to bunch together in a similar way near the Sun. Something is influencing these distant objects and astronomers are now using past observations to figure out where it is and how far away. The current estimates vary wildly and the analysis of the object's angular momentum presented here has not yet been factored in, but at least astronomers are now focused on the heart of the problem. Stay tuned...

© R.F.
November 10, 2015

Notes:

1. R.F."The Dark Star According to Physics",
http://www.angelismarriti.it/ANGELISMARRITI-ENG/REPORTS_ARTICLES/darkstar-
nemesis-vulcan-sundarkcompanion-according-to-physics.htm
2. Scantamburlo, L., L'ombra del Pianeta X, Borč Ltd., Rome, Italy, 2013.  See also his web site www. angelismarriti.it.   
3. Lloyd, A., The Dark Star: The Planet X Evidence, Timeless Voyager Press, Santa Barbara, CA, 2005.  See also his web site www. darkstar1.co.uk.
4. Warmkessel, B., See his web site www. barry.warmkessel.com.
5. Bur?a, M.,  "Distribution of gravitational potential energy within the solar system", Earth, Moon and Planets, 62, Aug. 1993, pp 149- 159.  This excellent study provided the estimates of potential energy for the planets used in Table 1.  The values for kinetic energy were computed by the author. 
6. Brosche, P., "Zum Masse-Drehimpuls-Diagramm von Doppel- und Einzelsternen", Zeitschrift für Astrophysik 286, 1962, pp 241- 252.  
7. Brosche, P., "Über das Masse-Drehimpuls-Diagramm von Spiralnebeln und anderen Objekten", Zeitschrift für Astrophysik 57, 1963, pp 143- 155.
8. Brosche, P., "The Mass-Angular Momentum-Diagram and the Black Hole Limit", Astrophysics and Space Science, 29, July 1974, pp L7- L8.  
9. Wesson, P., "Self-similarity and the angular momenta of astronomical systems - A basic rule in astronomy", Astronomy and Astrophysics 80, Dec. 1979, pp 296- 300. 
10. Wesson, P., "Gravitational interactions and the origin of the angular momenta of galaxies", Vistas in Astronomy 25, 3, 1981, pp 337- 426.
11. Wesson, P., "Clarification of the angular momentum/mass relation ( J = p M2 ) for astronomical objects", Astronomy and Astrophysics 119, 1983, pp 313- 314.
12. Gribbin, J., "The Search for Scale Invariant Cosmology", New Scientist 95, Sept 1982, pp 844-846. Very nice overview paper on the concepts behind Brosche's rule.
13. This relationship between mass and radius is not totally obvious, so a further comment may be helpful.  The mean density of an object is by definition equal to its mass divided by its volume.  The volume of a homogeneous object tends to increase in all three dimensions as it gets larger, so that volume as a whole increases on the order of R3.  For density of the object to follow an inverse square law, the mass therefore has to increase on the order of R for the mass-to-volume ratio to vary as the inverse of R2.     
14. For the purists, a note of clarification is called for. The angle between the radius of curvature and an inertial reference direction and the angle between the line connecting the two bodies and the same reference direction are clearly not the same.  If we establish a common origin and direction of measurement for the two angles, however, both angles go through exactly 360° in the orbital period and therefore both have the same mean value for a complete orbit.  Kepler would have been OK with this.  
15. One other issue should be made clear is that Kepler's third law expresses the period as being so many years per revolution, that is 2p radians. The apparent extra 2p in the equation is to express the angular rate in radians instead of increments of 2p radians. 
16. Trujillo, C., Sheppard, S., "A Sedna-like body with a perihelion of 80 astronomical units", Nature 507, Mar 27, 2014, pp 471- 474. 
17. Crockett, C., "Shadow planet: Strange orbits in the Kuiper belt revive talk of a Planet X in the solar system", Science News, Nov 29, 2014.
18. de la Fuentes Marcos, C., de la Fuentes Marcos, R., ?Extreme trans-Neptunian objects and the Kozai mechanism: signaling the presence of trans-Plutonian planets ?, Monthly Notices of the Royal Astronomical Society: Letters, 443 1, 2014, pp L59-L63. 
19. Khan, A., "Discovery of new dwarf planet hints at other objects in solar system", Los Angeles Times, Mar 26, 2014. On the discovery of the dwarf planet 2012 VP113 (Biden) by Chad Trujillo and Scott Sheppard, the article stated:  "The scientists noticed something else that seemed too odd to be a coincidence: Both Sedna and 2012 VP113 seemed to be making their closest approach to the Sun at similar angles. That could mean that there's a giant planet out there, tugging at both of their orbits in the same way. If so, this ghost planet could have a size of anywhere from 1 to 20 Earth masses, Sheppard said." 
20. Sample, I., "Dwarf planet discovery hints at a hidden Super Earth in solar system", The Guardian, Mar 26, 2014.   
21. Jenner, N., ?Two giant planets may cruise unseen beyond Pluto?, New Scientist, Jun 11, 2014. 
22. Iorio, L., "Planet X revamped after the discovery of the Sedna-like object 2012VP113", Monthly Notices of the Royal Astronomical Society Letters  444, Oct 11, 2014, pp L78- L79.
23. O'Toole, Thomas, "Mystery Heavenly Body Discovered", The Washington Post, Dec. 30, 1983.
24. Giampieri, G., Anderson, J., Lau, E., "Pioneer 10 Encounter with a Trans-Neptunian Object at 56 AU?", American Astronomical Society, DPS meeting #31, #26.04, 1999.
25. Cruttenden, W., The Lost Star, St. Lynn's Press, Pittsburgh, PA, 2005.


