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SPECULATIONS ON THE SUN'S DARK STAR COMPANION, ACCORDING TO PHYSICS

by R.F.

February 14, 2015

INTRODUCTION BY LUCA SCANTAMBURLO

The following article <<Speculations on the Sun's Dark Star Companion, according to Physics>> - unpublished so far - is written by R.F., an American applied mathematician who lives in USA and is interested in discussing Nibiru's approach and the possibile existence of a dark star, beyond the outer planets of the Solar System. He contacted me by e-mail a few months ago, after reading my articles in English language published on Internet. That's why he sent me his writings. I know his first name and surname but he kindly asked me to spread his study only with his initials. The following study is his second work on the subject of irregularities in the orbits of Uranus and Neptune (after his first article <<When Will Nibiru Get Here According to Physics?>> hosted since last year on my website):in this study he points out the possibile presence of a Dark Companion Star of our Sun, called by him with the expression of Vulcan, and he focus his attention on the Sun's angular momentum.
Of course there is no guarantee on these speculations, above all because we do not know all the astronomical data with enough throughness, and everything is based on clues come out in the course of the search for Planet X and Nemesis (a Dark Star, perhaps a brown dwarf) carried out by NASA, by U.S Naval Observatory and by other scientific institutions involved in the last century. The same existence of Nibiru beyond Neptune is not sure and accepted by modern Astronomy yet. The main idea of the author is that the Sun is involved in a cosmic dance with its invisible dark star companion: Vulcan (another name used after the well-known name of Nemesis). As the Author R.F. says in his scientific article,

  <<[...] One recent estimate even places the frequency of multiply related stars in the Milky Way as high as 85%. With such a high rate of occurrence and given the apparently low value of angular momentum observed in the solar system today, an interesting possibility is that our own star might be in a binary system itself, which raises the question about whether any evidence exists that can support or conversely rule out such a theory. A binary companion for the Sun could answer both puzzles of the solar system's and the Sun's angular momentum both being apparently too small>>.

The sad thing is that weeks ago I have forwarded this work to some astronomers (in Europe and in North America): nobody has found the time to answer, so far. But it does not matter, because if the Author is right - about the existence of Nibiru and of the Sun's dark star companion - sooner or later everyone will be involved in the biggest scientific and social revolution since the times of Galileo Galilei and Christopher Columbus. Another aspect touched by the Author is the following: is there any threat to the Earth posed by Vulcan?
In my opinion this point is connected to the existence of Nibiru, the so-called Planet of Crossing. As the matter of fact, R.F. (the Author) in his original research talks about the <<retinue of satellites which at perihelion could come careening through the inner solar system to create cataclysmic havoc on a scale difficult to imagine>>.
Please, if any astronomer at the moment is fully aware of this and of possibile unknown comet and meteor showers which could already got involved the Earth, contact me. Maybe we can warn the general public, to help everyone to be ready for the survival of humankind, and to accept the reality of Planet X and of his masters ad inhabitants: the Anunna, the Sumerian Gods who are probably returning.

Luca Scantamburlo
February 14, 2015, www.angelismarriti.it

[the following study has been updated by the Author R.F., and it contains appropriate corrections in interpreting Dr. Warmkessel's work - March 30th, 2015]

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

The Sun's Dark Companion According to Physics

by R.F.

Hints of a Dark Star

Astronomers have known about the irregularities in the orbits of Uranus and Neptune for almost two centuries now, since the time when French astronomer Joseph J.F. Lalande first detected, in May 1782, an anomaly on the motion of Uranus: the new planet discovered by Wilhelm Herschel in 1781, did not respect the time schedule predicted for it, during its motion. The responsible cause was the planet Neptune, discovered later on in 1846. Very soon astronomers realized that the presence of Neptune alone could not explain the motions of Uranus: the problem of a trans-Neptunian planet was born (the so-called Planet X, see the work by Percival Lowell). The suspicion early on was that some large celestial body lurking in the outer reaches of the solar system is most likely the cause, and with the discovery of Pluto in 1930 astronomers thought they might have found it, although at the time they had no way to determine Pluto's size to know for sure. But with the discovery of Pluto's moon Charon in 1978 (James W. Christy, U.S. Naval Observatory), astronomers were finally able to compute the mass of the two bodies, which they proved to be too small to account for the unexplained irregularities (Christy and R.S. Harrington). And so the search went on. NASA and the US Naval Observatory had already become interested in the problem in the early 70s, if not sooner, when they began releasing information to the press about their research plans in 1981. 

Exactly what the responsible object is has never been pinned down, at least officially, but in the early 80s it was suspected by government scientists and academia to be an unknown and larger-than-Earth-sized planet, which was called Planet X. One of the first articles describing NASA's intent to decipher the riddle appeared in Astronomy magazine in December 1981 entitled ?Search for the Tenth Planet? [1], in which a leading astronomer at the US Naval Observatory, Dr. Tom van Flandern, described NASA's plans to use the Pioneer satellite probes, conveniently launched in the early 70s, to locate the object. As the article stated:

<<Beyond Pluto, in the cold, dark regions of space, may lie an undiscovered tenth planet two to five times the size of Earth.  Astronomers at the US Naval Observatory (USNO) are using a powerful computer to identify the best target zones and a telescopic search will follow soon after... Van Flandern thinks the tenth planet may be between two and five Earth masses and lie 50 to 100 astronomical units (4.6 billion- 9.3 billion miles) from the Sun.>>

The June 28, 1982 issue of Newsweek added more detail to the story and an alternative to the Planet X theory with an article entitled ?Does the Sun Have a Dark Companion? [2], in which another solution to a perturbing influence was first officially offered by the writer's source, once again USNO astronomer Tom van Flandern:

