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was withdrawn the following week, the subject was dropped from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.
published on the American newspaper The Washington Post,
on Dec. 30th, 1983.
Photo by L.Scantamburlo, 2015
(Courtesy John DiNardo's archive)
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, 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.
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 m_{n},
linear velocity v_{n }and
radius of curvature r_{n}. The
total angular momentum of such a system is therefore a sum of terms
of the form m_{n}
r_{n} v_{n }. 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 v^{2} 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 M^{3/}^{2}
R^{1/2}. We're
almost there. For an object whose density varies as the inverse
of R^{2 }the mass of the object
varies as R, which applies throughout the system
[13]. Equivalently, M^{1}^{/}^{2 }varies
as R^{1/2}. Inserting this
equivalence into our running model, we see that the total angular
momentum is indeed on the order of M^{2}. 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
R^{2}). 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} m^{2} / sec ) ( 1.99 x 10^{30} kg )^{2} = 3.18 x 10^{44} kg-m^{2 }/ sec (2)
= 0.225 S-AU ^{2} / yr.
The standard accepted value for the solar system's total
angular momentum is
J_{T} = 3.21 x 10^{4}^{3} kg-m^{2}/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:
˝ v^{2} 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 p^{2}
a^{3} ) )^{˝
}
(6)
The virial theorem for our case can then be expressed as
J w - G M m / a = 0 (7)
J ( G (M + m ) / a^{3} )^{˝} - 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 p^{2} )^{˝}^{ } ) (10)
The semi major axis a of the orbit is computed from the
orbit's period T using Kepler's third law:
a = ( T^{2} G (M + m ) / 4 p^{2} )^{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.
Reproduction is allowed on the Web if accompanied by
the statement
© L. Scantamburlo - www.angelismarriti.it
Reproduced by permission.
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
J_{max} = 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
J_{max} = f ( G / v ) M ^{2}^{ }= (0.1 ) ( 6.67384 x 10 ^{-11} m^{3} kg ^{-1} s ^{-2} ) / ( 300 m /s ) ( 1.99 x 10 ^{30} kg ) ^{2}^{ }
^{ } = 8.8096 x 10 ^{46} kg-m^{2}/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 m_{1} m_{2} / a = 0
J ( G M / a^{3} )^{˝} ? G m_{1} m_{2} / 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
J_{max} = ( 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} m^{3} kg^{-1} s^{-2} ) ( 1.99 x 10^{30} kg )
= 2.361 x 10^{14} m = 1574 AU
And using this value we can solve for the orbital period as
T = ( 4 p^{2} a^{3 }/ 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 |
J_{max} |
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.
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
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
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
Youcanprint.it, Borč Srl, (Tricase, Lecce),
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
Youcanprint.it, Borč Srl, (Tricase, Lecce),
Apocalisse dallo Spazio. L'avvento di Nibiru e dei Vigilanti
by Luca Scantamburlo
Youcanprint, Borč Srl, (Tricase, Lecce), Italy
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
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
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.
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.