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I solved flyby anomaly of Galileo-I (1990)

November 15, 2011

Mohan, mdashf


chek the update (click here)

did a robust analysis of the flyby situation for Galileo-I, 19 pages,

a preliminary calculation and graphing gives a (exactly) similar graph to that obtained for Galileo, but th effect is perhaps 10 cm constant towards infinity, = 100 mm/s increament. In that case the about 1% noninertial effects of spinning earth may produce a 4 mm/s. I will try to include that later ….

UPDATE: 18/11/11; In my calculations for flyby of Galileo I had an error of about 0.4 in eccentricity coming from the angle of deflection. This changes semimajor axis by about 1100 kms and the “tradeoff impact” I defined from ~ -27000 to +21040 x R = 1.34 x 10^8 kms which is 447 light seconds = 7.45 light minutes = ~1 AU, that is by the time the satelite is as far as sun is. From that distance the instruments on earth will observe zero shift in frequency, at any frequency, which will start growing as the satelite approaches or recedes from that distance. ( I also had an error in one of the parameters I had defined which is why the tradeoff changed to a large value) But the important idea remains, that the shift we see due to general relativistic effect of earth’s gravity produces 100 mm/s (correct to a factor between fractional shift in frequency and that in velocity of satelite) This effect is present to infinite distance, and same everywhere. This is the earth’s potential at it’s surface: ~10^-9. The velocity shift is dependent on what frequency we use? (I haven’t reviewed that question) SO the noninertial effects of the spin of earth is what remains to be seen. I am studying this. This effect is slated at 1% in acceleration, hence it may very well produce 4% of static earth effect of 100 mm/s, the observed effect is at infinite 4 mm/s. The static effect in teh amount of the surface potential of earth is binding since we measure all effects from surface of earth itself, hence we must see this effect on our radars, if this has been corrected in teh Galileo flyby observations, we must look for non-inertial effects as has been suggested and pointed out.

(Below: a velocity adjusted plot of the obtained gravitational shift, I may not have done it correctly and may be missing a factor, to get fractional velocity shift I just divided the fractional frequency shift by 2, from derivative considerations, I get 10 cm/s constant increament away from perigee as you can see here. This is qualitatively same as the velocity shift of the Galileo. But their shift is only 4 mm/s which is about 4% of what we see here, so it may be coming from the spinning noninertialness of earth which is classically about 1% in acceleration, hence by principle of equivalence may be producing similar effect)

Flyby velocity anomaly from Galileo

Flyby velocity anomaly from gravitational shift of static earth for the Galileo-I hyperbolic trajectory, note: Perigee is at +/- 0.15 on x-axis, y-axis = speed in cm/s

1.2% spinning effect (classically) makes the 80 mm/s comedown to 9.6 mm/s, a factor of 1/2 (easily missed in transforming the fractional frequency and speed conversion can make this the observed 4 mm/s.

**The above is what I get if I do not have the 10^5 factor in the function which I did not have earlier. If this is indeed so even the static earth potential is producing an effect similar to teh flyby shon by Galileo. (I am reviewing)

(THE TRANSVERSE DOPPLER SHIFT FOR SUCH A SECONDARY OBJECT VELOCITY IS ONLY ~1 MHz WHEREAS THE GRAVITATIONAL REDSHIFT IS  SIGNIFICANTLY LARGER THAT ACCOUNTS FOR THE ANOMALY)

In this blog I have stated my analysis for the famous flyby anomaly which is observed for a number of satelites while flying by our planet. Here I have assumed the orbital energy of the satelite while passing by earth if not taken into account in the Gravitoelectric potential causes enough change in the observed frequency shift which at points near the earth ranges in the order of even the anomalous value itself.

Note1; I still need to remove one additional term from the actual binding energy(did in the calculations: scanned pages)

Note2; Gravitoelectric potential means: inertial General relativistic redshift/violet-shift due to earth on frequencies emitted by these satelites while passing through this field.