<<A 'dark companion' could produce the unforeseen force that seems to tug at Uranus and Neptune...  The best bet is a dark star orbiting at least 50 billion miles beyond Pluto.  It is most likely either a brown dwarf or a neutron star.  Others suggest it is a tenth planet...>>

That possibility of an orbiting quasi star became even more apparent later that year with an article in the November 1982 issue of Science Digest, written by Dr. J. Allen Hynek, professor emeritus of astronomy at Northwestern University. The article states that the orbital anomalies of Neptune and Uranus could be caused by either a planetary object 4 to 7 billion miles away or a much larger object such as a brown dwarf at a distance of about 50 billion miles [3]. Dr. Hynek said that NASA believed that data coming from the Pioneer 10 and 11 space probes, launched in 1972 and 1973 respectively, would allow scientists at NASA's research center in Pasadena, Cal., the Jet Propulsion Laboratory (JPL), to determine which of the two it is.

He also said that a telescope on board another satellite scheduled to be launched in January the following year would soon provide additional search capabilities:
<<... planetary scientists at NASA's Ames Research Center plan to use the Infrared Astronomy Satellite (IRAS) planned for launch next month, to try to find a brown dwarf in our solar system or even farther out in space.>> 
Interestingly, a diagram of the solar system in the article depicts both celestial objects, Planet X and the dark star, and the two space probes, Pioneer 10 shown as heading nominally in the presumed direction of the two proposed objects and Pioneer 11 heading in the opposite direction.  It was becoming apparent that authorities were considering the possibility of two large unseen objects in the solar system.    
That strategy apparently worked. IRAS was launched in January 1983, and later that year NASA made the  official announcement that the satellite had in fact detected the infrared signature of a large celestial body beyond the orbit of Pluto. An article proclaiming the event appeared on the front page of the prestigious Washington Post in December of that year: ?Mystery Heavenly Body Discovered? [4]. The article begins ?A heavenly body possibly as large as the giant planet Jupiter and possibly so close to Earth that it would be part of this solar system has been found in the direction of the constellation Orion by an orbiting telescope aboard the U.S. Infrared Astronomical Satellite (IRAS).... The mystery body was seen twice by the infrared satellite as it scanned the northern sky from last January to November, when the satellite ran out of the supercold helium that allowed its telescope to see the coldest bodies in the heavens. The second observation took place six months after the first and suggested the mystery body had not moved from its spot in the sky near the western edge of the constellation Orion in that time.? 
Further on the article states:
<<The most fascinating explanation of this mystery body, which is so cold it casts no light and has never been seen by optical telescopes on Earth or in space, and that it is a giant gaseous planet as large as Jupiter and as close to Earth as 50 trillion miles.>>
This number was corrected by the Post the following day to ?50 billion miles?, which is about 538 AU (astronomical unit, 1 AU being the distance from the Earth to the Sun). The article continues:
<<'All I can tell you is that we don't know what it is,' Dr. Gerry Neugebauer, IRAS chief scientist for California's Jet Propulsion Laboratory and director of the Palomar Observatory for the California Institute of Technology, said in an interview.>>
The researchers apparently had some idea of how big the object is though from the additional comments of  Dr. James Houck of Cornell University's Center for Radio Physics and Space Research and a member of the IRAS science team, who designed the IRAS instrumentation. In commenting on the fact that the object hadn't appeared to move over the six months of the two observations, he said <<This suggests it?s not a comet because a comet would not be as large as the one we?ve observed and a comet would probably have moved. A planet may have moved if it were as close as 50 billion miles but it could still be a more distant planet and not have moved in six months time.>> 
Additional information on the IRAS discovery came out in the national press the following summer. US News and World Report published an article in its September 10, 1984, issue entitled <<Planet X ? Is It Really Out There?>> in which Dr. Neugebauer is again quoted as having confirmed the previous year's observations [5]. The article was very clear that the astronomer believed that IRAS  had detected something large within the solar system:
<<Last year, the infrared astronomical satellite (IRAS), circling in a polar orbit 560 miles from the Earth, detected heat from an object about 50 billion miles away that is now the subject of intense speculation...  Scientists are hopeful that the one- journeys of the Pioneer 10 and 11 space probes may help to locate the nameless body.>>  
The questions raised in the article involved only what exactly the object is, not whether it was really observed or actually exists, which the article makes clear towards the end: 
<<Some astronomers say the heat-emitting object is an unseen collapsed star or 'brown dwarf' - a protostar that never got hot enough to become a star.  However, a growing number of astronomers insist that the object is a dark, gaseous mass that is slowly evolving into a planet.>>  
One week later in its next issue the magazine withdrew the article, which was the last official mention of the object to appear in the national media press to the best of this author's knowledge. Something had changed.
In early 1984 a series of peer-reviewed papers by NASA and a few universities began appearing in various science journals to parse the IRAS findings, most of which listed both Neugebauer and Houck as coauthors. One of the first of these indicated that the IRAS mission had accomplished the impressive task of surveying over 72% of the sky and imaging some 8709 infrared sources, of which NASA had highlighted nine in particular because they appeared in no existing catalogs of 'nonstellar sources' [6]. No mention whatsoever was made about the much closer object at 50 billion miles distance in the Orion constellation that IRAS had seen twice the previous year and splashed across the front page of the Washington Post only a few months earlier. 
The next telling paper came out a year later, in which NASA identified eight of the nine uncatalogued objects listed earlier as galaxies [7]. The ninth source was later declared to be a galaxy as well. Two astronomers from the University of Arizona's Steward Observatory took a bit longer to get the message and published  contrary papers in the mid 1980s based on their detection of radio signals from some of the nine sources, but they too soon came around to the official view that all nine of the sources had to be galaxies and there were no others in question. None of the NASA papers or those of the Steward Observatory ever mentioned the object announced on the front page of the Washington Post that December day in late 1983.       
That this object actually exists and that NASA has been analyzing it now for over 30 years appears to be close to certain. In official circles the object has been called a "hypothetical" gas giant living in the far reaches of the Oort cloud, which astrophysicists and NASA call Tyche, but NASA now claims in a more recent infrared satellite search for it they couldn't find it so therefore it must not exist. Other names being used run the range of Planet X, Nibiru, Nemesis, and Vulcan. We'll use the less common name Vulcan to clearly distinguish the Sun's distant companion from other conceptual possibilities such as Planet X or something else in the Kuiper Belt or Oort cloud. The following analysis explores some of the implications of this discovery from the standpoint of physics, but doesn't really depend on whether NASA spotted Vulcan or not. 


