Right Ascension of the Ascending Node (RAAN or Ω) illustrated with respect to the equatorial plane and the î direction. [Source]

Right Ascension (RA) describes the orientation of an orbit. The Right Ascension of the Ascending Node (RAAN or Ω) is a celestial coordinate; it’s the angle (along the $\hat{I}$ direction) of an object measured between the vernal equinox and the ascending node (the point where the object crosses the equator from the Southern Hemisphere to the Northern Hemisphere. Note that the ascending node lies on the intersection of the orbital plane and the equatorial plane.)

$$ \Omega = RA_0 + (15° \times (LTAN-12)) $$

Where:

• $\Omega$ [°] : Right Ascension of the Ascending Node (RAAN).
• $RA_0$ [°] : Right Ascension of the Sun on Launch Day
  • $RA_0 = 360° \times \frac{\text{no. of days since vernal equinox}}{365}$
• $LTAN$ [hrs after midnight] : Local Time at Ascending Node (based on the 24 hour clock, so it must be a number between 0 and 24)
  • Note that, for this calculation, the unit is in hours.
  • $LTAN = LTDN - 12$, where $LTDN$ is the Local Time at Descending Node. If the given LTDN is 10:30, use 10.5 in the calculations.
  • LTDN is with respect to sidereal time (i.e., time with respect to fixed stars on the theoretical celestial sphere), not solar time (time with respect to the position of the Sun in the sky) [[Source](https://en.wikipedia.org/wiki/Sidereal_time#:~:text=sidereal time is the angle%2C measured along the celestial equator%2C from the observer's meridian to the great circle that passes through the March equinox (the northern hemisphere's vernal equinox) and both celestial poles%2C and is usually expressed in hours%2C minutes%2C and seconds.)]. Therefore time given by the LTDN does not necessarily correspond to the time on Earth (e.g. 12:00 is not necessarily noon).

Assumptions:

• The inclination and eccentricity of Earth's orbit around the Sun is negligible
• Earth's axial tilt is negligible

With these two assumptions, we can say that the RA of the Sun varies linearly with time, starting from 0 degrees during the vernal equinox.

• Derivation

References:

Chapter 3 – The Classical Orbital Elements (COEs)

LTDN vs RAAN - FINCH - Confluence