It is estimated that approximately one third of the star systems in the Milky Way are binary or multiple, with the remaining two thirds being single stars.[1] The overall multiplicity frequency of ordinary stars is a monotonically increasing function of stellar mass. That is, the likelihood of being in a binary or a multi-star system steadily increases as the masses of the components increase. [2]
| Mass range | Multiplicity frequency | Average companions |
|---|---|---|
| ≤ 0.1 M☉ | 22%+6%−4% | 0.22+0.06−0.04 |
| 0.1–0.5 M☉ | 26%±3% | 0.33±0.05 |
| 0.7–1.3 M☉ | 44%±2% | 0.62±0.03 |
| 1.5–5 M☉ | ≥ 50% | 1.00±0.10 |
| 8–16 M☉ | ≥ 60% | 1.00±0.20 |
| ≥ 16 M☉ | ≥ 80% | 1.30±0.20 |
The mass of the sun is almost 4.4 × 1030 lbs (2 x 1030 kilograms), that is 4.4 followed by 30 zeros, equivalent to about 333,000 Earths. This value is defined as a solar mass (1.0 M☉) and becomes important when considering the masses of other stars which are, perhaps unsurprisingly, measured in terms of solar masses. [3]
Historical Multiplicity Analysis
The first modern, volume-limited multiplicity survey of solar-type stars was conducted by Duquennoy & Mayor (1991) [4], who studied 164 objects out to 22 pc by collating results from many separate studies and obtaining their own new radial velocity measurements. For the last two decades, this study has represented the “gold standard” of multiplicity surveys among field stars, against which most subsequent multiplicity surveys compared their results. Nonetheless, significant incompleteness potentially affected the conclusions of Duquennoy & Mayor, prompting Raghavan et al. (2010) [5] to revisit this topic with a volume- limited sample of 454 star out to 25 pc. In addition to a much cleaner sample, that survey built on much improved observational methods to reach a very high completeness rate (≈ 97%). This study now represents the most complete multiplicity study of a well-defined sample and likely will remain a standard reference for years to come.
Multiplicity Frequency Analysis
Instead of an average of 1.4 companion per primary star, Duquennoy & Mayor (1991) [4] and Raghavan et al. (2010) [5] demonstrated that CF MS 0.7−1.3 M⊙ = 62 ± 3%.
This means that for stars in the mass range of 0.7–1.3 M⊙, approximately 62% of them have at least one companion. The average number of companions per primary star in this mass range is 1.4.
Based on the information provided, it is estimated that approximately one third of star systems in the Milky Way are binary or multiple, while the remaining two thirds are single stars. It has been observed that the overall multiplicity frequency of ordinary stars increases as the masses of the components increase.
Specifically, for stars in the mass range of 0.7–1.3 M⊙, which includes our Sun with a solar mass of 1.0 M⊙, it has been demonstrated that the multiplicity frequency is 62 ± 3%. This means that approximately 62% of stars in this mass range have at least one companion, and the average number of companions per primary star is 1.4.
Considering these findings, it can be concluded that there is a statistically significant likelihood of our Sun having a companion.
Oriental Astronomy
Swami Sri Yukteswar, born on May 10, 1855, was a renowned Indian yogi, guru, and spiritual teacher. He was a disciple of Lahiri Mahasaya and played a pivotal role in introducing the ancient science of Kriya Yoga to the Western world. With his profound wisdom and deep understanding of spirituality, Swami Sri Yukteswar provided invaluable guidance to his disciples, including Paramahansa Yogananda, the author of the famous book “Autobiography of a Yogi.” Throughout his life, he emphasized the importance of harmonizing science and spirituality, bridging the gap between Eastern and Western philosophies.
Published in 1894 his book “The Holy Science” serves as a comprehensive guide to understanding the underlying principles of spirituality and its relationship with science.
A short discussion with mathematical calculation of the yugas or ages will explain the fact that the present age for the world is Dwapara Yuga, and that 194 years of the Yuga have now (a.p. 1894) passed away, bringing a rapid development in man’s knowledge.
We learn from Oriental astronomy that moons revolve around their planets, and planets turning on their axes revolve with their moons round the sun; and the sun, with its planets and their moons, takes some star for its dual and revolves round it in about 24,000 years of our earth - a celestial phenomenon which causes the backward movement of the equinoctial points around the zodiac. The sun also has another motion by which it revolves round a grand center called Vishnunabhi, which is the seat of the creative power, Brahma, the univeral magnetism. Brahma regulates dharma, the mental virtue of the internal world.