Solar System Formation
Whether a massive unseen object is looming in the outer reaches of the Sun's gravitational dominion or not, something about the solar system's configuration is a bit odd in any case.  The mass of the Sun contains over 99% 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 how can there be such a skewed imbalance between how much the Sun has versus how much the planets have? Jupiter has over 60% of the system's angular momentum by itself. The great minds of astronomy have never reached a consensus on why this is so or why an even larger disparity discussed below is also glaring. There's a deep puzzle about the solar system's arrangement, and this paper will attempt to unravel it.  

The most widely accepted model of the solar system's origin, believed to be at work throughout the galaxy, is called the nebular hypothesis, first proposed in the mid 18th century by the Swedish scientist-mystic Emanuel Swedenborg, followed shortly by philosopher Immanuel Kant and the French polymath Simon La Place. According to this model stellar planetary systems begin as a massive cloud of matter, mostly dust particles and molecular hydrogen, that coalesces into clumps under gravitational attraction, which causes them to collapse into a swirling mass that forms stars and planets. A refinement of this theory called the solar nebular disk model (SNDM) says that the evolving nebula also tends to form a flat protostellar accretion disk with thickness much less than its radius that rotates in the same direction and roughly in the same plane. This rotating mass has massive angular momentum, which earlier in the system's formation can be partially lost due to passing stars, stellar winds and various other dissipation processes, aided by the structure's weak gravitational field, but at some point the structure becomes stable.  At this point the angular momentum of the system remains approximately constant as long as the stellar system's mass stays intact and large external influences are minor. Angular momentum is always preserved in closed systems, so our own solar system would normally be expected to have most of its later stage angular momentum still intact.

The collapse of a molecular cloud into a protostellar disk goes through several stages which astronomers label Class 0 through Class III. Our focus and the research being leveraged involves the Class I stage, which begins at the end of Class 0, about 100,000 years after the start of collapse. At this point the accretion disk is formed, although a hazy envelope of mass still surrounds the disk and protostar. At the end of Class I the disk's surrounding envelope is largely dispersed, at least 80% of the mass of the future star is accreted into the newly formed protostar, and the disk itself has reached a certain dynamic stability.  From this point on the greater part of the disk's angular moment should be conserved in the evolving stellar system.  For stars of relatively low mass such as the Sun, the loss of angular momentum from the protostar / disk system due to stellar winds and other sources is normally small from this point on. In any case the system's angular momentum has to be at least as great as the amount that the solar system has today, an obvious premise we use throughout.     

One key goal of the following discussion is to come up with an approximate lower bound for the angular momentum of the Sun's disk at a representative point in its early development, which specifically does not encompass attempting to develop a model of the protostellar disk's dynamic behavior and evolution over time. To this end we opt to consider the protostellar disk at the end of the Class I stage and take the most conservative estimates available for each parameter used in our analysis, where ?conservative? is taken to mean that value which implies a low or lowest associated value for the disk's angular momentum among all accepted values. The approach draws heavily from data given in a recent survey paper by Williams and Cieza published in the 2011 issue of Annual Review of Astronomy and Astrophysics [8] in which the findings of a number of astronomical observational studies are described.  With the approach given herein some interesting inconsistencies with the current state of the solar system emerge, and the implications of these are used to assess a possible solution.       

The motion of material in a protostellar accretion disk is believed to closely follow a Keplerian rotation in which a subtle outwardly directed influence is embedded. If G is Newton's constant of gravity, then at a distance r from the accreting protosolar mass M, the angular velocity w of a thin ring of matter around the disk at that radius is described by the rule [9, 10]:

w = √G M / r³.                                                                  (1)

The rotation is circularized by gravity around the accreting mass and increases in speed with decreasing radius, producing an inwardly swirling flow of matter that rotates ever faster as it spirals in towards the protoSun. The accretion disk's angular momentum under the influence of the Keplerian rotation in general is fairly complex.  As matter swirls into the Sun it brings angular momentum with it which would normally increase the planetary system's angular momentum. But angular momentum is always conserved in a closed system, so the accretion disk has to do something else to reduce it somehow, which it does by 'transporting' it outwards. This process occurs because adjacent rings of material in the disk rotate at slightly different rates, which produces an outwardly-directed torque in the material of the disk. The torque generates heat and also transports matter away from the center of rotation, which increases the disk's angular momentum at the expense of the protostar's angular momentum. This effect accounts for at least part of the reason why the Sun's angular momentum accounts for only about 4% of that of the entire solar system, since the Sun can lose it to the transport process. The evolution of the accretion disk is a constant struggle between these two opposing effects, but amazingly at some point in time this conflict settles down and the ensemble angular momentum of the disk remains constant over time.  

