Wednesday, November 30, 2011
A Short Walk Through Some 19th Century Science
These links are from a somewhat random online stroll through some 19th Century scientific literature that I made this Tuesday.
On the Relative Motion of the Earth and the Luminiferous Ether, Albert A. Michelson, American Journal of Science, Nov, 1887
The relative motion of the Earth and the Luminiferous ether, Albert A. Michelson, American Journal of Science, Aug, 1881
A Contribution to Croll's Theory of Secular Climatal Changes, W.J. McGee, American Journal of Science, Dec, 1881
William John McGee - Wikipedia
James Croll - Wikipedia
James Croll in Context - jfleming, ICHM
Meteorology - Wikipedia
Letter on the change in the perihelion of Mercury - LeVerrier, Comptes Rendus, Sep, 1859
Charles Lyell - Wikipedia
Principles of Geology, Vol. I, Chapter XIII, Charles Lyell
Supplemental (Wed):
Joseph Adhemar - Wikipedia (very 2012)
Climate and Time in their Geological Relations, James Croll, 1875
Discussions on Climate and Cosmology, James Croll, 1885
Révolutions de la mer, Joseph Adhémar, 1842 (tipping point?)
Thursday, November 24, 2011
Relativity and Sun Grazing Orbits
One reason why I said that Relativity is juvenile is the postulate that the speed of light is a constant. Michaelson and Morley got a null result for measuring the effect of the Earth's motion on the speed. The changes throughout the year were less than the uncertainty of the error in the measurements. So to a high degree of approximation the speed of light for the Earth is a constant. Raising this to the level of an axiom though goes beyond the supporting evidence. It works just as well as an approximation as an axiom. Einstein may have claimed a little too much but the Theory of Relativity does make a good working hypothesis.
For a circular Sun grazing orbit the orbital speed is about 440 km/s and fraction of the speed of light β = v/c = 0.0015. For a highly elliptical Sun grazing orbit the perihelion speed is about 620 km/s and β = 0.0021. So the elliptical orbit is the worse case for bodies orbiting the Sun and the resulting increase in mass is only about 1 part in 500,000. Relativity only has a minor effect on an orbiting body but errors that accumulate over centuries are noticible.
For a circular Sun grazing orbit the orbital speed is about 440 km/s and fraction of the speed of light β = v/c = 0.0015. For a highly elliptical Sun grazing orbit the perihelion speed is about 620 km/s and β = 0.0021. So the elliptical orbit is the worse case for bodies orbiting the Sun and the resulting increase in mass is only about 1 part in 500,000. Relativity only has a minor effect on an orbiting body but errors that accumulate over centuries are noticible.
Wednesday, November 23, 2011
The Mature vs Juvenile World Views
While studying the Gaussian gravitational constant and its constant function I found that the function and the elliptical orbit remain unchanged in General Relativity's rotating (precessing) orbital plane. The equation for the orbit has an extra term, 3mu2 (u = 1/r), in GR. This shouldn't be surprising since motion in Relativity confuses space and time a little. The term is small compared to other terms in the equation of the motion and the solution for planetary orbits found in Eddington's The Mathematical Theory of Relativity used the classical orbit as a first approximation to derive a relativistic correction. Eddington also makes an assumption that the mass is constant since p = h2/μ is dependent on mass which relativistically varies with the speed at which a body is moving. For highly elliptical orbits the speed at perihelion near the Sun is much greater than at aphelion, the most distant point in the orbit, and we may expect the approximations made to break down a little.
But Relativity makes some assumptions like the speed of light being a constant which can be considered an approximation too. The results of Relativity are based on deduction which is more accurate than induction and it draws heavily on classical results such as the agreement of the form of the gravitational potential with classical theory. For a two-body problem in GR one has to work with two world lines and coupling their motion assumes a rigid rotor. This like all other constraints restricts the solution to the problem. The check on results is a correspondence principle which says that the results of GR should agree with classical results for low velocities and small masses but one is free to speculate about initial the possible changes that one can make.