When the sun in its revolution round its dual comes to the place nearest to this grand center, the seat of Brahma (an event which takes place when the Autumnal Equinox comes to the first point of Aries), dharma, the mental virtue, becomes so much developed that man can easily comprehend all, even the mysteries of Spirit. [6]
Channeled Material
The Cassiopaea Transcripts refer to a collection of alleged messages received through channeling by Laura Knight-Jadczyk and her husband Arkadiusz Jadczyk. The transcripts claim to originate from an extraterrestrial group called the Cassiopaeans, who are said to be highly evolved beings from the constellation of Cassiopeia. These channeled communications cover a wide range of topics including metaphysics, spirituality, cosmology, psychology, and conspiracy theories.
[…]
A: Since you have broached the subject: are you familiar with the “twin sun” theory?
Q: (L) No. What is it? (T) Referring to our sun and the possibility of Jupiter being a sun in the making?
A: No.
Q: (T) Okay, what is the twin sun theory?
A: Theory that the sun is really a double star.
Q: (L) Well, if it is a double star, how come we don’t see the other one? Where is the other one and why don’t we see it? (T) I don’t think I have ever heard of that, have you? (F) It seems vaguely familiar for some reason. (L) Is this factual, correct?
A: Wait a moment…
Q: (T) They are bringing in their twin sun expert… (L) No doubt! {planchette spins numerous times.}
A: Now, where were we?
Q: (L) We were talking about the twin sun phenomenon…
A: Theory.
Q: (T) Which is that sol is one part of a twin system?
A: Yes.
Q: (L) Is this theory correct?
A: Not yet, you are “jumping the gun.”
Q: (T) Okay. You brought this up. You indicated that, yes, our sun is one part of a dual system…
A: Yes…
Q: (T) …and that there is another star, another sun here…
A: Yes…
Q: (L) Can we see it? (T) Can we see it at this point in time?
A: Can you?
Q: (L) No. Not that I am aware of. Is the reason we cannot see it because it is always on the other side of the sun from us? That it orbits in such a way that we can never see it?
A: Orbits, yes, assuming it is there, however, we did not confirm that, now did we?
Q: (L) No, you didn’t, but you DID bring it up! (T) Let’s go back to the beginning. Okay, you brought up the twin sun theory and then indicated that our sun is one of two suns in this solar system…
A: Maybe.
Q: (T) The sun that we designate as sol is one member…
A: Maybe.
Q: (T) Is Sol two stars combined?
A: No. What is “dark” matter, and what are dark stars?
Q: (L) Are dark stars something like black holes?
A: No.
Q: (L) Are they… (T) I think it would be a star that has collapsed upon itself and the gravitational field of the star no longer allows light to escape.
A: No.
Q: (T) Okay, that is a black hole.
A: Yes.
Q: (T) Dark matter, that I have read about, is what the astronomical community calls all the loose stuff floating around out in the cosmos that must exist because of the equations, but they can’t see it.
A: Yes.
Q: (T) Would dark stars be part of this?
A: Yes.
Q: (T) So there is dark matter and dark stars?
A: Yes.
Q: (T) The dark matter they cannot see because it is dark.
A: Yes. How about “Brown stars?”
Q: (T) Okay, Brown stars I have heard of. There is yellow, red, blue, green… Okay, our star burns as a yellow star because of the matter it is composed of - hydrogen, etc.
A: Close.
Q: (T) Other stars burn different colors in the visible spectrum because of the make-up of the star…
A: Yes, but not “brown” ones.
Q: (T) Brown ones are not brown because of the make-up…
A: Okay, you have white, red, yellow, and blue…
Q: (L) Do those colors represent temperature?
A: Partially, but that is not the point!
Q: (L) Is Hale-Bopp a brown star?
A: No.
Q: (T) No. Hale-Bopp has nothing to do with this discussion. Hale-Bopp is just another flop. Okay, Brown stars. The colors we see are part temperature and part material… (L) How about the spectral shift which determines direction?
A: Yes, but that is not the point!
Q: (L) What is the point of a brown star? Black body…
A: No, silly! Try the most obviously apparent.
Q: (L) Well, is it one that just hasn’t started cooking yet?
A: Opposite.
Q: (L) Okay, it has burned so long it is about to run out of gas?
A: Yes.
