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3rd Kepler law, twin stars, centres, and semi major axis
I performed analysis of circular orbit case of twin binary stars.
I have realized the semi major axis in context of the 3rd Kepler law must be surprisingly considered wrt the other star and not wrt their barycentre, as I expected. Otherwise formula of the 3rd Kepler law does not match circular case requirement of equality of gravitational and centripetal acceleration. 3rd Kepler law formula is usually presented as G.(M1+M2) = 4.pi^2 . a^3/T^2 G is gravitational constant M1,M2 object masses a is semi major axis of orbit T is orbit period. For units of mass in Solar masses Ms, distance in AU and time in years, G/(4.pi^2 ) is approximately equal to 1. 1 + m/Ms = (AUs)^3/(years)^2 Let imagine our Solar system contains twin binary stars of same masses 1 Ms, on circular orbit with distance 2 AU, with the radius of orbits 1 AU. The gravitational acceleration caused by the other star is 1/4 of that acting on the Earth, as Sun stars have doubled distance 2 AU. But the orbit radius is the same as for Earth 1AU, so does centripetal acceleration for the same period 1 year. Therefore period must be doubled, as centripetal acceleration is proportional to 1/T^2. It leads to Kepler 3rd law equation in Ms, AU, year units 2 = 1^3/2^2 = 1/4 for case we take as semi major axis the circle radius as usually. This is obviously wrong. But if we take as coordination centre not the barycentre, but the centre of the other star, then a = 2R = 2AU. Then 2 = 2^3/2^2 = 2 and equality is reached. So for comparable objects it looks like G.(M1+M2) = 4.pi^2 . D^3/T^2 where D is maximum object distance. -- Poutnik ( the Czech word for a wanderer ) Knowledge makes great men humble, but small men arrogant. |
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3rd Kepler law, twin stars, centres, and semi major axis
Poutnik wrote:
I performed analysis of circular orbit case of twin binary stars. I have realized the semi major axis in context of the 3rd Kepler law must be surprisingly considered wrt the other star and not wrt their barycentre, as I expected. Otherwise formula of the 3rd Kepler law does not match circular case requirement of equality of gravitational and centripetal acceleration. […] (“twin binary stars” contains redundancy as both “twin” and “binary” refer to the number 2. Also, one system of a pair of stars orbiting each other is called a “binary _star_”.) Kepler’s laws (1609–1619 CE) are laws of *planetary* motion. They were never intended for binary star systems [Johannes Kepler did not know that such systems existed; the telescope had just been improved for planetary observation by Galileo Galilei in 1609, and it was William Herschel who first came up with the idea of a binary star in 1802]. They do not consider that the star a planet is orbiting is also moving, as they do not consider a system star–planet with a barycenter around which both star and planet are moving. [Kepler may not even have considered Sol merely a star of many others as that thought – worded as “the doctrine of the infinite universe and the innumerable worlds” – was formulated at first by Giordano Bruno in 1584, which was considered one heresy of several of Bruno’s by the Catholic Church, for which he was burned at the stake in 1600.] Therefore, as Isaac Newton (1687) showed already, Kepler’s laws are only an approximation of the observed planetary motion. We now know that even Newton’s laws are only a better approximation as (so far) only general relativity (GR), postulated by Albert Einstein in 1915, fully describes gravitational effects, including planetary motion. (GR could explain and predict the perihelion precession of Mercury to great precision; Kepler’s laws and Newtonian gravity could not.) As we have seen again recently with GW150914, effects predicted by GR become important when large masses like of that of stars are involved. Therefore, ISTM your calculations using Kepler’s laws to describe a binary star are mere cyclosophy. F'up2 sci.physics.relativity PointedEars -- Heisenberg is out for a drive when he's stopped by a traffic cop. The officer asks him "Do you know how fast you were going?" Heisenberg replies "No, but I know where I am." (from: WolframAlpha) |
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3rd Kepler law, twin stars, centres, and semi major axis
Dne 20/02/2016 v 17:55 Thomas 'PointedEars' Lahn napsal(a):
