Combined Gravitational and Second-Order Doppler Corrections for a Transported Optical Clock

A transportable clock does not stay in one potential, and it does not stay at rest. During a journey its rate carries both the gravitational redshift of its changing height and the second-order Doppler shift of its motion, and a geodesy result is only as good as the bookkeeping for both. This paper tabulates the two effects over realistic transport speeds and height changes, shows their ratio in a representative journey, and identifies the point at which trajectory recording, rather than clock performance, limits the measurement. The computed results, taken from the deposited script, are as follows in summary: The computed doppler_rows entry 1 speed_m_s is 10. The computed doppler_rows entry 1 doppler2_frac is 5.56e-16. The computed doppler_rows entry 2 speed_m_s is 25. The computed doppler_rows entry 2 doppler2_frac is 3.48e-15. The computed doppler_rows entry 3 speed_m_s is 50. The computed doppler_rows entry 3 doppler2_frac is 1.39e-14. The computed doppler_rows entry 4 speed_m_s is 80. The computed doppler_rows entry 4 doppler2_frac is 3.56e-14. The computed doppler_rows entry 5 speed_m_s is 250. The computed doppler_rows entry 5 doppler2_frac is 3.48e-13. The computed redshift_rows entry 1 dh_m is 10. The computed redshift_rows entry 1 redshift_frac is 1.09e-15. The computed redshift_rows entry 2 dh_m is 100. The computed redshift_rows entry 2 redshift_frac is 1.09e-14. The study is a computation from stated assumptions, not a measurement, and its figures should be read at the precision those assumptions support.