The following table provides a summary of the angular momentum of each primary body in the solar system, together with other relevant parameters, where the angular momentum of the Sun provided is due solely to its rotation and those of the planets are due strictly to their orbital motion. Specifically, the Sun is presumed to have no orbital angular momentum by conventional astronomy. Although countless smaller bodies are also members of the solar system, these listed in the table are believed to represent over 99.99% of the mass and angular momentum of the known solar system. As can be seen, Jupiter, Saturn and Neptune all have angular momenta much greater than the Sun's, and in fact Jupiter alone has over 60% of the angular momentum of the entire solar system. From this summary the total current angular momentum of the planetary solar system is found to be a little over 3.1 x 1043 kg-m²/sec, which is almost 30 times that of the Sun alone. 

Table 1.0 Angular Momentum in the Solar System

The angular momentum of the solar system's accretion disk can best be understood by considering a thin band around the Sun made up of a circular cross-sectional increment of the disk's material. The band's angular momentum is simply the expression from classical physics, m_r r² w, where m_r is the mass of the band, r is the distance from the center of rotation and w is the band's angular rate of rotation. Because w is simply the rate from equation (1) above, the band's angular momentum Jr can be written by combining the two equations; the term with the radical is known as the specific angular momentum, the angular momentum per unit mass.

    J_r = m_r r² w = m_r (G M r)^1/2 .                                                                 (2)

But mr also varies with radius because the density of the bands drops off with distance and also because the bands get larger as their distance from the Sun increases.  Density is believed to vary as r to the power -p where p according to Williams and Cieza is always less than 1.5, which we take here as the most conservative density profile that drops off the fastest, thereby producing the least angular momentum for our model. That said a small increment dm of the disk's total mass m is computed by taking the disk's density to use the above density rule times its height y, assumed to be roughly constant, its length 2?r and thickness dr, where all constants in the calculation are collected together in the single constant K:  

   dm = k r^-3/2 2 p r y dr = K r^-1/2 dr

The total mass of the disk is then computed by integration:

              m              R

   m = ∫ dm = ∫ K r^-1/2 dr   =  2 K √R - 2 K √R_{o ≈} 2 K √R  

          m_o             R_o

      

The second term is omitted as insignificant due to the relatively small size of Ro and therefore the mass of the disk is found to increase as the square root of the disk's radius for this model.     
To compute the angular momentum J of the accretion disk with mass m, we integrate the angular momenta of all the incremental bands that make up the disk from the surface of the protostar to the outer edge of the protostellar disk

              m                        R

    J =  ∫ √G M r dm = ∫√ G M r K r^-1/2 dr                                                                 (4)

            m_o                        R_o

 

    = ½ 2K √R √G M R - ½ 2 K √R_o √G M R_o ≈ ½ m √G M R       (5)

This last expression follows from collecting together the parameters for the disk's total mass m  from equation (3), excluding that within the protoSun's radius, and from the fact that the lower limit Ro is the radius of the protoSun and again can be ignored in the following analysis since its value is several orders of magnitude smaller than the outermost radius of the solar system R. The solution of interest becomes only the first term of the integration.  It's interesting that even though the angular velocity of the disk decreases with increasing R from equation (1), the corresponding angular momentum increases linearly when we take the variation of mass into account. To be very clear, the above equation represents the instantaneous angular momentum of the non solar mass contained in the protostellar accretion disk once the Class I stage has ended.

The most controversial part of the calculation is picking realistic values for the constants used in the computations, in particular that of the early solar system's effective radius. To do this we use the research of Williams and Cieza again. To make the numbers easier to work with we use "solar coordinates" to express the various quantities with most of the time, where S is the unit of mass expressed as multiples of solar mass, yr is the integer number of years and AU is astronomical units, one AU being the distance from the Earth to the Sun.

The mass of a Class I protostellar disk in theory can be as high as 20% of the mass of the protostar's mass and even greater. Williams and Cieza quote an astronomical study of 20 Class 0 and 1 protostars, however, in which the mass was seen to vary from 0.02 to 0.10 solar masses with a median value of 0.04, although the masses of the stars were not provided. The lower value is the more conservative, and this is the one selected for the study. A lower value of disk mass is not all that unreasonable in any case because the researchers site another study that indicates the protostar typically gains half its mass from the accretion disk in only about 7% of the 500,000 years of its Class 0 and I lifetime.   

The radius of a protostellar disk varies considerably and is difficult for astronomers to assess reliably in any case because of the disk's tenuous structure in its outer regions that causes it to emit only very weakly. Several studies have been performed of protostellar disks in the Orion constellation, however, one of which had a sample of 125 disks for which the researchers inferred a median radius of 75 AU. Another study of Orion protoplanetary disks found that radii varied from 50 to 194 AU, so because 50 AU was the smallest identified, in the interest of angular momentum conservatism we'll use 50 AU.

So, with these numbers we estimate a lower bound for the angular momentum Jp of the protostellar accretion disk corresponding to the values for mass and disk radius indication, and using the interesting fact that in the units being used GM equals (2p)²:

   J_p = ½ ( .02 S ) (6.28 )? 50 = 0.444 S-AU ² / yr.                 (6)

In comparison, the known angular momentum of the solar system's 0.14% non solar mass is known to be

   J = 3.1 x 10 ⁴³ kg-m²/sec = .0219 S-AU ² / yr.                    (7)

Adding in the Sun's angular momentum due to rotation, the angular momentum of the entire known solar system is therefore not much larger:

   J_T = 3.21 x 10 ⁴³ kg-m²/sec = .0227 S-AU ² / yr.               (8)