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©  L. Scantamburlo - www.angelismarriti.it
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Appendix -  Angular Momentum and Binary Stars


The choreographed motion of celestial systems described by the virial theorem and Brosche's rule is surprising in its broad degree of generality, but for binary systems in particular it has additional implications. The quadratic relationship of a celestial system's angular momentum with mass described by Brosche's rule is a statistical tendency that arises from the balance of energy when the system is in equilibrium, as we saw above.  There is no requirement that any given body in the system has to move a certain way in relation to the system or that the system as a whole has to adhere to the rule to any particular degree of precision. The drift towards equilibrium that accounts for Brosche's rule ultimately traces back to the property of all isolated systems to move towards a state of equilibrium in accordance with the second law of thermodynamics. In the late 80s physicist Joel Tohline at Louisiana State University published two papers that developed an interesting corollary to Brosche's rule specifically for binary systems, which show that nature doesn't have as much liberty in picking orbits as previously thought. Using arguments from the theory of adiabatic gases, the astronomer was able to prove that for every non-accreting binary system in equilibrium there is a maximum value of angular momentum that the system cannot exceed [1A, 2A]. The maximum value is determined by the mass of the two objects and the approximate temperature under which the system formed. For such a system with total mass M, Tohline showed specifically that the maximum angular momentum is given by the formula

    Jmax = f ( G / v ) M 2

where G is Newton's gravitational constant, v is the characteristic velocity of sound and f is a dimensionless constant equal to 0.1. The velocity of sound here is that associated with the medium in which the system formed under the influence of gravity, which increases linearly with the temperature of the medium. In examining a large set of spectroscopic binaries, physicist Virginia Trimble at the University of California showed that virtually all the binaries examined with a period greater than 1000 days formed in temperatures below 1000 K, which implies v should have been about 300 m/sec during the formation [3A].

If we assume that the mass of the solar system M is approximately equal to the mass of the Sun, which we showed above is a very good approximation based on what  we know, then the maximum value of the orbital angular momentum of the Sun and its dark star according to Tohline's rule is given by

    Jmax = f ( G / v ) M 2 = (0.1 ) ( 6.67384 x 10 -11 m3 kg -1 s -2 ) / ( 300 m /s ) ( 1.99 x 10 30 kg ) 2

    = 8.8096 x 10 46 kg-m2/sec = 62.32 S-AU 2 / yr,


which is about 400 times larger than the value estimated using Brosche's rule.   In other word's the Solar system's angular momentum could possibly be much larger and still be consistent with its currently presumed total mass.

By knowing the maximum allowed angular momentum, we can estimate other parameters about the associated orbit as well, including the semi major axis and period of rotation.   Considering an arbitrary binary system with masses m1 and m2  and sum M, we have that the system's angular momentum from equations (7) and (8) become

    J w ? G m1 m2 / a = 0


    J ( G M / a3 )˝ ? G m1 m2 / a = 0


Solving these two equations for J without the simplifying assumption used in equation (9), we get

   J = ( m1 m2 / (m1 + m2 ) ) ( G M a ).˝

If J is at its maximum, then the equation above has a minimal value for the semi major axis when the two masses are equal, from which it follows that

    Jmax = ( M / 4 ) ( G M a ).˝

Combining this expression with that of Tohline gives 

    f ( G / v ) M 2 = ( M / 4 ) ( G M a )˝


which can then be used to solve for the semi major axis as

   a = 16 (f / v ) 2 G M.