I do not see the reason why did you hijacked my post to SPR, as nature of topic is not SR/GR related. (“twin binary stars” contains redundancy as both “twin” and “binary” refer to the number 2. Also, one system of a pair of stars orbiting each other is called a “binary _star_”.) Oh, your usual obsessive nitpicking arrogance... Find the help of a psychologist, as life around of you must be challenging. Twin is adjective describing stars of a binary star system, while binary is adjective describing their relation. Most of stars of binary systems are not twins, while twin stars are most probably not a part of the same binary system. The plural can stay, as the topic is focused on individual stars rather than on the system. Kepler’s laws (1609–1619 CE) are laws of *planetary* motion. They were ......... Church, for which he was burned at the stake in 1600.] I am quite familiar with this history. Therefore, as Isaac Newton (1687) showed already, Kepler’s laws are only an approximation of the observed planetary motion. But it is not fully the case of the 3rd law with sum of masses. It is applicable as well for planet-moon systems and binary star systems. See e.g. http://www.astro.caltech.edu/~george...Ay20-Lec4x.pdf Kepler’s Laws, Binaries, and Stellar Masses. GR effects for stars of Sun mass at distance 2 AU are very minor, and negligible wrt to accuracy of data. -- Poutnik ( the Czech word for a wanderer ) Knowledge makes great men humble, but small men arrogant. |
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3rd Kepler law, twin stars, centres, and semi major axis
Poutnik wrote in sci.physics, sci.astro, and sci.physics.relativity:
Dne 20/02/2016 v 17:55 Thomas 'PointedEars' Lahn napsal(a): I do not see the reason why did you hijacked my post to SPR, as nature of topic is not SR/GR related. Nothing was "hijacked […] to SPR", especially not "your post". You have crossposted without Followup-To (which is unwise at best) to sci.physics and sci.astro, and I have set Followup-To to the newsgroup where I think that this discussion belongs. You might disagree, but that does not give you the right to throw insults. (“twin binary stars” contains redundancy as both “twin” and “binary” refer to the number 2. Also, one system of a pair of stars orbiting each other is called a “binary _star_”.) Oh, your usual obsessive nitpicking arrogance... Find the help of a psychologist, as life around of you must be challenging. Ad hominem. Twin is adjective describing stars of a binary star system, while binary is adjective describing their relation. Nonsense. https://en.wikipedia.org/wiki/Binary_star Most of stars of binary systems are not twins, while twin stars are most probably not a part of the same binary system. It is the modern definition of a binary star (system) that it is actually two stars revolving around a common center of mass. You are confusing *double* stars with *binary* stars and *twin* stars. A “twin star”, by contrast, would be *one* star that has approximately the same characteristics as another star (cf. “twin planet”) where nothing would be implied about their orbits. But your definition above proves that you were not referring to that. The plural can stay, as the topic is focused on individual stars rather than on the system. More nonsense. Kepler’s laws (1609–1619 CE) are laws of *planetary* motion. They were ........ Church, for which he was burned at the stake in 1600.] I am quite familiar with this history. Obviously you *were* not. And you have not only distorted, but completely destroyed the context of my statement. Learn to quote. Therefore, as Isaac Newton (1687) showed already, Kepler’s laws are only an approximation of the observed planetary motion. ^^^^^^^^^^^^^^^^ But it is not fully the case of the 3rd law with sum of masses. ^^^^^^^^^^^^^^^^^^ I beg your pardon? It is applicable as well for planet-moon systems No doubt about that, although there appears to be no official (IAU) definition what distinguishes a planet–moon system from a double planet system. and binary star systems. But the error increases with mass and decreases with minimum distance. See e.g. http://www.astro.caltech.edu/~george...Ay20-Lec4x.pdf Kepler’s Laws, Binaries, and Stellar Masses. Interesting. So much more… surprising that you have not cited this in your OP, but mispresented those findings as your own. GR effects for stars of Sun mass at distance 2 AU are very minor, There are stars