Q: (T) It is running out of matter that gives it color?
A: Why did we put “brown” in quotes?
Q: (L) Because a brown star is assumed to be one that is burning out, but is the opposite?
A: No.
Q: (L) Because… I don’t know.
A: How easy is it to see brown against a black background?
Q: (T) Not easy at all! That is why they can’t see the dark matter…
A: That is why scientists dubbed it “brown.”
Q: (L) How does brown star connect to twin sun theory?
A: Guess!
Q: (T) Okay. Let’s talk about this. For some reason we have to work through this to maintain free will. (L) Is this star small enough that it is orbiting with the planets?
A: No, we are leading you to something, if you will be patient.
Q: (L) Okay, lead on… we want to know about this. (T) They just asked if we understood what a brown >star was.
A: Do you?
Q: (L) What is the significance of the brown star?
A: Dark star.
Q: (L) It is a dark star… okay…
A: If it is there.
Q: (L) Well, will you put me out of my misery and tell me? (T) Wait, a dark star is dark because it doesn’t give off light. It is still a star, and acts like a star…
A: Yes. And if it has an elliptical orbit… would it, maybe, like, “come and go?”
Q: (T) What science, astronomy, has described as double stars, are two stars that are close together with some sort of interactive orbit. But that is not necessarily the only way two stars can exist.
A: Close. As you perceive from your vantage point. But how would you like to embark on a bicycle trip between them?
Q: (L) So the ones that we are aware of and see can be so far apart that there can be a lot between… (T) So our astronomers have not recognized this possibility?
A: Yes they have.
Q: (T) They know, but don’t talk about it. So, we may have, in this theory, a dark star orbiting…
A: And what would happen if you did?
Q: (L) Well…
A: And it, like, comes and goes?
Q: (T) Like every 3600 years?
A: Maybe.
Q: (T) And maybe this dark star also has some planets orbiting it?
A: Ok, change of direction: Oort cloud and comet cluster and sun twin occasionally passing through the former like a bowling ball through pins.
Q: (L) How does the dark star passing through the Oort cloud relate to the comet cluster?
A: Cause and effect.
Q: (L) So, the comet cluster is caused by the dark star smashing through the Oort cloud? (T) Well, not necessarily smashing, but passing close enough for its gravity to change things… (L) But they said “like a bowling ball through pins.” (T) The gravity of that star would cause the comets to be flung in all directions. (J) Is the earth like a pin? A bowling pin? (L) No, that’s the Oort Cloud.
A: Explain what the Oort Cloud is, Laura.
Q: (L) Does everybody know what the Oort Cloud is? (J) I don’t. {explanation of Oort cloud.} Okay, so we are looking at something…
A: Now, think of how your Biblical prophecies speak of a very terrifying period, followed by an apparent renewal of normalcy, followed by the “End.”
Q: (T) We learned about a “comet cluster” very early on. Now you are going beyond that and giving us a reason as to why this comet cluster…
A: Laura, please clarify exactly what the prophecies say.
Q: (L) Well, actually, “The Noah Syndrome” is primarily prophetic exegesis with a little science >thrown in to help clarify why such occurrences must be along a “natural” order. [Discussion of primary passages in “Noah.”] In the prophecies it is said: “If those days were not shortened, would no flesh be saved. But, for the elects’ sake, will those days be shortened.” Now, can you tell me what this means?
A: Not yet.
Q: (T) Where are we? We have a dark star flinging comets like bowling pins…
A: Your Biblical prophecies speak of a period of terror and chaos followed by calm, and then, unexpectedly, amidst seeming overwhelming peace and renewal and prosperity, the end.
Q: (L) Okay, elucidate, please.
A: No, you elucidate, please!
Q: (L) Okay, I would say that this “end” is the passing through the “realm border,” the “wave.”
A: Yes, but don’t you know of the prophecies we are referencing?
Q: (L) Are you talking about the Epistle of Peter where it says the heavens will end with a thunderous crash and the earth will burn up with fire?
A: No.
Q: (L) Thank God! Are you talking about the book of Daniel where a number is given regarding the time period between the one event and the other?
A: Close.
Q: (L) Okay. I know where to go to find it and study it. So, we are talking about the number 6.3, as in years?
A: Yes, but that is not correct.
Q: (T) The number of years is not correct. (L) Okay, is it possible, since you have brought this up on other occasions, through technology, to either skip this event by creating a means of travel to the next realm, or to create a parallel universe…
A: No.
