The Moon's sub-solar point is that point on the Moon's surface where the Sun would appear at the observer's zenith (straight up). At that point, there are no shadows, so determining where this point lies is a little tricky. Since the Moon can be observed to be up to 6 degrees above (or below) the ecliptic plane as observed from the Earth's surface, the location of the sub-solar point is not simply along the Moon's equator from the apparent center. There can be a significant parallax because of the Moon's orbit inclination.

We already have the phase angle, which is coincidentally that angle from the Moon's center to the sub-solar point, assuming that the Moon is lying right on the ecliptic plane. In all other instances, the location of the sub-solar point can be estimated by using the Moon's angle above (or below) the ecliptic plane.

At 9:22 p.m. May 23, 2010, the Moon was 5.844444 degrees BELOW the ecliptic plane as seen from the observer's location on the Earth.

This means that the sub-solar point of the Moon will be displaced by approximately 5^o.8444444 below the apparent center point determined in Step 1.

Therefore, the apparent angle of the sub-solar point from the Moon's apparent equator will be:

l_ss = -5^o.8444444

Recall that this angle is BELOW the apparent Moon's equator, and therefore will have a negative sign.

The apparent longitude of the sub-solar point from the apparent center of the Moon is simply the phase angle:

m_ss = 49^o.690216

The angle from the apparent center of the Moon to its sub-solar point can be calculated by using the following equation:

g_ss = cos^-1 ( cosl_ss cosm_ss )

This value should turn out to be slightly larger than the phase angle:

g_ss = 49^o.942405