in binary star systems that have masses greater than the Sun and a minimum distance from each other less than 2 AU. For example, Eta Carinae A has ≈120 to {170 to 200} M☉, and Eta Carinae B has 30 to 80 M☉ [1], while the minimum distance between the stars is estimated to be 1.6 AU [2] (the orbits are highly eccentric, which is why it took so long to recognize that Eta Carinae is actually a binary star system). https://en.wikipedia.org/wiki/Eta_Carinae and negligible wrt to accuracy of data. Either do the math or cite evidence in which the math is done. F'up2 news:sci.physics.relativity PointedEars ___________ [1] Kashi, A.; Soker, N. (2010). "Periastron Passage Triggering of the 19th Century Eruptions of Eta Carinae". The Astrophysical Journal 723: 602. arXiv:0912.1439. Bibcode:2010ApJ...723..602K. doi:10.1088/0004-637X/723/1/602. [2] Madura, T. I.; Gull, T. R.; Owocki, S. P.; Groh, J. H.; Okazaki, A. T.; Russell, C. M. P. (2012). "Constraining the absolute orientation of η Carinae's binary orbit: A 3D dynamical model for the broad [Fe III] emission". Monthly Notices of the Royal Astronomical Society 420 (3): 2064. arXiv:1111.2226. Bibcode:2012MNRAS.420.2064M. doi:10.1111/j.1365-2966.2011.20165.x. -- A neutron walks into a bar and inquires how much a drink costs. The bartender replies, "For you? No charge." (from: WolframAlpha) |
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3rd Kepler law, twin stars, centres, and semi major axis
Dne 20/02/2016 v 23:07 Thomas 'PointedEars' Lahn napsal(a):
Poutnik wrote in sci.physics, sci.astro, and sci.physics.relativity: Dne 20/02/2016 v 17:55 Thomas 'PointedEars' Lahn napsal(a): I do not see the reason why did you hijacked my post to SPR, as nature of topic is not SR/GR related. Nothing was "hijacked […] to SPR", especially not "your post". You have crossposted without Followup-To (which is unwise at best) to sci.physics and sci.astro, and I have set Followup-To to the newsgroup where I think that this discussion belongs. You might disagree, but that does not give you the right to throw insults. Follow-to is not mandatory, default behavior is following to crossposted groups. Similarly ReplyTo in mails is not mandatory, with default reply to the sender. Twin is adjective describing stars of a binary star system, while binary is adjective describing their relation. Nonsense. https://en.wikipedia.org/wiki/Binary_star Most of stars of binary systems are not twins, while twin stars are most probably not a part of the same binary system. It is the modern definition of a binary star (system) that it is actually two stars revolving around a common center of mass. You are confusing *double* stars with *binary* stars and *twin* stars. A “twin star”, by contrast, would be *one* star that has approximately the same characteristics as another star (cf. “twin planet”) where nothing would be implied about their orbits. But your definition above proves that you were not referring to that. No, you have misinterpreted my words. You can see there is no redundancy. 2 stars being twins and being bonded in binary system are 2 indenpendent pieces of information. Kepler’s laws (1609–1619 CE) are laws of *planetary* motion. They were ........ Church, for which he was burned at the stake in 1600.] I am quite familiar with this history. Obviously you *were* not. And you have not only distorted, but completely destroyed the context of my statement. Learn to quote. You should do so at the first place, responding to the OP. BTW, it was general comment to the paragraph. Everybody can review the full version. IF I was reacting to particular statements, I would mention them. Therefore, as Isaac Newton (1687) showed already, Kepler’s laws are only an approximation of the observed planetary motion. ^^^^^^^^^^^^^^^^ But it is not fully the case of the 3rd law with sum of masses. ^^^^^^^^^^^^^^^^^^ I beg your pardon? ...obsessive nitpicking arrogance... It is applicable as well for planet-moon systems No doubt about that, although there appears to be no official (IAU) definition what distinguishes a planet–moon system from a double planet system. It does not really matter, it can be generalized to all 2 body systems, where GR effects can be still beglegted. and binary star systems. But the error increases with mass and decreases with minimum distance. I do not object and I am aware of it. But I suppose the evaluate the masses first by the Kepler law from observed periods and