   = 16 ( 0.1 / 300 m/s ) 2 ( 6.67384 x 10-11 m3 kg-1 s-2 ) ( 1.99 x 1030 kg )

    = 2.361 x 1014 m = 1574 AU

And using this value we can solve for the orbital period as


   T = ( 4 p2 a3 / G M ) ˝

     = 62450 yrs.


This value of T is then the shortest period a binary system in equilibrium of the above values for mass M and angular momentum Jmax the system can have. Even if we keep the total mass and therefore angular momentum of the binary system the same, any other allocation of total mass between the two stars drives the semi major axis and period higher dramatically. If M is the mass of a binary system and M1 and M2 are the masses of the two stars, then the following table uses the above analysis to predict the associated period T.  

M1 / M

M2 / M

Semi Major Axis a

Period T

Jmax

0.500

0.500

1574 AU

62450 years

62.32 S-AU 2 / yr

0.900

0.100

12145 AU

1338442 years

62.32 S-AU 2 / yr

0.990

0.010

1003724 AU

1005591362 years

62.32 S-AU 2 / yr


The angular momentum is so large to begin with that the period has to go up extremely fast as mass gets small to maintain the system's net total angular momentum. Tohline's extremely high upper bound for the solar system clearly plays little role in predicting anything about the Sun's dark companion. 

© R.F.
November 10, 2015

Reproduction is allowed on the Web if accompanied by the statement
©  L. Scantamburlo - www.angelismarriti.it
Reproduced by permission.

References:
1A  Tohline, J. , Christodoulou, D., ?Star Formation Via the Phase Transition of an Adiabatic Gas." The Astrophysical Journal, 325, 1988, pp 699 -- 714.
2A  Tohline, J., ?The J vs. M Relationship for Binary Stars?, in McNally, D., Highlights of Astronomy, Volume 8, Kluwer Academic Press, 1989, pp 137 ? 138.  
3A  Trimble, V., ?The angular momentum-vs-mass relation for spectroscopic binaries?, Astrophysics and Space Science, vol. 104, 1984, pp 133- 143. 

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COMMENT AND FURTHER REFERECENS
by Luca Scantamburlo

I suggest to the public to read or just to refer to the following writings and books:

The Scientific Search for a Missing Planet. SCIENTIFIC ARTICLES AND STUDIES ON PLANET X. IS IT NIBIRU/MARDUK VENERATED IN MESOPOTAMIA?

by Luca Scantamburlo,
February 22, 2008, www.angelismarriti.it
Article in English language

Alla ricerca di Nibiru. Forze occulte del papato nell'epoca del contatto
Youcanprint.it, Borč Srl, (Tricase, Lecce), Italy
PAPERBACK and EBOOK edition

by Luca Scantamburlo

Fist edition in Italian language, May 2014

Nel segno di Nibiru. Dalla Mesopotamia ai segreti vaticani

by Luca Scantamburlo
Youcanprint.it, Borč Srl, (Tricase, Lecce),
Italy
PAPERBACK and EBOOK edition
New edition, June 2013
First edition (The American Armageddon), with Lulu.com, Lulu Press, Inc.,USA, 2009


L'ombra del Pianeta X. Storia del Decimo pianeta, fra Servizi segreti ed insider

by Luca Scantamburlo
Youcanprint.it, Borč Srl, (Tricase, Lecce),
Italy
PAPERBACK and EBOOK edition
First edition in Italian language, May 2013

Apocalisse dallo Spazio. L'avvento di Nibiru e dei Vigilanti

by Luca Scantamburlo
Youcanprint, Borč Srl, (Tricase, Lecce), Italy

PAPERBACK and EBOOK edition
New and revised edition in Italian language, Jan. 2015
First edition with Lulu.com, Lulu Press, Inc., Lulu Enterprises, Inc., USA,October 2011


Other interesting and well written books on the subject are the following:

Planet X and Pluto

by William Graves Hoyt, The University of Arizona Press
Tucson, Arizona, 1980


The Elements of Astronomy
by Edward Arthur Fath, McGraw-Hill Book Company,
New York and London, 1944





1 statute mile (sta.mi.) = 1.61 km
1 nautical mile (knot)  = 1.853 km (1.15 sta.mi)
1 Astronomical Unit (A.U.) = 149597870 km.


Reproduction is allowed on the Web if accompanied by the statement
©  L. Scantamburlo - www.angelismarriti.it
Reproduced by permission.



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