This value is an order of magnitude smaller than the above theoretical ?minimum? value Jp, which in a nutshell is the problem at hand. Over 95% of the solar system's minimal protostellar angular momentum is now missing. In a study cited by Williams and Cieza that was performed in 2009 on eleven protostellar disks, the researchers found that the specific angular momentum, i.e. the angular momentum per unit mass, of all of them fell in the range of 10^15.4 m²/ s to 10^16.9 m²/ s.  Using the specific angular momentum is nice on the one hand in that it removes the relative effect of mass which serves to make comparisons easier.  But it also adds a degree of subtlety that warrants a note of caution, because it tends to compress differences.  The specific angular moment of Jupiter for example is 1.02x1016 , but that of all the planets taken together is 1.11x1016. Even though Jupiter has 62% of the planetary angular momentum, it has over 98% of the planetary specific angular momentum. Similarly, the lower bound for the specific angular momentum corresponding to Jp computed in equation (6) is 1.57 x 1016, only 40% larger than that of all the planets taken together even though Jp is 20 times larger than J, the angular momentum of all the planets together. The main message to note here is that although Jp  falls in the middle of the range found in the study, so does the specific angular momentum of the planetary solar system. Both numbers are ?reasonable? in the sense of falling in the range of actual observations. From this sample alone we cannot conclude that the solar system's angular momentum looks odd.     

Whatever subtle dynamics the solar system may have gone through during its evolution that caused it to lose more than 95% of its angular momentum, perhaps much more, however, this massive loss has no explanation in conventional astronomy, the above study notwithstanding. The loss is excessive by any measure, but in all likelihood is probably considerably larger yet, because of the extremely conservative approach used to estimate the lower bound derived for the solar system's protostellar angular momentum at the end of its Class I stage.  But could it be that the original angular momentum didn't get lost at all and is still somehow hiding in the solar system? A body of sufficient mass in a long-period orbit could possibly account for the apparent loss.   


A Binary Alternative
The galaxy's organization is now known to be very different in some ways than astronomers had long presumed. Of the stars nearest our Sun over half have been shown to be in binary or higher multiple systems, a fact which is apparently true throughout the galaxy.  One recent estimate even places the frequency of multiply related stars in the Milky Way as high as 85%. With such a high rate of occurrence and given the apparently low value of angular momentum observed in the solar system today, an interesting possibility is that our own star might be in a binary system itself, which raises the question about whether any evidence exists that can support or conversely rule out such a theory. 

A binary companion for the Sun could answer both puzzles of the solar system's and the Sun's angular momentum both being apparently too small. Binary systems of low mass stars can have orbital angular momenta three orders of magnitude larger that of the solar system and five orders greater than the spin angular momentum of the Sun, which rotates at a rate typical of low-mass stars. If we assume that a hidden solar twin does exist, then its additional angular momentum may be able to explain why the Sun's and solar system's is so small. Maybe the solar system's angular momentum didn't really go away, it was just transferred to something we can't see. The following analysis sheds light on just how big such a hidden twin might be.


Orbital Analysis
If we know the approximate orbital period and eccentricity of a body in orbit around the Sun, Vulcan if you will, and we ignore minor perturbations induced by the planets, then we can compute its Vulcan's orbital semi major axis and mass uniquely, provided that we have some idea of the solar system's ?original? angular momentum, which we now have in the form of our lower bound estimate. The particular period and eccentricity to use comes from various sources, but there are not many that have been proposed, and so we examine a few of these. To begin with, we describe the methodology to be used. 

We need to know how to calculate key attributes about the binary system's orbit including the semi-major axis of its orbital ellipse, its angular momentum and the mass of the Sun's dark twin. For the semi-major axis, using Kepler's third law we know that the period T of two objects in orbit around one another satisfies the following relationship:

   T ² = 4 p² a³ / G (M + m)              (9)

T is the orbital period of the orbiting body around the Sun,
M is the mass of the larger, central body, here taken to be the Sun
m is the mass of the second object
G is the Newton's gravitational constant and,
a is the semi-major axis of the orbital ellipse.

If a is expressed in astronomical units (AU's), M and m are in multiples of solar mass and T is in years, then M is unity, m is a fraction (of the Sun's mass), the other constants can be suppressed and we can recast the above equation to calculate the semi-major axis as 

   

   a = ( T² (M + m) )^1/3                   (10)

Another key expression we need is called the specific relative angular momentum h, which is the total orbital angular momentum per unit mass of two orbiting bodies. In a closed system this quantity is constant, a direct consequence of Kepler's second law, which says that a line joining a planet and the Sun sweeps out equal areas in equal time intervals. For h, this statement is basically another way of saying that angular momentum is conserved. There are a number of equivalent ways to express h, but the one most useful for us is given as:   

 

   h = ( 2p / T ) a² √ 1 - e²        (11)

Using h and standard astrophysics the total orbital angular momentum J of the two bodies relative to their center of mass is then h times the ?reduced mass? of the bodies. Note that this is the total orbital angular of the dark companion and the Sun, since the Sun now has an orbit around its companion as well. Because at least 95% and most likely much more of the solar system's orbital angular momentum is unaccounted for, the analysis ignores the relatively small planetary angular momentum and for convenience of calculation assigns that of the entire accretion disk to the Sun's dark companion. The primordial disk not only produced a small dark star with angular momentum, but in the process induced orbital angular momentum in the Sun as well in its role as the larger member of a binary system.  Accordingly, J is then the combined orbital angular momentum of both bodies, and is given by:

    J_p = h M m / (M + m ).          (12)

For a given period and eccentricity the above equations can be used to solve for the mass and semi major axis of the dark companion uniquely.  By fixing the eccentricity, orbital period and angular momentum of the solar system, the orbital semi major axis and mass of the hidden companion form a unique matched set defined by equations (10), (11) and (12).  Specifically, when a star with known mass has an object in orbit around it with known period and eccentricity, then that object's mass and semi major axis are determined uniquely. To compute these parameters from the equations directly is a bit messy, however, but fortunately there is a simpler way.  If we solve for m in equation (12), we can create a recursive algorithm to determine both quantities using a simple bootstrap scheme. The desired expression for m is

   m = J_p M / (M h - J_p )                         (13)

By setting m to zero initially and using equations (10), (11) and (13) in that order recursively, both parameters can be readily calculated. Once the initial value for m is computed in equation (13) the next iteration simply uses that value of the dark object's mass in equation (10) in the next iteration. Notice that each iteration also produces an updated version of the semi-major axis a, which is used for other calculations within that iteration. The algorithm converges quickly; two iterations are normally sufficient to compute both parameters of interest, m and a, to three or more decimal places.     

