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The Effect of Solar Gravitational Potential On GPS Clocks Tom Van Flandern & Thomas B. Bahder Army Research Laboratory PAWG, Colorado Springs 1998 August 19 (last two slides updated 2002 March 30) This presentation will probably involve audience discussion, which will create action items. Use PowerPoint to keep track of these action items during your presentation In Slide Show, click on the right mouse button Select “Meeting Minder” Select the “Action Items” tab Type in action items as they come up Click OK to dismiss this box This will automatically create an Action Item slide at the end of your presentation with your points entered.

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What affects clock rates? SR: motion slows clocks by the factor (typically < ) GR: gravitational potential slows clocks by the factor (typically ) N.B. neglected ( )

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GPS satellite clock rates (ns/day) Orbital speed slowing:7,200 Gravitational potential slowing at GPS orbital altitude: 14,400 Gravitational potential slowing on Earth’s geoid: 60,300 Net: GPS satellites run faster than clocks on geoid by: 38,700

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Question: Why is solar potential effect not included in GPS? Predicted effect: 12-ns amplitude, 12-hour period GPS raw pseudo-ranges show unexplained 12-hour periods correlated with Sun direction Hoffman (1963): Clocks in “freefall” should show no effect. But counter-examples exist.

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What does not affect clock rates? Cyclotron experiments with meson decay times show that acceleration as such does not affect clock rates even at g. F = m a and equivalence principle: If acceleration does not affect clocks, gravitational force cannot either.

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Inside a uniform spherical shell, force is zero, potential is constant, clocks slow Alley experiment: speed & altitude affect clocks, but not acceleration (e.g. when plane circles)

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Fixed and accelerating clocks with same speed and height tick at same rates Suspension/free fall doesn’t affect clock rates

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Compared to pole, clocks on equator slow due to higher rotation speed, but speed up due to weaker gravitational potential. These effects cancel. Result: All clocks at sea level tick at the same rate!

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Satellite clocks tick slower than clocks at infinity because speed and potential effects accumulate. Also, as satellite falls toward perigee, potential gets stronger and speed increases. Again, the two clock-slowing effects accumulate.

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As Earth and satellite fall toward perihelion, all their clocks slow in unison relative to distant pulsar clocks. The two slowing effects accumulate.

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To Sun But as the satellite orbits, do the two clock-slowing effects from gravitational potential and velocity in the Sun’s field cancel or accumulate? predicted amplitude of solar potential effect: 12 ns (3.6m)

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To Sun Or if Earth is not present but the satellite follows the same path as before using rocket propulsion, do the two solar clock- slowing effects cancel or accumulate? predicted amplitude of solar potential effect: 12 ns (3.6m)

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Alley experiment: Clocks flown over south pole to New Zealand South pole in winter was always farther from Sun than Earth center No effect of Sun’s potential was seen.

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Solar clock-slowing computation Sun’s potential at Earth’s surface is 14 x stronger than Earth’s potential. Mathematica applied to GR metric equation to compute satellite proper time in Sun’s field for any orbit. Lorentz transformation applied to convert from SSB coordinate time to ECI coordinate time. Result: potential and velocity effects from Sun on Earth-satellite cancel!

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Explanation: As Earth orbits Sun (center path), aphelion satellite positions must travel at higher speed to get from A to B, and perihelion satellite positions must travel at slower speed to get from C to D. A B C D This “centripetal potential” is speed difference added by Sun’s presence.

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Summary of clock-slowing effects Motion free Motion forced

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Conclusions Solar potential effect does not exist in GPS data; motion is “forced” Unexplained 12-hour periods correlated with Sun direction must have some other explanation No unresolved relativity issues remain at the 1-meter level for GPS

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