Caption: "An image of the Japanese Sun Goddess Amaterasu emerging from a cave."

The synodic day is what is ordinarily meant by a day in astronomy: the axial rotational period of a planet relative to its host star.

The synodic day formula for uniform circular motions in a common plane is given below.

The synodic day formula is

  P_day = P_orbital*P_axial/(P_orbital - P_axial)  ,

where P_orbital is the orbital period of a planet and P_axial is the planet's axial rotational period relative to the observable universe.

If P_day < 0, then the axial rotation of the planet relative to its host star is retrograde.

The synodic day formula is a special case of the synodic period formula derived and explicated in file synodic_period.html.

Credit/Permission: Utagawa (Shunsai) Toshimasa (1866--1913), 1887 (uploaded to Wikimedia Commons by Rocco Pier Luigi (AKA User:Moroboshi), 2005) / Public domain.
Image link: Wikimedia Commons: File:Amaterasu cave.JPG.
Local file: local link: synodic_period_day.html.
File: Orbit file: synodic_period_day.html.