Q: (L) Well one thing we can establish…
A: What would happen if the brown star that is the sun’s twin were to get close enough to be illuminated by the sun?
Q: (T) Well, if it were close enough to be illuminated, the obvious result is that it would be SEEN. People would panic…
A: Yes.
Q: (T) Governments would fall…
A: And terror and chaos. And when it departs again?
Q: (L) Everything will seem to be fine! But, they won’t realize that the Oort cloud has been hit! Oh, sugar!
A: And then what?
Q: (L) It is not the Oort cloud or the comets that is going to cause all this terror and carrying on, it is going to be the seeing of the illuminated brown star, which will go away, and then no one will see what is coming! And this IS talked about in both the Bible and Nostradamus - but it was incomprehensible before! Okay, how long will it take the comets to get from the Oort cloud to here?
A: Let us just say that the cluster travels much faster than the usual cometary itinerary.
Q: (T) And this is because they are traveling in the wake of a large sun sized gravity well…
A: And we have spoken of the comet cluster before, and we have told you that this time, it rides the >Wave.
Q: (L) Is the wave the energy from this brown star?
A: No.
Q: (L) The Wave is the Wave. (T) So, the dark star is going to come through the Oort cloud, and it doesn’t have to get too close. Any star that gets that close is TOO close. Its gravity will propel these comets in our direction. And, on top of that, they are being propelled by the Wave from behind, so they are being both pulled and pushed.
A: No.
Q: (L) Are they being “kicked” and then they get on the wave that is already on the move?
A: Yes. This time. You have had the comet cluster before in antiquity, but the wave was last here >aeons ago.
Q: (L) Is this wave a gravity wave?
A: Interrelated.
Q: (L) Okay, now… (T) Well, the wave is a form of energy. (L) Yes, they once told us that it was >“hyper-kinetic sensate.”
A: Realm border, this is your quantum factor, Laura, so plug it into “Noah” accordingly, and check out the results.
Q: (L) Does this brown or dark star have planetary bodies of its own, other than sharing planets with Sol?
A: No.
Q: (T) Okay. No correlation to Sitchin and his “planet Nibiru.” (L) If the wave is the quantum factor for the transition of the Solar System, what is the factor for the transition of an atom?
A: Electrons emit what?
Q: (L) Photons… (T) An electrical charge of some kind. (L) I thought that an electron didn’t have any parts… that it is an elementary particle…
A: Right…
Q: (L) What the heck does an electron emit? (T) An EM field?
A: What did we say about gravity? What did we say?!?
Q: (T) What was it? (L) That gravity collects… (T) What you were reading earlier…
A: Read it.
Q: [Reads segment from previous session about gravity.] (L) Is gravity emitted by an electron?
A: Yes.
Q: (L) What…
A: Electromagnetically.
Q: (L) What is it that causes a quantum transition? A collecting and dispersing of gravity. What is it >about an atom that causes it to collect or disperse?
A: How does the electron fit into the equation of the “atom?”
Q: (L) Well, it orbits the nucleus… (T) Planets orbit the Sun. (L) The energy that an electron collects, or any other part of an atom collects, that causes a quantum transition, comes to it from outside of it? A wave?
A: How many electrons orbit the nucleus?
Q: (L) It depends on the atom. The number of electrons determine what an atom is an atom of… the number of particles it has orbiting…
A: Yes.
Q: (T) And they orbit in specific shells…
A: Yes… And how does the sun relate to this macro- dynamically?
Q: (L) Okay, something from the nucleus of the atom acts upon the electrons to cause a transition?
A: How many electrons orbit the nucleus?
Q: (L) Of the sun? Are you asking how many electrons orbit the nucleus of the Sun? Different atoms have different numbers…
A: Of any atom?
Q: (L) Okay, it varies.
A: From what to what?
Q: (L) From one to somewhere in the nineties or hundreds…
A: And what determines the number?
Q: (L) Well, that is a damn good question! (T) What makes one atom helium and one atom oxygen? How do they know how to become what they are?
A: No.
Q: (T) Well, what determines the number?
A: Is it the composition of the nucleus?
Q: (L) Yeah. That’s right. We forgot. What causes or determines the number of protons or whatever in the nucleus?
A: What composition would cause the orbiting of one electron?
Q: (L) One proton?
A: Now, think macro-dynamically.
