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Alternative theories being tested by Gravity probe B 
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#181
Apr2107, 02:33 PM

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PF Gold
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[tex]\Omega_{fd} = \frac{GI}{c^2R^3}[\frac{3\underline{R}}{R^2}(\omega.\underline{R})  \omega][/tex] Garth 


#182
Apr2107, 06:30 PM

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I also need a value for I, which is the moment of inertia for the Earth. I can calculate assuming a solid sphere; but distributions of mass are not uniform, so this is only an approximation. The vectors are [itex]\omega[/itex], which is in a fixed direction along the Earth's axis, and R, which is the location of the probe. This part has a maximum value over the poles, the dot product is just a product of magnitudes. The direction is the opposite of [itex]\omega[/itex] by the sign differences, so at the poles but in brackets has magnitude [itex]2\omega[/itex]. But over the equatot, the dot product drops to zero and the magnitude is [itex]\omega[/itex] in the opposite direction. The suggests a mean magnitude of [itex]\omega/2[/itex]. I think. Using [itex]I = 0.4MR_e^2[/itex] as a solid sphere, I get [tex] 0.2*GMR_e^2R^{3}c^{2}\omega [/tex] as a magnitude. My spreadsheet gets very roughly in the ball park with this, at 0.049 But I'm still out by much too much. Any GPB experts can help me out? Thanks  Sylas 


#183
Apr2207, 04:35 AM

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Followup to my previous post! I have now found a figure for the moment of inertia of the Earth, and am able to get close to the OLD value for the framedragging effect. But I'll go through it all from the beginning with new numbers.
To get accurate values, I have tried to use data to five or six figures of accuracy, or whatever is available. I use SI units, unless explicitly given otherwise. I've checked out the following sources.
Geodetic effect The geodetic effect is 1.5*(GM)^{1.5}c^{2}r^{2.5}, where r is the semilatusrectum of the probe orbit. This is the semimajor axis times (1e^{2}), as indicated in a previous post. The radius of the orbit of GPB is measured as 7027.4 km (semimajor axis). The eccentricity is e = 0.0014, so the semilatus rectum is 7027.386 km. Adler and Silbergeit use 7028 km with a circular orbit; this was calculated before launch. The value of GM (gravitational constant times earth mass) is known with great precision; much more than either G or M individually. The value of G*M is 3.986004418*10^{14} m^{3}s^{2} in IERS. NIST currently gives G as 6.67428*10^{11}, which would give M as 5.9721864*10^{24}. The NASA page gives Earth mass as 5.9736*10^{24}; this corresponds to G = 6.6727*10^{11}. To convert from radians per second to milliarcseconds per year, the factor is 6.50908*10^{15}. That uses a tropical year of 365.24219 days (IERS). The speed of light is 299792458 The only meaningful source of error here is in r, the radius of the orbit. The largest value for the geodetic precession requires r to be small, and the conversion factor to be large. The conversion factor for milliarcseconds/year uses the length of a tropical year, being 365.24219 days; there's no basis for using anything greater. The calculation is [tex] 1.5 * (3.986004418 * 10^{14})^{1.5} * 299792458^{2} * (7.027386 * 10^6)^{2.5} * 365.24219 * 86400 * 360 * 3600 * 1000 / 2 / \pi [/tex] This gives the geodetic precession as 6603.77 milliarcseconds/year. I can't see any possible way to make this any bigger. To make matters worse, Adler and Silbergeit also give a correction factor to take account of the Earth's oblate shape. This factor is given as: [tex]1\frac{9}{8}*J_2*(R/r)^2)[/tex] Here R is the radius of the earth and J_{2} is the quadrupole moment. For the radius of the Earth, Adler and Silbergeit use 6378 km, NASA gives 6378.1 km, and the IERS gives 6378.1366 km. For J_{2}, Adler and Silbergeit use 1.083*10^{3}, and the IERS gives 1.0826359*10^{3}. The accuracy here will not matter much. This factor reduces the geodetic prediction, by (11.00*10^{3}), to give a final prediction of 6597.14 milliarcsec/year How anybody ever got 6614.4 I don't know. There is an additional precession due to the Sun; but this is in a different plane, and is going to have more effect on framedragging. It is discussed in Adler and Silberguit as well, with a magnitude of 19 milliarcsec/yec, but mostly perpendicular to geodetic precession. Frame dragging As derived above (and Adler and Silbergeit confirm) the magnitude of the effect works out to be [itex]GJr^{3}c^{2}\omega / 2[/itex] The moment of inertia for a solid body is [itex]J = kMR^2[/itex], where k is a "moment of inertia ratio". This ratio is 0.4 for a uniform sphere, but it will be more if the mass density is greater near the surface, and less if the mass density is greater near the center. Adler and Silbergeit use k = 1/3.024 = 0.3307. The NASA fact sheet gives k = 0.3308. Using the value of GM from IERS, GJ would be kGMR^{2}, which is about 5.3640*10^{27}, using the NASA value for the radius R of the Earth. The IERS gives a value for J directly, which is 8.0365*10^{37}; and with their value of G as 6.6742*10^{11}, this gives GJ = 5.3638*10^{27}. The length of a sidereal day 23.9345 hours (NASA sheet) so the rotation velocity ω is 7.2921 * 10^{5} rad/sec. In IERS it is 7.292115*10^{5}. The framedragging effect is [tex] GJr^{3}c^{2}\omega/2 [/tex] This works out to [tex] 5.3638*10^{27} * (7.0274 * 10^6)^{3} * 299792458^{2} * 7.292115*10^{5} * 365.24219 * 86400 * 360 * 3600 * 1000 / 2 / \pi / 2[/tex] which gives 40.81 milliarcsec/year, As before, there is a correction factor; this time equal to [tex]1+\frac{9}{8}*J_2*(R/r)^2(1\frac{3}{7}*MR^2/J))[/tex] This works out to [itex]12.97*10^{4}[/itex], which brings the prediction back down to 40.80 milliarcseconds/year. This close to the 40.9; but now I don't know how they are obtaining 39. Cheers  Sylas 


