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.