The dependence of the results of Relativity on classical physics makes it appear somewhat juvenile. It doesn't appear to be able to stand on its own as one would expect of a more mature theory. The initial assumptions that one can make have a lot of room for error that the correspondence with the classical limit can't narrow down. The same is true for Quantum Mechanics which has a similar approach to the problem and has to deal with uncertainties of its own. In QM one seems to be solving for theories to fit the facts. The results can sometimes be confusing and contradictory which the Schrodinger's cat thought experiment tried to point out. In the quantum world one can never have a clear picture of what is happening and one tends lose track of events.
One's comprehension may be limited but that doesn't mean that theory itself is bounded by one's limits. Are we being held back by a juvenile world view or is there still room for change. We probably shouldn't forget that change comes from within. And additionally we need to narrow down what is "good conduct" in Science. That might be a sign of movement towards a more mature approach.
But Relativity makes some assumptions like the speed of light being a constant which can be considered an approximation too. The results of Relativity are based on deduction which is more accurate than induction and it draws heavily on classical results such as the agreement of the form of the gravitational potential with classical theory. For a two-body problem in GR one has to work with two world lines and coupling their motion assumes a rigid rotor. This like all other constraints restricts the solution to the problem. The check on results is a correspondence principle which says that the results of GR should agree with classical results for low velocities and small masses but one is free to speculate about initial the possible changes that one can make.
The dependence of the results of Relativity on classical physics makes it appear somewhat juvenile. It doesn't appear to be able to stand on its own as one would expect of a more mature theory. The initial assumptions that one can make have a lot of room for error that the correspondence with the classical limit can't narrow down. The same is true for Quantum Mechanics which has a similar approach to the problem and has to deal with uncertainties of its own. In QM one seems to be solving for theories to fit the facts. The results can sometimes be confusing and contradictory which the Schrodinger's cat thought experiment tried to point out. In the quantum world one can never have a clear picture of what is happening and one tends lose track of events.
One's comprehension may be limited but that doesn't mean that theory itself is bounded by one's limits. Are we being held back by a juvenile world view or is there still room for change. We probably shouldn't forget that change comes from within. And additionally we need to narrow down what is "good conduct" in Science. That might be a sign of movement towards a more mature approach.
Wednesday, November 16, 2011
The Gaussian Gravitational Constant
In Theoria Motus (1809) Gauss gave a formula which he stated was a constant for all bodies orbiting the Sun. This quantity, k, has come to be known as the Gaussian gravitational constant. The formula Gauss gives is,
ΔA/Δt, the areal velocity, is the area ΔA swept out by the body's radius in time Δt, and is a constant for a single body according to Kepler's 2nd law. p is the parameter in the formula for the elliptical orbit of a body and m is the body's mass given in solar masses. This is based on a two-body solution for an orbit about the Sun and gravitational perturbations by other bodies are ignored. In the two-body problem one can replace the mass of a body with its reduced mass. M presumably represents the mass of the Sun.
The Moon and the Earth are gravitationally bound so their motion is coupled and we would have to use the sum of their masses in the formula to get it work properly. But does M truly represents the mass of the Sun if we have to treat gravitationally bound together or is this only an approximation? The other planets are gravitationally bound to the Sun so in treating this as a two-body problem we are splitting the solar system into two parts, the Earth-Moon system and the rest of the Solar System with the total mass being M'(m1 + m2) where m1 is the mass fraction of the other bodies in the Solar System and m2 is the mass fraction of the Earth-Moon system. Instead of 1+m on the left we would have m1 + m2 = 1 and a mass term would not be present in the formula. Even though the fractions were different for each body and the "Sun" the sum of the two masses would have the same value, M'. Gauss' formula would then be approximately correct since the mass of Jupiter is about one thousandth that of the Sun. The formula that Gauss gave may work better for single orbits but over long periods of time one may notice a discrepancy with the perturbations of the orbits being a complication.