The following table provides the calculated value of m using this methodology for a range of orbital scenarios of interest. The values m and a computed for each scenario completely compensates for the "missing" angular momentum defined as the difference between the known and presumed theoretical value used in the computations given in column four of the table. The measure of Vulcan's mass in terms of Jovian masses in column nine, Jupiter's mass being 0.0009546 S, is interesting, and gives maybe a clearer idea of the range of possible sizes of the Sun's dark companion. 

The scenarios examined center around four particular orbital periods. The first of these is 3661 years proposed for Planet X by historian Zecharia Sitchin and examined by a number of researchers including me. The period of 5000 years is the approximate period proposed by physicist Barry Warmkessel [11] based on his analysis of comets and other phenomena. The mass for the dark star he proposes to be less than 0.0005 solar masses. The third period derives from NASA's announced detection of an object in 1983 at a distance of some 50 billion miles (537 AU). We don't really have any way to know where the object was in its orbit since NASA only published the one number, but if we assume arbitrarily that the distance away then was about half the length of its orbit which equals the length of its semi-major axis, then the resulting period turns out to be a little under 12500 years. The last period of interest is 26000 years proposed by researcher Walter Cruttenden [12], which matches the time the Earth's precession takes to complete a full 360° cycle. Cruttenden proposes that the Sun's dark companion contributes to the precession and that its orbit might be synchronized with the precession, which interestingly enough seems to be slowing down. (This writer strongly disagrees with this view, by the way.) The mass for the dark star he proposes as .06 solar masses. The eccentricities of .54 and .90 were selected because one researcher is pushing very hard for the former, and the latter induces a cometary orbit for comparison. Also, visual binary stars have been found to have eccentricities of about .50 on average anyway, so the former number is close to one we might expect anyway. Other values can readily be evaluated using the computational methodology provided above.

Table 2.0 Characteristics of the Dark Companion

To close the gap between the solar system's currently presumed angular momentum and an associated ?original? value computed from data based on astronomical observations of protostellar disks in the Milky Way, Table 2.0 indicates that an object of several Jovian masses is required. Orbits with larger periods clearly reduce the required size of the object as expected.  It's interesting that if such an object is several thousand AU distance and emits primarily in the infrared, it could indeed be very difficult to observe and its effect on the solar system is apparent primarily through its angular momentum. The above argument is interesting and credible, but is it really true? There may be a way to determine its validity by direct measurement, however. If an object of several Jovian masses is adding angular momentum to the solar system, then the Sun can tell us. 


Perturbing the Sun
Knowing roughly what Vulcan's orbit looks like, we can now probe the question regarding to what extent if any the Sun is being influenced by Vulcan's presence. According to the laws of celestial mechanics two bodies orbiting each other are really both orbiting a fixed point in space between them, the location of their common center of mass. Letting M and m be the mass of the Sun and Vulcan respectively and rM and rm be the two distances from the center of each body to the common center of mass, then the following classical relationship holds:

   M r_M = m r_m                                                                       (14)

Both rM  and  rm certainly vary as the two bodies orbit each other, but the variation occurs in a way such that equation (14) is always satisfied. If we solve for rM in the above expression we get the distance from the Sun's center of mass to the center of its orbit:

   r_M = r_m m / M.                                                      (15)

At aphelion, the point that the two bodies are farthest apart, rM  and  rm  are both at their maximal values and at that point have the following relationship to the semi major axis of Vulcan's orbit:

   a (1 + e ) = r_M + r_m                                                       (16)

Combining equations (15) and (16) appropriately, we arrive at

   r_M = a ( 1 + e ) m / ( M + m ).                           (17)

To get some idea of how the Sun is moving about in its dance with Vulcan, we can assume for the sake of discussion that Vulcan is orbiting somewhat like the orbit we guessed at to model NASA's IRAS observation of 1983. To do this we assumed an eccentricity of .54, an orbital period of 12500 years and a semi major axis a of 538 AU. If we consider the corresponding value of Vulcan's mass from Table 2.0 computed as 0.0036 S and the other numbers as indicated immediately above, then on applying equation (17) rM is found to be about 2.97 AU for this case. This implies that the Sun would have an orbit about the common center of mass with a maximum radius three times Earth's orbital radius around the Sun. With longer periods and / or a larger mass for Vulcan, the Sun's orbital radius would be appropriately larger.  Presuming Cruttenden's 26000 yr period, the eccentricity also at 0.54 (he actually proposed 0.038), the orbit's semi major axis of 879 AU and a computed mass for Vulcan of 0.0063 S, then rM  becomes 8.47 AU, which is larger than the orbital radius of Jupiter! An interesting thing to note is that as with this case, the orbits of the Sun and Vulcan never intersect. Vulcan appears to have the orbital character of a very out-sized planet. 

These equations also allow us to take a closer look at some of the earlier proposed orbits for
Vulcan in a different light. For Dr. Warmkessel's proposed configuration of Vulcan with period of 5000 years, eccentricity 0.54, semi-major axis of 292 AU and mass of 0.0005 S, then rM is only 0.22 AU using equation (17), well within the orbit of Mercury. The problem with this scenario is that the mass is too small to account for the solar system's missing angular momentum. If the mass figure is actually close to being correct then either the eccentricity has to be significantly smaller, the period considerably larger or both.