#184
Apr2207, 08:21 AM

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That's a very impressive piece of work, thank you Sylas.
Why don't you email Alex Silbergleit at: gleit@relgyro.stanford.edu with these questions? And then let us know the answer of course. Garth 


#185
Apr2207, 02:51 PM

P: 38

Thank you for your efforts to help clarify several points. I still cant decode the axis informations from the plot named Torque modeling example: motion of gyro 3 in the L10028 poster. Can someone who attended the APS conf or GPB expert help us ? regards, F HC 


#186
Apr2307, 08:09 AM

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1) A time dependent polhode precession due to the gyros not being exactly spherical. The time dependency is modelled by a dissipation of kinetic energy over time. 2) A misalignment torque due to a variation of electric potential over the surface, which can arise due to the polycrystalline structure. It can be affected by presence of contaminants and is modelled as dipole layer. The patch fields are present on rotor and housing walls and cause forces and torques between these surfaces. On the Torque modeling example: motion of gyro 3 in the L10028 poster the legend is very unclear, but the xaxis is EW orientation milliarcsec/yr and the yaxis I believe is Polhode phase angle error. Garth 


#187
Apr2307, 12:30 PM

P: 38

and were told that an upgrade will soon be available on their site with clearer plots... For the time being i dont see how i could read the amount of relativistic motion from this plot where the dashed line represents the estimated relativistic motion. I also asked how the error plot was obtained on the error poster ... They also told me that the audio file of their presentations will soon be available online Best regards, F HC Regards, F HC 


