This paper discusses the methodology necessary to measure the Hubble constant Ho to a high degree of accuracy based upon Doppler tracking of spacecraft in the solar system. Using this methodology with available published data we determine a model independent value of the Hubble constant for the current epoch in the solar system to be Ho = 2.59 \pm 0.05 x 10^-18 (s^-1) or as 79.8 \pm 1.7 (km/s/Mpc). We calculate the direct effect of the Cosmic Redshift on Doppler tracking of spacecraft in the solar system. It is shown that with current tracking systems, such as NASA's Deep Space Tracking Network, when the return trip light time of the Doppler signal exceeds a certain threshold, imposed by the stability of the frequency standard, the effect of the Cosmic Redshift is coherently conserved in the returning Doppler signal. We demonstrate that in an underdetermined orbit, one determined by line of sight Doppler alone, that if this Cosmic Redshift term is not accounted for, the orbit determination program (ODP) miscalculates the actual recessional velocity of the spacecraft from the measured recessional velocity causing a mismatch between the actual and the predicted trajectory of the spacecraft. One consequence is that the ODP will generate Doppler residuals, the difference between the actual trajectory and the predicted trajectory which show an anomalous force. When this effect is integrated in long arc solutions, it can grow to considerable magnitude. We show that the ODP residuals uniquely separate the Cosmic Redshift term from velocity Doppler sources and that the solution can provide an accurate determination of Ho.
A major effort of astronomical research is to estimate the value of the Hubble constant H₀ over a wide range of distances. Figure 1 is a summary graph of the estimate of H₀ from the 2001 HST Key study (1). While advances in new technologies and methodologies allow the measurement of H₀ to greater and greater distances, the ability to measure H₀ accurately at very short distances has not been investigated. This may be due to the fact that traditional astronomical methods allow for an estimation of H₀ only at distances greater than several Mpc. *email: allen.joel.anderson@gmail.com, allen.j.anderson@fysast.uu.se In a recent paper, Riess,et.al. (2) stress the need to determine the zero point value of H₀ for the Cepheids in order to define better the value of H₀ at all distances. Figure 2 summarizes the measurement of H₀ for the Cepheids from the HST study (1). At the moment there is no short range model independent value of H₀. As we will discuss, celestial mechanics experiments are currently sufficiently accurate that the effect of the Cosmic Redshift causing a spectral shift in the Doppler tracking of spacecraft is measurable. If the direct effect of a Cosmic Redshift on the Doppler measurement of spacecraft is not accounted for, the actual measurement of the Cosmic Redshift is either placed in other available parameters being fit by the orbit determination program (ODP) or left as a badly fit trajectory solution leading to the appearance of misfit Doppler residuals and anomalous forces.
Literature currently reports instances of anomalous and badly fit trajectories of spacecraft tracked by NASA’s Deep Space Tracking Network (DSN) (3), ( 4), ( 5), (6) but the analysis of the direct effect of the Cosmic Redshift on the Doppler signal itself and how this would impact the predicted trajectory produced by the ODP has not been well described. These reported spacecraft anomalies generate discussion among a wide portion of the scientific community, but leave us with a less than adequate understanding of a solution to the problem (7).
The Cosmic Redshift, the Doppler Tracking Signal and the Hubble Constant Celestial mechanics has traditionally relied upon measurements of positions of planets measured against the background of stars. Together with GM based equations of motions, they provide the basis for the celestial reference frame. In earlier measurements, no electromagnetic data of spectral measurements was utilized. More recently, data from Doppler tracking of spacecraft, basically a spectral measurement of electromagnetic radiation, places spacecraft into a solution of the celestial reference frame. While traditional celestial mechanics has no need to incorporate the Cosmic Redshift (the FLRW metric causing a spectral shift in electromagnetic radiation), Doppler tracking does. Although it is stated that current orbit determination programs are fully relativistic, they do not, in fact, incorporate this implied FLRW redshift correction to spacecraft Doppler measurements.
The FLRW metric for the Cosmic Redshift in its simplest form can be written as follows:
Equ. 1 1+ z = a (now)/ a (then) Here a is the FLRW metric scale factor. The ratio of this term in seconds of time and its effect on z provides a measure of the magnitude of H₀. z is an electromagnetic spectral shift in wavelength and is directly dependent upon the difference in seconds in time between two measurements: that is, the difference between two space metrics at different times from each other. This difference is a purely spectral shift, representing the photon accommodating to a change in the space metric. Thus, H₀ is simply a scale factor of the rate at which the photon changes its wavelength in seconds of time.
From this the Cosmic Redshift effect for a fractional change of frequency of an electromagnetic signal in time is: Equ. 2 -Δƒ/ƒ = t x H₀ Here t is the time difference between send and receive in seconds of time , which is equivalent to the value of the return trip light time (RTLT) of the DSN. H₀ used here is the standard approximate value of: H₀ = 2.5 x 10⁻¹⁸ (s¯¹).
A successful measurement of this fractional change of frequency would yield a precise value of H₀ at the current epoch. In fact, this would be the intent of any experiment to measure H₀ in this manner.
Figure 3 plots this relationship together with an estimate of the DSN’s ability to measure fractional frequency Doppler shifts. Since the 1970s (8) the DSN’s ability to make this measurement is confined by the stability of the frequency standard, measured as Allen variance in time, and is displayed here (9). In addition, this basic limit shows that for a RTLT greater than a certain fraction of an AU, currently around 0.1 AU, the Cosmic Redshift is incorporated into the Doppler signal and conserved in the tracking data. In other words it impacts the measured Doppler signal.
For most spacecraft, disturbances of several types reduce the ability to model the trajectory sufficie
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