For Dr. Cruttenden's proposed Vulcan characteristics with period of 26000 years, eccentricity 0.038, semi-major axis 878 AU, and mass of 0.06 S, the value of rM becomes an astonishing 51.58 AU! The orbit of the Sun is comparable to the orbit of Pluto, and the nearly circular orbit implies that for his scenario Vulcan's perihelion is 845 AU. If we take the angular momentum into account and solve the orbital equations for a, h and m accordingly, then again Dr. Cruttenden's mass for Vulcan seems excessibely high. But again, we don't really know how large the missing angular momentum actually is, just its minimum size. In any case Cruttenden's orbital scenario appears to be very unlikely.            

A crucial implication of this portion of the analysis is that it uncovers the Sun's missing angular momentum. That portion of the Sun's angular momentum due to its orbital dance with Vulcan is actually embedded in equation (12), which represents the sum of both bodies' orbital angular momentum about the pair's common center of mass. The portion due strictly to the Sun's orbiting motion is the counterpart to Vulcan's orbital angular momentum. The angular momentum of the two bodies individually in the following two equations produce equation (12) when summed together [13]:

   J_sun = h M m² / ( M + m )² = J_p m / ( M + m )          (18)

   J_vul = h m M² / ( M + m )² = J_p M / ( M + m )          (19)

An interesting aspect of these equations is that the angular momenta of the Sun and Vulcan can both be calculated without knowing anything about the orbit! All we need are Jp and the mass of each body. Using these two equations another interesting relationship is now apparent:

   J_sun = J_vul m / M                                                          (20)

The value of Jsun is entirely determined by knowledge of Jp , M and m, which depend on the overall arguments used in the discussion. Even though we can't know the exact size of Jsun because Jp is only approximated and the calculation of m depends on an assumed orbit, we can get a rough idea about the range of Jsun  from the various cases examined in Table 2.   For comparison purposes, the Sun's rotational angular momentum given in Table 1.0 is equivalent to .0007771  S-AU2/yr in the solar-based units we're using. In the table the lowest and highest estimates of Jsun are .0013 and .0047 S-AU2/yr , which are respectively 1.7 and 6.1  times more than the known value of the Sun's rotational angular momentum.  If a large body is orbiting the Sun, the Sun's orbital angular momentum has to increase as a result according to equation (20), the dominant influence by far being Jp, the approximate lower bound for the angular momentum of the protostellar accretion disk. However large Jp may turn out to be as our knowledge of planetary evolution improves, the Sun's missing angular momentum has finally been accounted for, at least theoretically.

One final idea to consider is that if the Sun is actually scribing an elliptical arc through space, this motion is detectable!  Given that the Sun is a distance r from the dark star at any particular point in time, then the instantaneous angular velocity of the Sun's orbit according to celestial mechanics is 

   

    Omega = (G (M + m) / r ³ ) ^½                                        (21)

If the Sun is rotating through space at this rate, then so is the Earth.  For an orbital period of let's say 12500 years as an example, the associated angular rate of the Sun turns out to be about one degree of arc every 35 years on average, or about 1.7 arc minutes per year.  Modern instrumentation can certainly sense this rate and distinguish it for the other motions the Earth is undergoing. Mercury's perihelion precession, used as one of the key benchmarks of general relativity, is 0.9 arc minutes per year, which current technology has determined to four decimal places for at least the past two decades.  Within a month or so we should be able to have a very good idea about whether a dark star is dragging the Sun around or not.


Discussion
As we've argued, something fairly large is missing from conventional astronomy's description of the solar system, perhaps something that NASA has been watching for several decades.  The solar system's angular momentum is simply too small assuming even the most conservative estimates of that which its accretion disk must have had in the protostellar past, given the non-solar mass and angular momentum in the solar system we have today. There are no obvious ways to account for the difference between the measured and theoretical momentum values without some fairly large object in orbit around the Sun supplying it. The Kuiper Belt and Oort Cloud are both too short of mass by current best estimates. Examining a variety of different orbital scenarios proposed by various researchers suggests that in fact it takes an object with a mass larger than Jupiter's to supply the missing momentum in all the cases examined, given that its period is at least 5000 years based on current theories. NASA on several occasions affirmed that the object exists and even claimed to have imaged the object in the infrared frequency band in 1983 using the IRAS research satellite.  That this object exists and that NASA has been carefully watching and analyzing it for over 30 years is all but certain, even though nothing has come out officially since the mid 80s.  Based on this analysis what else can we infer?   

We looked at orbital periods that various researchers have proposed  for Vulcan based on a various arguments, although none that used angular momentum specifically. This approach doesn't claim where the object should be located in space but rather assuming that it has a given period and eccentricity the study has showed about how massive the object has to be for a lower bound of the radius of the protostellar accretion disk. To be very clear, a higher value of the radius implies a larger mass for Vulcan. As with the body's orbital period, we also of course don't know what its eccentricity is either, and so we examined two. The average eccentricity of visual binary stars is known to be about 0.50, but one writer seems so sure that his eccentricity is right at 0.54 that we used that number instead of 0.50. The other eccentricity selected was 0.90 which produces a cometary orbit with an elongated semi-major axis. These two eccentricities address two of the more interesting orbital regimes.

