Local solar time relates to the location of the sun. Local solar noon is the time when the sun is highest in the sky. Local time varies because the Earth's orbit is an ellipse rather than a circle, and because of the way we divide time zones into hourly intervals and because of daylight savings time.
Find the local solar time meridian. This is used to adjust for where you are on the Earth. It is 15 degrees_, which is the difference between local time and Greenwich mean time in hours. For example, in New York City with daylight savings time it is 4 hours earlier than Greenwich mean time, so this is 15_-4 = -60.
Find the number of days since the start of the year; call this d. For example, today is June 18th, which is 168 days since the start of the year.
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Subtract 81 from d and multiply by 360/365. In the example, 168-81 = 87 and 87*360/365 = 82.85. Call this B.
Calculate 9.87_sin(2B) - 7.53_cos(B) - 1.58_sin(B). In the example, sin(2B) = sin(165.7) = 0.247; cos(B) = cos(82.85) = 0.128 and sin(B) = 0.992. 9.87_0.247 - 7.53_0.128 - 1.58_0.992 = 2.437 - 0.964 - 1.567 = -0.094. Call this E.
Calculate the time correction factor. This is 4_(local solar time meridian - longitude) + E. In the example, the longitude for New York City is 73 degrees west, or -73 degrees. So this is 4_(-60-(-73)) -0.094 = 4_(-60+73) - 0.094 = 4_13 - 0.094 = 51.806. This is given in minutes.
Add the time correction factor to the local time to get local solar time. The local solar time in New York City on June 18, 2011 is 51.806 minutes later than local time.