distances. Then, if combination of masses and distance calls for GR application, that is is done. See e.g. http://www.astro.caltech.edu/~george...Ay20-Lec4x.pdf Kepler’s Laws, Binaries, and Stellar Masses. Interesting. So much more… surprising that you have not cited this in your OP, but mispresented those findings as your own. I have not cited as I have found it AFTER your response. But I was aware of this application even before I have found it. GR effects for stars of Sun mass at distance 2 AU are very minor, There are stars in binary star systems that have masses greater than the Sun and a minimum distance from each other less than 2 AU. For example, Eta Carinae A has ≈120 to {170 to 200} M☉, and Eta Carinae B has 30 to 80 M☉ [1], while the minimum distance between the stars is estimated to be 1.6 AU [2] (the orbits are highly eccentric, which is why it took so long to recognize that Eta Carinae is actually a binary star system). https://en.wikipedia.org/wiki/Eta_Carinae I am aware about Eta Carinae. and negligible wrt to accuracy of data. Either do the math or cite evidence in which the math is done. If you think about it more, you will agree, considering accuracy of the periods and distances. Generally, not limited to binary stars, there are cases where usage of Newton gravity is not justified and there are cases where usage of GR is not justified. -- Poutnik ( the Czech word for a wanderer ) Knowledge makes great men humble, but small men arrogant. |
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3rd Kepler law, twin stars, centres, and semi major axis
Poutnik wrote:
Dne 20/02/2016 v 23:07 Thomas 'PointedEars' Lahn napsal(a): Poutnik wrote in sci.physics, sci.astro, and sci.physics.relativity: Dne 20/02/2016 v 17:55 Thomas 'PointedEars' Lahn napsal(a): I do not see the reason why did you hijacked my post to SPR, as nature of topic is not SR/GR related. Nothing was "hijacked […] to SPR", especially not "your post". You have crossposted without Followup-To (which is unwise at best) to sci.physics and sci.astro, and I have set Followup-To to the newsgroup where I think that this discussion belongs. You might disagree, but that does not give you the right to throw insults. Follow-to is not mandatory, That depends on the server that one uses; there are servers that would not allow crossposting without Followup-To, or crossposting to more than three newsgroups. There are newsreaders that would issue a warning on attempt. At least it is the polite and recommended behavior to respect the organization of Usenet into *topics*, and set Followup-To accordingly. Likewise, it is the polite and recommended behavior to trim one’s quotes to the parts that one is referring to. You did neither. default behavior is following to crossposted groups. Which is why it is a bad idea. “No posting is relevant to more than a handful of newsgroups. If World War III is announced, it will be announced in news.announce.important.” –Peter da Silva (translated) See also “no spam” in http://www.eternal-september.org/index.php?language=en&showpage=terms. Similarly ReplyTo in mails is not mandatory, with default reply to the sender. The important difference being that the e-mail is still directed to only a few people or one person. Most of stars of binary systems are not twins, while twin stars are most probably not a part of the same binary system. It is the modern definition of a binary star (system) that it is actually two stars revolving around a common center of mass. You are confusing *double* stars with *binary* stars and *twin* stars. A “twin star”, by contrast, would be *one* star that has approximately the same characteristics as another star (cf. “twin planet”) where nothing would be implied about their orbits. But your definition above proves that you were not referring to that. No, you have misinterpreted my words. You can see there is no redundancy. 