ΔA/Δt, the areal velocity, is the area ΔA swept out by the body's radius in time Δt, and is a constant for a single body according to Kepler's 2nd law. p is the parameter in the formula for the elliptical orbit of a body and m is the body's mass given in solar masses. This is based on a two-body solution for an orbit about the Sun and gravitational perturbations by other bodies are ignored. In the two-body problem one can replace the mass of a body with its reduced mass. M presumably represents the mass of the Sun.The Moon and the Earth are gravitationally bound so their motion is coupled and we would have to use the sum of their masses in the formula to get it work properly. But does M truly represents the mass of the Sun if we have to treat gravitationally bound together or is this only an approximation? The other planets are gravitationally bound to the Sun so in treating this as a two-body problem we are splitting the solar system into two parts, the Earth-Moon system and the rest of the Solar System with the total mass being M'(m1 + m2) where m1 is the mass fraction of the other bodies in the Solar System and m2 is the mass fraction of the Earth-Moon system. Instead of 1+m on the left we would have m1 + m2 = 1 and a mass term would not be present in the formula. Even though the fractions were different for each body and the "Sun" the sum of the two masses would have the same value, M'. Gauss' formula would then be approximately correct since the mass of Jupiter is about one thousandth that of the Sun. The formula that Gauss gave may work better for single orbits but over long periods of time one may notice a discrepancy with the perturbations of the orbits being a complication.
Tuesday, November 8, 2011
Loading Orbital Elements in Distant Suns
Loading orbital elements into astronomy programs can be challenging at times. I have an older one, Distant Suns, that works quite well but is in need of an update. The HORIZONS ephemeris provides the osculating orbital elements for an object. This provides enough information to specify the orbit if the definitions are known. The process of loading the elements for 2005 yu55 into Distant Suns is shown below and the program shows the its location at the time of closest approach to the Earth. The time shown at the bottom of the image is UT.
click for clearer view
The HORIZONS User Manual gives the following definitions for the elements,
Not all of these elements are needed to specify an orbit. For example the ascending node can be used instead of the longitude of perihelion since the latter is just the sum of the ascending node and the argument of perihelion. Another drop down menu would have been more useful. This method was used for the choice between periapsis and the semi-major axis which specify the size of the orbit.
The HORIZONS User Manual gives the following definitions for the elements,
Not all of these elements are needed to specify an orbit. For example the ascending node can be used instead of the longitude of perihelion since the latter is just the sum of the ascending node and the argument of perihelion. Another drop down menu would have been more useful. This method was used for the choice between periapsis and the semi-major axis which specify the size of the orbit.
Monday, November 7, 2011
Tracking 2005 YU55
2055 YU55 is rapidly approaching the Earth. The closest approach is about 0.0021374 AU Nov 8 23:30 UT. It is very faint being a 12th magnitude object so a good telescope is required to see it.
One can follow its approach using the JPL Small-Body Database Browser. This site requires Java to view the plot of the orbit. Center on the Earth and zoom in to follow the relative motion. One can also generate an ephemeris for 2005 YU55 using links on the page.
The conditions for the fly-by favor Europe but after sunset on the US east coast one might be able to track it as it moves towards the Moon. The New Moon is on Nov 10 and may complicate viewing.
One can follow its approach using the JPL Small-Body Database Browser. This site requires Java to view the plot of the orbit. Center on the Earth and zoom in to follow the relative motion. One can also generate an ephemeris for 2005 YU55 using links on the page.
The conditions for the fly-by favor Europe but after sunset on the US east coast one might be able to track it as it moves towards the Moon. The New Moon is on Nov 10 and may complicate viewing.
Friday, November 4, 2011
Asteroid 2005 YU55 Close Encounter Nov 8, 2011
2005 YU55, a 400 meter asteroid, will pass within the distance of the Moon of the Earth on Nov 8, 2011. It will pass within 0.85 lunar radii (325,000 km) of the Earth and 239,000 km of the Moon. It is considered a Potentially Hazardous Object because of the proximity of its orbit to that of the Earth's and the chance of an impact although that is not likely to happen in the near future.
Subscribe to:
Posts (Atom)