Q: (L) Well, you once said that the sun is a window, or transition point to another density. Are you saying that the nucleus of an atom is also a window?
A: What we are saying is the sun is a proton and its twin is an electron!
Q: (L) Well, I am still trying to get at… the wave causes transitions in the macro-cosmic atom, what causes the microcosmic atom… what causes a quantum jump? What accumulates in an atom that causes it to transition? (T) Is it a case of accumulation, or something being given off?
A: Completion of Grand Cycle.
Q: (J) It just is. (L) No, no…
A: And who says that the Sun’s twin appears every 3600 years?
Q: (L) Okay, we have the 3600 year comet cluster cycle, the Sun twin is another cycle altogether, and then we have the wave, which is a Grand Cycle. So, we have three things causing a transition in nature?
A: Like “biorhythms.”
Q: (T) And we have a triple bad day coming up! Or a good day, depending on which way you look at it.
A: Bad day if you are John D. Rockefeller, good day if you are Mahatma Gandhi.
Q: (L) So, does something like a three cycle event also occur in sub-atomic transitions, like the biorhythm?
A: Yes.
Q: (L) So, there are three factors to be considered… or more than three?
A: Either, or.
Q: (L) Can it be a random, arbitrary number of events?
A: If you wish.
Q: (L) Is it partly the observer that adds one of the factors? Consciousness?
A: Yes.
Q: (T) So, we see three separate cycles coming together here…
A: Everything reflects macrodynamically and microdynamically. We suggest you absorb for now; and, fear not! For it is not imminent!
Good Night.
End of Session [7]
Tom Montalk’s Twin Sun Calculations
For a while I have been researching the Planet X idea and its many variants, from the theories of Sitchin to Zetatalk, Barry Warmkessel, Richard Muller, and Velikovsky. And of course the version explored in the Cassiopaean transmissions which are close to Richard Muller’s theory of a dark star companion that swings by every 26-27 million years. The twist is that via its highly elliptical orbit, it punches through the Oort cloud and sends a trailing cluster of comets into the solar system which then establish a 3600 year orbital cycle, periodically pelting the inner solar system.
Here is a analysis conducted in 2003 that explores the plausibility of this scenario [8]
This is a short report containing my mathematical analysis of the data provided by the Cassiopaea session transcripts on the subject of the twin sun and associated effects. I find the Cassiopaean’s statements to be most logical and comprehensive concerning what extrasolar objects may be affecting earth later this decade.
LKJ proposed that the Maunder minimum and passage of the twin sun may be related. All I did was plug in the data points into standard physics equations to see if they correlated. The Cassiopaeans called the twin sun the “brown star”, but I will refer to it as the Twin, to give it a name separate from Richard Muller’s terminology which relates to a similar but slightly different concept.
Here is thorough listing of the data points I used, derived directly from the Cassiopaean transcripts:
- Twin has 56% mass of the sun
- Its nuclear “fire” went out long ago
- Oort cloud is 510 billion miles away
- It has already passed through the Oort cloud
- Punching through the Oort cloud, it drags with it a cluster of comets
- Orbit is flat elliptical
- Orbital period is 27 million years
- Dinosaurs died out 27 million years ago, not 64 million as commonly assumed
- Twin’s closest approach to our sun is no closer than the orbit of Pluto
- Comet clusters from past encounters still exist, there are many
- Some of these clusters have “already made the circuit”, others yet to come
- Most of these clusters have an orbit of roughly 3600 years
- New comet cluster travels faster than usual comets due to following gravity well of Twin.
- In July 1998, it was located in the constellation Libra
- Gravity field of twin increases the gravity field of sun during close approach
- Sun’s increase in gravity, via principles not presently known in orthodox physics, causes it to suppress solar flares, and thus diminish sunspot numbers
- Twin’s gravity field would only slightly flatten the sun’s otherwise spherical shape
- Comet clusters orbit in “spirograph” rather than strictly elliptical orbits
- Most of the approaching clusters will be seen coming from the direction of the Magellanic clouds, southern hemisphere
- New comet cluster will appear as single body
- Metal content of some comets: nickel and cobalt
- Melted comet debris acts as a conductor, possessing magnetic and electric properties: can initiate and channel solar flares toward earth
Here are additional details furnished by LKJ and further researched by members of the associated QFS:
- One comet need not hit earth to cause damage: a barrage of comet fragments or smaller comets can do the same but not leave a huge impact crater. Also see Velikovsky and McCanney’s work for non-impact cataclysmic effects of comets.