2 stars being twins and being bonded in binary system are 2 indenpendent pieces of information. I have not misinterpreted your words. You have claimed that “twin stars are most probably not a part of the same binary system.” Lack of evidence notwithstanding, as “twin star” usually means something else than you think it does, your statement is nonsense. You can invent your own terminology, but you should not be surprised if that leads to misunderstandings and others frown when you insist on that only your private terminology is the correct one. Therefore, as Isaac Newton (1687) showed already, Kepler’s laws are only an approximation of the observed planetary motion. ^^^^^^^^^^^^^^^^ But it is not fully the case of the 3rd law with sum of masses. ^^^^^^^^^^^^^^^^^^ I beg your pardon? ..obsessive nitpicking arrogance... /Ad hominem/ again because you cannot face the fact that your statement is wrong. In fact, both you and the resource that you cite claim that the approximation of Kepler’s laws *holds* in the case of binary stars. It is applicable as well for planet-moon systems No doubt about that, although there appears to be no official (IAU) definition what distinguishes a planet–moon system from a double planet system. It does not really matter, it can be generalized to all 2 body systems, Then there was no reason to mention this as a special case, was there? where GR effects can be still beglegted. YSCIB. GR effects for stars of Sun mass at distance 2 AU are very minor, There are stars in binary star systems that have masses greater than the Sun and a minimum distance from each other less than 2 AU. For example, Eta Carinae A has ≈120 to {170 to 200} M☉, and Eta Carinae B has 30 to 80 M☉ [1], while the minimum distance between the stars is estimated to be 1.6 [AU 2] (the orbits are highly eccentric, which is why it took so long to recognize that Eta Carinae is actually a binary star system). https://en.wikipedia.org/wiki/Eta_Carinae I am aware about Eta Carinae. So you were deliberately misrepresenting the facts, i.e. lying? Because Eta Carinae, for example, shows that you are proceeding from a false assumption, namely that GR would never be needed to describe binary star systems *properly*. and negligible wrt to accuracy of data. Either do the math or cite evidence in which the math is done. If you think about it more, you will agree, considering accuracy of the periods and distances. IOW, you are not able or willing to substantiate your claims. Figures. Score adjusted. F'up2 sci.physics.relativity PointedEars -- Q: What did the female magnet say to the male magnet? A: From the back, I found you repulsive, but from the front I find myself very attracted to you. (from: WolframAlpha) |
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3rd Kepler law, twin stars, centres, and semi major axis
On Saturday, February 20, 2016 at 6:30:02 PM UTC+8, Poutnik wrote:
I performed analysis of circular orbit case of twin binary stars. I have realized the semi major axis in context of the 3rd Kepler law must be surprisingly considered wrt the other star and not wrt their barycentre, as I expected. Otherwise formula of the 3rd Kepler law does not match circular case requirement of equality of gravitational and centripetal acceleration. 3rd Kepler law formula is usually presented as G.(M1+M2) = 4.pi^2 . a^3/T^2 G is gravitational constant M1,M2 object masses a is semi major axis of orbit T is orbit period. For units of mass in Solar masses Ms, distance in AU and time in years, G/(4.pi^2 ) is approximately equal to 1. 1 + m/Ms = (AUs)^3/(years)^2 Let imagine our Solar system contains twin binary stars of same masses 1 Ms, on circular orbit with distance 2 AU, with the radius of orbits 1 AU. The gravitational acceleration caused by the other star is 1/4 of that acting on the Earth, as Sun stars have doubled distance 2 AU. But the orbit radius is the same as for Earth 1AU, so does centripetal acceleration for the same period 1 year. Therefore period must be doubled, as centripetal acceleration is proportional to 1/T^2. It leads to Kepler 3rd law equation in Ms, AU, year units 2 = 1^3/2^2 = 1/4 for case we take as semi major axis the circle radius as usually. This is obviously wrong. But if we take as coordination centre not the barycentre, but the centre of the other star, then a = 2R = 2AU. Then 2 = 2^3/2^2 = 2 and equality is reached. So for comparable objects it looks like G.(M1+M2) = 4.pi^2 . D^3/T^2 where D is maximum object distance. -- Poutnik ( the Czech word for a wanderer ) Knowledge makes great men humble, but small men arrogant. I posted on 17/2/16 in sci.physics: 'Science has not grasped the significance of Kepler's third'. Sam Wormley replied: "Kepler's third law (from the early 1600s) gave us the relative sizes of the orbits, T^2 ~ a^3. Isaac Newton's version of Kepler's third law T^2 = (2π)^2 a^3 / G(M+m) where (2π)^2/G is just a constant of proportionality. Orbits of earth and solar system satellites today are described by their Keplerian Elements. https://en.wikipedia.org/wiki/Orbital_elements Science not only grasped the significance of Kepler's third law, but makes use of it 24/7". Both Sam and Herr Spitzohr make sense, you do not. |