#188
Apr2307, 12:48 PM

P: 38

1) The dissipative polhod motions must after some time determine the actual rotation axis of the four gyroscopes. its direction should be a priori different and arbitrary for each one: has this information been given at the conference: the raw "initial" axis direction for each Gyro i.e. after stabilisation (when the polhode dissipative effects become small) 2) If it is true that there is a force proportional to a misalignment... eventually this force should impose an alignment therefore a preferred direction relative to the stator and if the latter is fixed with respect to distant stars (is it?) this should prevent any relativistic motion at some level... Sorry for these probably naive questions but since i couldnt attend the conference... Regards F HC 


#189
Apr2307, 03:09 PM

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Yes,
The unexpected signals have complicated the matter somewhat for the GPB and that explains why it is taking so long to produce the results. Whereas they were expecting some polhode motion, the new thing was for this to be time dependent. Therefore it has taken some time for them to model this effect correctly. However, the polhode motion is only a minor effect which is only significant when the most accurate readings are required especially for the framedragging effect. However what has nearly spoiled the experiment was the misalignments experienced because of electrostatic patch effects. With the spacecraft rolling about its axis pointing towards the guide star every 77.5 seconds they found a misalignment of up to 1 arcsec/deg/day, potentially larger than the relativity precessions they were looking for. In their words: The first thing they have to do is to model is patch effect correctly using the geometry and rate of change data to isolate the 'noise' from the signal without using the expected relativity answer in the process. As to the detailed answers to your questions they haven't given much away, because I think they are not sure at present of that answer. They say that it is very much "a work in progress". We will have to wait until the end of the year when all will be revealed, I trust, in the promised published papers. However I must add that I find Francis Everitt's final comment "Always be suspicious of the news you want to hear." intriguing. It is almost as if he does not want to believe in their results.... Garth 


#190
Apr2407, 10:46 AM

P: 731

Garth,
Whilst I've read (and unfortunately only understood part) of this thread, it's always been with interest. I've refrained from posting since most of the questions I have about the data you've been posting (thanks, by the way) Garth is that the questions tend to be answered anyway. Your recent post on the gravity probe B results (sorry your theory didn't work out), and subsequent discussions, are particularly an interesting read from a beginners point of view as it's easy to pick up knowing what's left. It may be a tall ask, but if you've time, could you offer a brief, simplified post (even a PM?) on the basic large scale changes/implications these theories may have? any other outrageous features you're proficient in are always worth noting! thanks. 


#191
Apr2507, 12:51 AM

P: 1

The Stanford website mentions on the first phase GPB results dated 14th April 2007:
“….. the data from the GPB gyroscopes clearly confirm Einstein's predicted geodetic effect to a precision of better than 1 percent. However, the framedragging effect is 170 times smaller than the geodetic effect, and Stanford scientists are still extracting its signature from the spacecraft data. The GPB instrument has ample resolution to measure the framedragging effect precisely, but the team has discovered small torque and sensor effects that must be accurately modeled and removed from the result.” Can anyone answer whether the result of the framedragging effect, derived after removal of the much larger torque and sensor effects, can still be considered to be a result obtained from a “controlled” experiment? 


#192
Apr2707, 01:29 PM

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The GPB team are working very hard to model and determine the unexpected time dependent polhode and patch effect torques on the gyro rotors to within 0.1 mas so they can be subtracted from the raw data. They claim they are doing so without using the expected results in the process, thus keeping GPB a controlled and objective experiment. However I have two questions: 1. How do you know that you have allowed for all other effects that might be affecting the result? In other words the tendency in any experiment is to keep subtracting sources of 'noise' until the remaining signal is what you expect, and then stop. If the result is not what is expected then you might discover another unknown source of error. 2. Might there be a degeneracy in the modelling of these torques? i.e. Having fitted the time dependent data well to one solution and use that to obtain the final result, might there be another solution that also fits the noise data well that produces a different result? Garth 