- Maunder minimum lasted from 1645 – 1710, a period of virtually NO sunspot activity
- Astronomer by name of Flamsteed discovered “Flamsteed’s Star” (aka Supra Tau) in the constellation Cassiopeia in 1680.
- Modern astronomers think his sighting was an error, because even if it was a supernova, there are no traces of it today at that location
- This star may have been the Twin, illuminated by the Sun as it neared perihelion
- Another possible date of perihelion is 1630, when two suns during the daytime were sighted – however, I don’t think this has as much validity as Flamsteed’s star of 1680.
Quick Analysis of the above data:
Mass of Twin, brown star
The Twin is a cool white dwarf. It is not technically a brown dwarf, because brown dwarfs are stars that never “lit up”, while white dwarfs are old burnt out stars. The laws of astrophysics indicate that brown dwarfs and white dwarfs have a narrow allowable range of mass, each having its own range. The Twin, having a mass 56% that of the sun, is within the range allowed by astrophysical laws (brown dwarf mass limit, Chandrasekhar limit) for a white dwarf. White dwarfs are typically around the size of earth, average 60% mas of the sun, and have very low luminosity due to their small size
Dinosaurs died out 27 million years ago
Muller’s book “The Nemesis Theory” documents the archaeological evidence behind this, except the dating method is off. Isotopes can change in concentration with solar radiation bombardment during cataclysmic times. The date of 65 million is incorrect, and should be shifted to 27 million. This makes a big difference: with 65 million, subtracting 27 twice equals 11 million years, which would indicate the Twin is far away from the sun and there’s nothing for us to worry about. But if the date is actually 27 million years, then the Twin is near us.
Comet cluster travels in spirograph fashion
Correct, because they originate from outside the solar system and are gravitationally affected by the planets during swing through, therefore they won’t have the same orientation of orbit twice, thus tracing out a spirograph pattern
Mathematical analysis of data
Semimajor axis of elliptical orbit
Mass of 0.56 solar masses and 27 million years for orbital period allows one to calculate the semimajor axis (half of the length of an ellipse comprising the orbit):
G(m₁ + m₂)P² = 4π²a³
- Where G = gravitational constant,
- m₁ = mass of sun (kg),
- m₂ = mass of Twin (kg),
- P = orbital period (sec), and
- a = semimajor axis (meters).
Converted to astronomical units (AU), the semimajor axis = 104,323 AU.
Eccentricity and semiminor axis
Perihelion, taken to be the distance of Pluto’s average orbit (39.4 AU), allows one to calculate eccentricity, and hence the semiminor axis:
Dp = a(1 - e)
Where Dp = perihelion distance,
e = eccentricity.
The eccentricity is: 0.99962236.
b² = a²(1 - e²)
Where b = semiminor axis.
Converted to AU, the semiminor axis = 2867 AU.
36.4 : 1 ratio.
Equation of elliptical orbit
Using the above figures, the equation of the ellipse can be written in polar coordinates, that is: radius from focal point of ellipse as a function of angle.
r = a(1 - e²) / (1 + e cos θ)
- Where r = radius,
- θ = angle.
In AU, the equation is:
r = 104,332(1 - 0.999622362) / (1 + 0.99962236 cos θ)
Time to orbit partially around ellipse
Kepler’s Law says that equal areas of an ellipse are swept out in equal times. Therefore, if it takes 27 million years to sweep out the full area of the ellipse ((πab)), then to sweep out a lesser area will require a proportionately lesser amount of time.
It is possible to calculate the sectional area ((A_s)) swept by the radius as it points from one angle to another:
As = ½ ∫[θ1 to θ2] r² dθ
or
As = ∫[θ1 to θ2] ½ [a (1 - e²) / (1 + e cos θ)]² dθ
This integration is done numerically to give a nearly exact result.
Once (A_s) is known, the time to traverse that section ((T_s)) is the simple ratio:
Ts = P As / (πab)
Example: if the Sun is located at the right focus point of the ellipse, then the time required for the Twin to traverse the section consisting of all area to the right of the sun (in other words, going from angle -90° to 90°) is 118 years. This is approximately how long it would take the Twin to swing around the solar system.
The Maunder Minimum (1645-1715) lasted 70 years. This correlates with the 118-year -90° to 90° sweep time. Using the time period of 70 years, (A_s) can be calculated, and hence it is possible to figure out what angle the Twin sun may have been when the Maunder Minimum began. Answer: ± 68.4°.
