CALC-094

Station keeping against sunlight, inverted from the 1978 design's own propellant budget

Feeds CLM-0885. Answers QST-004. Basis: dimensionless mass fraction, and m/s of delta-v.

Why it was needed. VC04's depth target is L2, which asks for the governing parametric relationship with its inputs and their ranges. Station-keeping propellant over a multi-decade life is that relationship, and CLM-0330 states that solar radiation pressure counteraction is 8.12 of the 9.98 percentage points - 81 per cent of the whole budget. Inverting the published mass fraction gives the delta-v it implies, which is a figure no source states and which can be compared directly against a conventional satellite.

Relationship.

m_p/m_0 = 1 - exp(-(dv_annual x T) / (Isp x g0))        g0 = 9.80665 m/s2

Inputs.

Working.

dv = -Isp x g0 x ln(1 - m_p/m_0),  then divided by T = 30 years

total    9.98 pct   ->  13,403.7 m/s over 30 yr  ->  446.8 m/s/yr
SRP      8.12 pct   ->  10,796.4 m/s over 30 yr  ->  359.9 m/s/yr
non-SRP  1.86 pct   ->   2,393.6 m/s over 30 yr  ->   79.8 m/s/yr

Convention, stated because the two routes disagree by 8.9 per cent. The non-SRP figure is computed from its own stated mass fraction, 1.86 points of m_0, and NOT by subtracting the SRP delta-v from the total. Subtracting gives 2,607.3 m/s, 86.9 m/s/yr, because MASS FRACTIONS REFERENCED TO A COMMON INITIAL MASS DO NOT SUBTRACT UNDER THE ROCKET EQUATION - each fraction is taken against the same m_0, while successive burns are not. The mass-fraction route is the correct one here because CLM-0330 states 8.12 and 9.98 against the same spacecraft mass. THIS 8.9 PER CENT IS NOT THE CROSS-CHECK BELOW AND MUST NOT BE READ AS IT - the two numbers are close in size and unrelated in meaning.

Cross-check, from a different table of the same report. CLM-0331 gives 93,000 kg/yr of propellant. Over 30 years that is 2,790,000 kg against a 30.47e6 kg dry mass, or 9.16 per cent, against CLM-0330's 9.98 per cent. Agreement within 9 per cent, from cells that were not used above.

Result. The SRP term alone requires 360 m/s/yr, 7.2 times the approximately 50 m/s/yr that is a conventional geostationary satellite's ENTIRE station-keeping budget. The residual non-SRP term, 80 m/s/yr, lands on the same order as that conventional figure, which is the check passing rather than a separate finding.

What this does not establish. Solar-radiation-pressure delta-v scales with area-to-mass ratio, and the register holds an SRP propellant figure for exactly one design. It does not transfer.

Cite this