#193
Apr2707, 01:51 PM

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In the BD theory a minimally connected scalar field is added to the GR field equation that has the effect of perturbing the GR spacetime and therefore freelyfalling particle and photon geodesics, but does not otherwise interact with them. As these perturbations have not been discovered the scalar field must be very weakly connected to matter. The presence of the scalar field affects the cosmological solution and cosmic evolution. In SCC that scalar field is now nonminimally connected and interacts with particles inducing a scalar field force on particles but not photons. The scalar field force exactly compensates for the perturbation of spacetime in vacuo, and SCC freelyfalling particle and photon geodesics are the same as those of GR. The coupling constant [itex]\lambda[/itex] was equal to unity. The theory passed all the tests GR does, up to but not including the GPB geodetic precession prediction. It had interesting cosmological consequences as well as predicting the Pioneer anomaly discussed elsewhere on the Cosmology Forum. I can see my way clear to a general self creation theory in which [itex]\lambda[/itex] is left as an unknown variable. The geodetic prediction becomes [tex]\Omega = [(1  \lambda/3)6.6 + 0.25][/tex] arcsec/yr. (I have found an extra 0.25 arc/sec/yr precession due to cosmological time dilation (clock drift) that makes my original prediction 4.65 arcsec/yr not 4.4 arcsec/yr.) Unfortunately the theory then predicts the total mass density parameter for the universe to be [tex]\Omega_T = \frac{1}{3\lambda}[/tex], so if [itex]\lambda[/itex] is small a lot of DM and DE is required and an attractive feature of the original theory is lost. I will post more when I have published. Garth 


#194
Apr2807, 07:48 AM

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I have now (modestly!) included my modified General Theory of Self Creation Cosmology (GSCC) which leaves the [itex]\lambda[/itex] parameter undetermined.
Note that the results published at the April APS meeting in Jacksonville include a modified GR prediction with one decimal place less accuracy. The reason for this modification is not clear, however the modified GR predictions are included in this post. The running now stands:
The predictions are now: GPB Geodetic precession (NorthSouth)
We await the GPB gravitomagnetic frame dragging precession (EastWest) result.
Garth 


#195
Apr2807, 12:40 PM

P: 2,043

Garth, all the respect for your theory, but didn't your theory predict 4.4096 arcsec/yr? How come it now shows a number much closer to the experimental outcome?



#196
Apr2807, 01:18 PM

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The original theory was highly determined with [itex]\lambda = 1[/itex]. It had many interesting features, which I have been discussing on that thread and it made a particular prediction for GPB, which appears not to be verified. Therefore the day after that announcement was made on the 14th April I sat down and looked again at the theory. I realised that there was a cosmological clock drift time dilation to take into account that adds another 0.25 arcsec/yr, but which wasn't enough to save the theory. I therefore generalised the theory by leaving [itex]\lambda[/itex] undetermined. This meant losing some of the attractive features of the model but still preserving several others. The result is the prediction posted above and I am now writing up the new General Theory of Self Creation Cosmology for publication. If the final results for GPB are exactly those predicted by GR then I will finally say goodbye to SCC! Garth 


#197
Apr2907, 10:20 AM

P: 23

The smaller the value of [tex]\lambda[/tex], the higher the value of [tex]\Omega_T[/tex] will be. Can you tell us which is the highest value of [tex]\Omega_T[/tex] you would consider acceptable ? This would provide a minimum acceptable value of [tex]\lambda[/tex] and a maximum acceptable value for the geodetic precession. Paul 


#198
Apr2907, 03:15 PM

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In GSCC if [tex]\lambda \neq 1[/tex] then ideally [tex]\lambda[/tex] ~ 1/3, which would give [tex]\Omega_T[/tex] ~ 1, concordant with the standard model, however that would be too high a value of [tex]\lambda[/tex] for the present geodetic precession. I say "present" because I think we ought to wait for the final analysis before being sure what that reading actually is. They have to accurately model the polhode and patch effect torques accurately and unambiguously first. The present error bars on the geodetic measurement allow for [tex]\lambda[/tex] < 0.14 which gives [tex]\Omega_T[/tex] > 2.33. It then depends on whether that could be concordant with the WMAP data and exactly how much DE and DM would be required and plausible. Garth 


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