Plugging this into the radial equation tells how close the Twin was to the sun when the Maunder Minimum began. Answer: 57.6 AU. It would then approach over the next 35 years until reaching perihelion at 39.4 AU.
Interestingly, Flamsteed sighted his star in 1680. This is the exact middle of the Maunder Minimum. If the Twin caused the Maunder Minimum, then the middle of that period would correspond to the Twin’s perihelion. Also, being at perihelion, it would be brightest and therefore more easily discovered. Flamsteed recorded the brightness of the star to be 6th magnitude.
Sweep across the sky:
If the Twin was at perihelion in 1680 and located at the position of Flamsteed’s star, and the Cassiopaeans said in 1998 that it was then located in Libra, does that match with the equations above?
Average angular distance between Flamsteed’s Star and Libra is 130° – 140°. This can be determined using an astronomy program, or by using the average center of Libra and the coordinate of Flamsteed’s star (August 16, 1680 R.A. 23h 21m 55s; Dec +58h 32.3m).
Calculation: Time period in question: 1998 – 1680 = 318 years. This gives proportionate area swept ((A_s)) via the simple ratio equation. From that, angle of sweep can be calculated. Answer: 135°, exactly as predicted.
This sweep across the sky, if correct, allows one to figure out the Twin’s orbital plane inclination to the ecliptic.
Sweep since 1998:
Between July 1998 and July 2003, the Twin has moved .27 degrees from its supposed position in Libra. Because this is a small distance, it is assumable that the Twin is still in Libra at this time.
If the assumed distance for 1680 is 39.4 AU, its present distance would be 272.8 AU.
Oort cloud intersection:
The Cassiopaeans indicated that the Oort cloud is 510 billion miles away, or 5483 AU. How long does it take the Twin to travel from the Oort cloud to perihelion?
From the sun, the radius of 5483 AU and the equation for Twin’s elliptical orbit intersect at two points: where the Twin sun punches through the Oort cloud, and where it exits. The first intersection gives (\theta_1 = 170.4°) and perihelion is set at (\theta_2 = 0°). The sectional area (A_s) between them allows travel time to be calculated: 24,867 years.
Since then, another 318 years have elapsed, bringing the figure up to 25,167 years. Curiously, this divides by 3600 years almost exactly seven times. This suggests that old and new comet clusters not only have similar orbital periods, but they tend to converge upon the solar system simultaneously as a form of resonance. Thus, we can expect to see multiple comet clusters sighted, especially from the Magellanic clouds (southern hemisphere) as the Cassiopaeans suggested.
Curious that Stromlo, the only observatory in the southern hemisphere perfectly equipped to see these coming burned down in a freak Australian brush fire.
What else to calculate – or, what things I did not have time or resources to figure out:
- Present magnitude: Since even Neptune and Uranus are hard to see via reflective light from the sun, if Flamsteed saw the twin at +6 magnitude, then if it were outside Pluto’s orbit, it would have to be radiating its own light. Calculations are needed to determine its current magnitude based on this.
- Orbital inclination: What is the angle of inclination between the Twin’s orbit and the ecliptic, given that at perihelion it was at the location of Flamsteed’s star and at present is in Libra?
References:
- Gould, A. (2016). Most Milky Way Stars Are Single. Harvard University. Link
- Duchêne, G., & Kraus, A. (2013). Multiplicity Frequency of Solar Mass Stars: A Consistent Picture from Spectroscopic Surveys and Asteroseismology. The Astrophysical Journal, 763(2). Link
- Space.com. (n.d.). What Is a Solar Mass?. Link
- Duquennoy, A., & Mayor, M. (1991). Multiples systems among solar-type stars. Astronomy and Astrophysics, 248, 485.
- Raghavan, D., McAlister, H. A., Henry, T. J., Latham, D. W., Marcy, G. W., et al. (2010). A Survey of Stellar Families: Multiplicity of Solar-type Stars. The Astrophysical Journal Supplement Series, 190(1), 1.
- Yukteswar, S. (1894). The Holy Science (pp. 7-8).
- Knight-Jadczyk, L., & Jadczyk, A. (1996, August). Cassiopean Transcripts - Session 3 August 1996. Retrieved from Link
- Montalk, T. (2003). Twin suns and comet swarms: Possible evidence of a hidden planet in our solar system. Retrieved from Link