Exact circular and cubic fits with fixed inner RV phase differ by less than 1.8 seconds at 130 WASP-4 transits

We tested a cubic expansion of the long-period radial velocity signal in WASP-4. We fitted 130 distinct public transit times from 2007–2025 and 37 radial velocities from 2007–2020. We compared a cubic velocity and its integrated light delay with an exact circular orbit. The primary fits hold the inner radial velocity phase fixed and fit separate linear timing ephemerides. They use a 25-second timing error floor and fixed extra radial velocity noise. The timing predictions differ by at most 1.739 seconds, with a root mean square difference of 0.792 second. Joint chi-square is 128.511 for the cubic fit and 127.704 for the exact fit. Both have nine free parameters. The exact circular period is 25.645 years. Local conversion of the cubic coefficients gives 27.720 years. A conditional 95 percent circular-period interval is 21.35–34.42 years. A free intrinsic period-change term gives a conditional interval of -1.85 to +1.88 ms/yr. Noise and data choices change these intervals. An eccentric search reaches the period boundary with zero intrinsic change. Thus, circular-period limits do not apply to unrestricted orbit shapes. A nonlinear inner-phase check changes the circular timing predictions by at most 0.072 second. For this selected sample and these weights, the cubic expansion changes the timing curve by much less than the assigned timing errors. The result does not exclude general orbital decay. We preserve source extracts, selection ledgers, code, profiles, figures, and offline reproduction instructions.

WASP-4 · transit timing · radial velocity · light travel time · orbital decay · circular orbit · Taylor expansion · variable projection · reproducible research