Confirmed(!!): I (might) have solved flyby anomaly of Galileo-I (1990) the satelite which had made a lot of news around 1990-1994 when I was in highschool. If I rememeber it was all over the news. I do not remember exactly which one, the flyby or the Eda or the 951 Gaspra or Jupiter itself. If its later I was into precollege. (I didn’t recall until I reviewed about this satelite yesterday). Turns out that if I assume Galileo-I to be at any altitude, a secondary in circular motion much like say a GPS satelite, then by correcting for it’s energy of flight in it’s actual trajectory which is a hyperbolae, at 2 GHz frequency, I would get a redshift of 86 mHz when the flight is at 1500 kms above earth surface. (this value will change when I make correct corrections, this was based on incorrect determination, which I fixed, see the scanned pages)

The hyperbolae I have computed from the available flight data of Galileo. Also one needs to know exact frequency used for Doppler measurement, it is only stated they do the measurement at S-band (which I reviewed in another source of information to be: 2 to 4 GHz) and X-band (another source of information: 8 to 12 GHz).

{{{

statements on {{{…}}} here have been corrected, see emboldened description below

Note that the exact height where violet and red shifts will be traded is not R/2 (50% of earth radii) but (1-alpha)x50% of earth radii, alpha=0.376. That is if the satelite comes to a height of 31.2% the earth radii we will have zero shift. Below this height there will be violet shift and above redshift. This is 0.312*6371 ks = 1988 kms.

SO the above says it is actually not a redshift but a violet shift for heights below 1988 kms. eg If you go slightly above this distance you will again get a redshift, at 2 GHz, a value of 139 mHz.

One therefore takes a range of altitude and integrates over this range to get any residual shift. Actually since the satelite is falling at 13.7 km/s it goes a great distance in a minute, 822 kms. that in itself corresponds to a large change in frequency shift.}}}

The above calculation was wrong: After correctly determined it turns out this depends on alpha, alpha_mu: but in a much interesting way, this blows the height where redshift and violet-shifts are traded to be: ~ -27, 000 kms which is 2.45 times farther than the impact parameter of Galileo (I haven’t touched the impact parameter calculation available on wikipedia, but you can check my calculations now in the appended scanned pages, So we have an interesting situation, when we go from Perigee to that distance, 2.45 x impact we reach the level of zero shift in frequency due to the earth’s presence, I call this distance “tradeoff impact” because that is where there is a tradeoff between red and violet shift. If we integrate between +ve and -ve values of this distance we may get the exact velocity offset we are seeing for Galileo)

For such frequency (S-band or X-band, unspecified frequency) they measure a shift of 66 mHz. They do not mention the altitude where they do this. If this is Perigee as mentioned (956 kms) then I get around 195 mHz and at height 1500 kms I get 86 mHz. So I will get 66 mHz exactly a within another 100 kms or so, in height. This will change once we know the actual frequency used and the range of altitudes where the shift is 66 mHz. In that case one can easily average and get exact values.

NOTE above we have corrected the values by a correctly determined calculation, in the scanned pages.

Note that I have a lot of calculations that I had to do, eg determining the hyperbolae as per obtained parameters mentioned here: http://en.wikipedia.org/wiki/Flyby_anomaly, I get very close to velocity at infinite, if I use the impact parameter as the infinite distance of the secondary. I will upload detailed calculations later when my scanner gets back to life. (scanner has gotten back to life so if You have been hating me for all the uploads I have been doing on this blog site, I am sorry talk to the hand or blame it on the scanner)

Note that in summary Graviattional red/violet shift induced by earth’s gravitational field accounts for the anomaly by correctly adjusting for the positive energy the secondary has along its trajectory. This may not be the exact amount for anomaly since other small perturbations can also be accounted. But this is clearly the biggest factor that accounts for such. (all we need to do now is to take an exact way to integrate our result to see if we have indeed found the answer for the anomaly for Galielo-I, for other satelites I will do later)

Two publication that are relevant to this anomalous effect: (iii. I haven’t read yet)

i. http://arxiv.org/PS_cache/arxiv/pdf/0907/0907.4184v1.pdf

ii. http://www.hindawi.com/archive/2009/807695/

iii. http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.2779v1.pdf

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