I suggest that there could be a ‘crucial experiment’ to investigate, due to some information I came across a few years ago, about 1992.
This may prove Larson’s mathematics re the perihelion precession to be superior to the current method. Some years back there was a detailed paper by K.V.K. Nehru, (not included), showing the two methods, side by side, and the conclusion was that there was not enough discrepancy between the two methods for one theory to prove more accurate, and this pivoted on the lack of ‘decent’ eccentricity of the orbit.
To put it more succinctly, there are two limiting cases for eccentricity, being zero and unity. If b approaches a, being the semi-minor axis and semi-major axis resp., then we have the ellipse transform into a circle, whose eccentricity is zero. On the other hand, when b approaches zero, the eccentricity approaches unity, and the ellipse elongates to what would look like a straight line. In a circle we have a perimeter coming out at 4a/2 = 2a
In the following example, we have eccentricity, e = 99/101 and my approximation to the perimeter using an elliptic integral of the second kind.
To start at the beginning I heard a talk about a multi-purpose observatory (probe) called GRANAT, so it sounded. It was developed for Venus probes. It weighs 4 tons (or perhaps tonnes). I forget what the earth weighs so I cannot complete the solution of this problem, although there was no problem stated; I merely realized the opportunity to make a problem for the sake of proving the superiority of R.S. over current methods. Its closest approach to earth is 2,000 miles and its furthest distance is 200,000 miles. This is sufficient information to determine the ellipse, with the Earth at a focus F, where:-
x2/a2 + y2/b2 = 1 Cartesian
x = a.sin theta, y = b.cos theta Cartesian Parametric
where a = 101,000
b = 20,000 and the distance of the focus F from the centre is 99,000 and when brought under the integral sign in the manner to determine arclength and integrated between 0 and PI/2 reduces the elliptic integral of the second kind to 4a(1 - 0.3)/2 (approx.), due to the high eccentricity, and evaluating we obtain about 445,000 miles as the path of the probe and all this in just under 100 hours so we have an average orbital speed of 4,450 mph.
So if one calculates with greater accuracy than me and extends the problem to account for the mass of the Earth, then an accurate prediction of the flight path, based on R.S. should show a better description, than that predicted to date, provided someone amongst us can access official data on GRANAT.
Granat was sent up by a consortium of member countries, including France, USSR, Denmark and Bulgaria about 10 years ago, (about December 1989).
I hope that you guys at the conference or via the web will find the time to pool your talents and compare the two methods and prove R.S. to be more accurate for describing this orbit.
Two papers from the web incorporated below.
GRANAT was a Russian dedicated X-ray/gamma ray astronomy mission in collaboration with other European countries. Launched on 1 December 1989, Granat operated for almost 9 years. After an initial period of pointed observations, Granat went into survey mode in September 1994.
Archive : PHEBUS Gamma-Ray Burst Catalog
Part of the NASA OSS Structure and Evolution of the Universe theme.
Granat was launched on 1 December 1989 aboard a Russian PROTON rocket.. It was placed in a highly eccentric 96 hour orbit with an initial apogee of 200,000 km and a perigee of 2000 km. Over time the orbit circularized so that by 1991 the perigee had increased to 20,000 km. Three days of the four day orbit were devoted to observations. After an initial period of pointed observations, Granat was placed in survey mode in September 1994, when the attitude control gas was exhausted. Granat ceased transmissions on 27 November 1998.
With seven different instruments onboard, Granat was designed to observe the Universe at energies ranging from X-ray to Gamma-Ray. Specific instruments were meant to study Gamma-Ray bursts and other transient X-Ray sources while others were intended to image X-Ray sources near the Galactic Center. One instrument (WATCH) was designed to monitor the sky continuously and alert the other instruments to new or interesting X-Ray sources.
The SIMGA telescope was a collaboration between CESR (Toulouse), CEA (Saclay), and IKI (Moscow). It covered the energy range 30-1300 keV with an effective area of 800 cm2 and a maximum sensitivity field of view of ~5x5. The maximum angular resolution was 10 arcmin. Its imaging capabilities were derived from the association of a coded mask and a position sensitive detector based on the Anger camera principle.
The ART-P and ART-S instruments were both the responsibility of the IKI institute in Moscow. The ART-P instrument covered the energy range 4-60 keV for imaging and 4-100 keV for spectroscopy and timing. There were 4 identical modules of the ART-P telescope. Each consisted of a position sensitive Multi-Wire Proportional Counter (MWPC) together with a URA coded mask. Each module had an effective area of ~600 cm2. The field of view was 1.8° x 1.8°. The angular resolution was 5 arcmin. The instrument could achieve a 1m Crab sensitivity in an 8 hour exposure. The maximum time resolution was 4 ms. The ART-S instrument covered the energy range 3-100 keV. The field of view was 2° x 2°. The instrument consisted of 4 detectors based on spectroscopic MWPCs. The effective area was 2400 cm2 at 10 keV, 800 cm2 at 100 keV. The time resolution was 200 microseconds.
The PHEBUS experiment was designed by CESR (Toulouse) to record high energy transient events in the range 100 keV - 100 MeV. It consisted of 2 independent detectors with their associated electronics. Each detector consisted of a BGO crystal 78 mm in diameter by 120 mm thick, surrounded by a plastic anti-coincidence jacket. The 2 detectors were arranged on the spacecraft so as to observe ~4pi steradians. The burst mode was triggered when the count rate in the 0.1-1.5 MeV energy range exceeded the background level by 8 sigma in either 0.25 or 1.0 seconds. There were 116 energy channels.
Four WATCH instruments, designed by the Danish Space Research Institute, were in operation on the Granat observatory starting in January 1990. The instruments could localize bright sources in the 6-180 keV range to within 0.5? using a Rotation Modulation Collimator. Taken together, the 3 fields of view of the instruments covered ~75% of the sky. The energy resolution was 30% FWHM at 60 keV. During quiet periods, count rates in 2 energy bands (6-15 keV, 15-180 keV) were accumulated for 4, 8, or 16 s, depending on memory filling. During a burst or transient event, count rates were accumulated with a time resolution of 1 second into 36 energy channels.
The KONUS-B instrument, designed by the Ioffe Physical-Technical Institute in St. Petersburg, consisted of 7 detectors distributed around the spacecraft. They responded to 10 keV - 8 MeV photons. They consisted of NaCl(Tl) scintillator crystals 200 mm in diameter x 50 mm thick, with a Be entrance window. The side surfaces were protected by a 5 mm thick lead layer. The burst detection threshold was 5×10-8 - 5×10-7 erg/cm2, depending on the burst spectrum and rise time. Spectra were taken in two 31 channel PHAs. The first 8 spectra were measured with 1/16 s time resolution. The remaining spectra had adaptive time resolutions depending on the count rate. The range of resolutions was 0.25 s - 8 s. The KONUS-B instrument operated from 11 December 1989 until 20 February 1990. Over that period, the “on” time for the experiment was 27 days. Some 60 solar flares and 19 cosmic gamma-ray bursts were detected.
The French TOURNESOL instrument consisted of 4 proportional counters and 2 optical detectors. The proportional counters detected 2 keV - 20 MeV photons in a 6°? x 6° field of view. The visible detectors had a field of view of 5° x 5°. The instrument was designed to look for optical counterparts of high-energy burst sources, as well as performing spectral analysis of the high-energy events.
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