Conclusion
The basic premise of this study is that angular momentum is missing from the solar system in modern astronomy's standard reckoning. The Sun's angular momentum is disproportionately small compared to that of planets, and that of the entire solar system is much too small relative to conservative estimates of the amount that the accretion disk that spawned it must have had. Something is missing from the reckoning, and that something may just be a large body of several Jupiter masses orbiting the Sun well beyond the orbit of Pluto. The above analysis showed that a body of about two to ten Jovian masses with a variety of different orbits can account for both irregularities. Furthermore, there is even evidence from the early 80s that the IRAS satellite probe very likely imaged the object in 1983. The analytical argument is difficult to dismiss and hard observational evidence may exist as well in NASA archives. Time will tell but there is no other solid theory on the table that does all these things, although conventional astronomy does in fact have a somewhat tortured argument to explain them [14]. The authors are even somewhat apologetic in stating that they have no real evidence for the explanation they propose.      

The angular momentum missing from the solar system is consistent with the existence of a binary companion of the Sun. What is particularly surprising is that the mass of companion doesn't vary drastically in size for a variety of presumed values of the companion's orbital period and eccentricity. In almost all cases examined the size of the object is well under 1% of the Sun's mass, falling between 3.0 and 11.2 Jovian masses ? not small by any means but the smallest possible given the presumed orbital characteristics and the lower bound for the angular momentum of the solar system's protostellar accretion disk that spawned the our planetary system. The hidden object could just as well be much larger. What this means remains to be seen, but the likelihood that our Sun does have a hidden Jovian-sized companion now seems much more likely. The angular momentum apparently missing from the solar system can be provided by a large body although having considerably less than 1% of the Sun's mass.

One of the most fascinating aspects of this theory is that it's testable.  If the Sun is scribing an elliptical orbit through space then so is the Earth and all the other planets, and current instrumentation can sense it. A similar measurement was done with Mercury's perihelion precession several decades ago. If we really want to know if the solar system's missing angular momentum is due to a large distant object in orbit around the Sun, we have the technology to determine it. The only question is do we have the will. And that very much remains to be seen.      

The real question of the hour is does Vulcan pose a threat to the Earth? The best answer for now is it depends. All of the orbits examined or that are reasonably likely imply that Vulcan's perihelion is well outside the orbit of Pluto, but even if that's so an object of several Jovian masses coming that close to the Sun would be sorely felt throughout the solar system you can be certain. Possibly a larger threat is that Vulcan may well have its own retinue of satellites which at perihelion could come careening through the inner solar system to create cataclysmic havoc on a scale difficult to imagine. The other major threat of such an object is its likely ability to drive any number of comets hurtling in our direction. As more about Vulcan's orbit and properties becomes known, the threat assessment will become more realistic. For now, stay tuned.

© R.F.
February 14, 2015
Last updating, Febr. 25, 2015 - March 30, 2015
[with appropriate corrections and comments in interpreting Dr. Warmkessel's work.]

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

References:

1. ?Search for the Tenth Planet?, Astronomy, Dec. 1981, p 17.
2. ?Does the Sun Have a Dark Companion??, Newsweek, June 28, 1982, p 83. 
3. Hynek, J. Allen, ?Mysterious Planet X?, Science Digest, Nov. 1982, p 42.
4. O'Toole, Thomas, ?Mystery Heavenly Body Discovered,?, The Washington Post, Dec. 30, 1983, pg A1.
5. ?Planet X ? Is it Really Out There??, US News & World Report, Sept. 10, 1984, p 74. 
6. Houck, J., Neugebauer, G., et. al., ?Unidentified Point Sources in the IRAS Minisurvey?, The Astrophysical Journal, 278, March 1984, pp L63- L67.  This article identifies each of the nine point sources by name and coordinate location. 
7. Houck, J., Neugebauer, G., et. al., ?Unidentified IRAS Sources:  Ultrahigh-Luminosity Galaxies?, The Astrophysical Journal, 290, March 1985, pp L5- L8.
8. Williams, J., Cieza, L., ?Protoplanetary Disks and Their Evolution?, Annual Review of Astronomy and Astrophysics 49, 2011, pp 67- 117.  
9. Beskin, V., et. al., Accretion Disks, Jets and High Energy Phenomena in Astrophysics, Springer Verlag, Berlin, 2003.
10. Benacquista, M., An Introduction to the Evolution of Single and Binary Stars, Springer Verlag, Berlin, 2013.  Many of the ideas herein are described in Chapter 2 of this excellent source.  
11. Kuehne, H., Warmkessel, B., ?Vulcan's New Orbital Parameters?,
http://www.barry.warmkessel.com/2002Paper.html.
12. Cruttenden, W., ?Calculations?,  
http://www.binaryresearchinstitute.org/bri/research/calculations/size.shtml  He also suggests other orbital periods as well that corresponds to the length of the precession cycle in times past.
13. Kleppner, D., Kolendow, R., An Introduction to Mechanics, McGraw-Hill, New York, 1973. 
14. Rozelot, J., Neiner, C. ed., The Rotation of Sun and Stars, Springer Verlag, Berlin, 2010, pp 1- 14. 
15. Nibiru, Mystery Heavenly Body Solved?  2013, https://www.youtube.com/watch?v=R_iHAhU_gE4  This is an interesting video on the early publications about the IRAS mission findings. 
16. The Binary Star of the Sun, 2013,
https://angeliinastronave.wordpress.com/2013/02/26/la-stella-binaria-del-sole-parte-1-di-2-lombra-del-pianeta-x-
dietro-la-pioggia-di-meteoriti/
An interesting web article on the Sun's alleged binary companion.
17. Cruttenden, W., The Lost Star, St. Lynn's Press, Pittsburgh, PA, 2005.
18. Lloyd, A.,  The Dark Star:  The Planet X Evidence, Timeless Voyager Press, Santa Barbara, CA, 2005.

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, Borè 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, Borè 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, Borè 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, Borè 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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