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3rd Kepler law, twin stars, centres, and semi major axis
Dne 21/02/2016 v 03:21 Peter Riedt napsal(a):
I posted on 17/2/16 in sci.physics: 'Science has not grasped the significance of Kepler's third'. That is false. Sam Wormley replied: "Kepler's third law (from the early 1600s) gave us the relative sizes of the orbits, T^2 ~ a^3. That is true. Isaac Newton's version of Kepler's third law T^2 = (2π)^2 a^3 / G(M+m) where (2π)^2/G is just a constant of proportionality. That is true. Orbits of earth and solar system satellites today are described by their Keplerian Elements. I would not call them Keplerian. https://en.wikipedia.org/wiki/Orbital_elements Science not only grasped the significance of Kepler's third law, but makes use of it 24/7". That is true. Both Sam and Herr Spitzohr make sense, you do not. -- Poutnik ( the Czech word for a wanderer ) Knowledge makes great men humble, but small men arrogant. |
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3rd Kepler law, twin stars, centres, and semi major axis
On Saturday, February 20, 2016 at 6:30:02 PM UTC+8, Poutnik wrote:
I performed analysis of circular orbit case of twin binary stars. I have realized the semi major axis in context of the 3rd Kepler law must be surprisingly considered wrt the other star and not wrt their barycentre, as I expected. Otherwise formula of the 3rd Kepler law does not match circular case requirement of equality of gravitational and centripetal acceleration. 3rd Kepler law formula is usually presented as G.(M1+M2) = 4.pi^2 . a^3/T^2 G is gravitational constant M1,M2 object masses a is semi major axis of orbit T is orbit period. For units of mass in Solar masses Ms, distance in AU and time in years, G/(4.pi^2 ) is approximately equal to 1. 1 + m/Ms = (AUs)^3/(years)^2 Let imagine our Solar system contains twin binary stars of same masses 1 Ms, on circular orbit with distance 2 AU, with the radius of orbits 1 AU. The gravitational acceleration caused by the other star is 1/4 of that acting on the Earth, as Sun stars have doubled distance 2 AU. But the orbit radius is the same as for Earth 1AU, so does centripetal acceleration for the same period 1 year. Therefore period must be doubled, as centripetal acceleration is proportional to 1/T^2. It leads to Kepler 3rd law equation in Ms, AU, year units 2 = 1^3/2^2 = 1/4 for case we take as semi major axis the circle radius as usually. This is obviously wrong. But if we take as coordination centre not the barycentre, but the centre of the other star, then a = 2R = 2AU. Then 2 = 2^3/2^2 = 2 and equality is reached. So for comparable objects it looks like G.(M1+M2) = 4.pi^2 . D^3/T^2 where D is maximum object distance. -- Poutnik ( the Czech word for a wanderer ) Knowledge makes great men humble, but small men arrogant. I posted: Science has not grasped the significance of Kepler's third. Sam Wormley Kepler's third law (from the early 1600s) gave us the relative sizes of the orbits. T^2 ~ a^3 Isaac Newton's version of Kepler's third law T^2 == (2π)^2 a^3 / G(M+m) where (2π)^2/G is just a constant of proportionality. Orbits of earth and solar system satellites today are described by their Keplerian Elements. https://en.wikipedia.org/wiki/Orbital_elements Science not only grasped the significance of Kepler's third law, but make use of it 24/7. |
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3rd Kepler law, twin stars, centres, and semi major axis
Dne 21/02/2016 v 03:23 Peter Riedt napsal(a):
Kepler's third law (from the early 1600s) gave us the relative sizes of the orbits. T^2 ~ a^3 Isaac Newton's version of Kepler's third law T^2 == (2π)^2 a^3 / G(M+m) where (2π)^2/G is just a constant of proportionality. Orbits of earth and solar system satellites today are described by their Keplerian Elements. https://en.wikipedia.org/wiki/Orbital_elements Science not only grasped the significance of Kepler's third law, but make use of it 24/7. The point is, Kepler laws are laws just by name, being empirical rules as consequence of the Newton gravity law. Not the otherwise. -- Poutnik ( the Czech word for a wanderer ) Knowledge makes great men humble, but small men arrogant. |
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