About/About the Math

Calculation of Local Daylight Time

Local Daylight Time is a new system of civil time designed to upgrade current time systems, eliminating the discontinuities of time zones and seasonal clock changes while retaining the benefits of Daylight Saving with reduced variation in sunrise time. The time is calculated for each unique location and date as explained below.

The basis of Local Daylight Time (LDT) is Coordinated Universal Time (UTC1). The LDT for a given location and date is found using an offset from UTC summed with the actual UTC. All times are in units of hours.

The LDT offset has several terms. First, a constant offset of 0.5 hours is added, half of the Daylight Saving or Summer Time offset used by many systems of Standard Time. This constant reduces the average difference from existing systems. The added half hour shifts the mean sunrise (and sunset) at equinox to be 6:30 instead of 6:00 (neglecting atmospheric refraction). Social preference for a later sunrise time is evident in both the use of Daylight Saving time and the tendency to skew time zones to the west from their original positions.

Next, an offset component based on longitude is included as used in mean solar time. Dividing the longitude in degrees by 15 (°/h) produces an offset in hours that is proportional to the ratio of 24 hours to a full rotation of 360 degrees. A continuous offset is produced, eliminating time zone discontinuities. The longitude offset polarity is positive for locations west of zero degrees, up to a maximum of less than 24 hours. Longitude coordinates are expressed as

The geographic coordinates and date are used to calculate the local apparent sunrise time. This is a complex calculation but it is available from NOAA, both in Excel and Javascript. The PHP language includes a built in sunrise function2. The use of sunrise time introduces a latitude term that follows the illumination map and thus improves tracking between clock time and daylight.

A Local Daylight Time offset term reduces seasonal sunrise time variation. However, since the day length at higher latitudes may be zero or 24 hours, an adjustment to this offset component is required. A suitable function can be derived from the equation of the length of the day. Adding the difference between the ideal mean sunrise (6am) and the actual sunrise would cause sunrise to occur at the same time every day. Because sunrise may not occur at all above the polar circles, an adjustment function reduces this offset term at higher latitudes, allowing full correction (adj=1) only at the equator, and reducing sunrise adjustment to zero at high latitudes.

If the adjustment becomes negative, the value is kept at zero so that no sunrise time correction is applied.

Finally, to avoid a daily discontinuity due to change in the offset, the sunrise offset is calculated both for the current date and the next day. The difference is then linearly interpolated and spread over the current day so that the offset changes continuously.

The complete equation for Local Daylight Time is this:

It should be noted that including a sunrise term causes the length of each day to vary, as the sunrise time varies seasonally. Sunrise time varies with the date at all latitudes, but the daily change of the LDT Offset is small. The interpolation term varies continuously, so the LDT clock runs slightly fast or slow as needed to achieve the next offset value. This achieves the Daylight Saving effect without the one hour discontinuities.

1UTC includes the use of leap second to correct for the changing rate of the earth's rotation. While leap seconds introduce discontinuities that are contrary to the continuous character of LDT, this flaw is accepted as a necessary compromise given the need for an agreed basis time reference. The infrequent one second shift is imperceptible relative to the daily change of the natural day. The leap second discontinuity may be avoided as done by Google

2PHP and other sunrise functions include a term for atmospheric refraction to account for the difference between the 90 degree hour angle and the hour angle of the apparent sunrise. For simplicity, LDT adopts a constant value of 90.833 degrees for the sunrise zenith. Atmospheric refraction also affects the latitude offset adjustment function, which is based on the ratio of the tangent of latitude to the tangent of the polar circles.

NB Constants - Atmospheric refraction varies with conditions but 50' is widely used for sunrise and is a good approximation. The obliquity of the ecliptic varies over time but this has an effect of less than one offset minute over the next 1,000 years. 6:30 for the target sunrise time is somewhat arbitrary, but fits well with present time systems.

© Copyright Keith Thomas 2013