A MATHCAD
ELECTRONIC BOOK

Sexagesimal (angles or time)
to decimal

Astronomical
Formulas: Sunrise, Sunset, and Twilight

The conversion function is given by

Reprinted from Astronomical
Formulas by Martin Zombeck. Copyright (c) 1997 by MathSoft, Inc. All
rights reserved.

where

a = integral number of degrees or hours

b = integral number of arc minutes or minutes of
time

c = integral number of arc seconds or seconds of
time

Introduction

This document calculates, for latitudes between 65 degrees
North and 65 degrees South and any date in the latter half of the 20th century,
the times of sunrise, sunset , and civil, nautical, and astronomical
twilight . The calculated times refer to
locations with a clear horizon and a standard refraction correction for normal
meteorological conditions. The calculated times are accurate to within 2
minutes; higher accuracy is seldom justified since local meteorological
conditions will cause the actual time to vary.

Unit
Definitions:

Twilight is caused by
the scattering of sunlight by the upper layers

of the Earth's atmosphere. There are different definitions of
twilight.

Astronomical twilight begins at sunset (ends at sunrise) and
ends (begins) when the Sun's center is 18 degrees below the horizon. Civil
twilight begins (ends) when the Sun's center is 6 degrees below the horizon;
nautical twilight begins (ends)

when the Sun's center is 12 degrees below the horizon.

Decimal (angles or time) to
sexagesima l

We must first define some intermediate
functions:

The method of the Almanac for
Computers (United States Naval Observatory) is used to calculate the
times.

The definitions for sunrise, sunset, and twilight for an
observer at sea level are given in terms of the zenith distance z (p /2 - altitude) of the center of
the Sun:

where d_h is the angle or time in decimal
notation.

Sunrise and Sunset

90 deg 50 arc min

The conversion function is then given by

Civil Twilight

96

Nautical Twilight

102

Astronomical Twilight

108

Sunrise or Sunset is defined for a geocentric altitude of -50
arc minutes; 34 arc minutes is to correct for atmospheric refraction and 16 arc
minutes is to correct for the Sun's semidiameter.

For an observer at a height h meters above sea level the
following correction should be added to the zenith distances above:

2.12h1/2 arc
minutes

The following formulae are used to compute the
phenomena:

M = 0.985600t - 3.289 degrees

l = M + 1.916sin M + .020sin 2M + 282.634 degrees

tan a = 0.91746tan l

sin d = 0.39782sin l

x = cos H = (cos z - sin d sin f)/(cos d cos
f)

T = H + a - 0.065710t - 6.622 hours

UT = T + L

where

• M =
Sun's mean anomaly

• t = approximate time of
phenomenon in days since 0 January, 0h UT

• l = Sun's true longitude

• a = Sun's right ascension

• d = Sun's declination

• H =
Sun's local hour
angle

• T =
local mean time of phenomenon

• f = latitude of observer

• L = longitude of observer (east is negative; west
is positive) in hours

• UT =
Universal Time of
phenomenon

Method

Using the above formulae, the procedure from the Almanac for Computers is as
follows:

1. Calculate an initial
value for t from

t = N + (6h +
L)/24

(for morning phenomenon)

t = N + (18h +
L)/24

(for evening phenomenon)

where N is the day of the year.

2. Calculate M and
l.

3. Solve for a, noting that a is in the same quadrant as
L.

4. Solve for d.

5. Solve for H; the correct quadrant for H is
given by the following rules:

rising phenomenon: H = 360 - arccos x

setting phenomenon: H = arccos x

6. Calculate
T.

7. Calculate UT. To convert to the local time
see the standard time conversions in Section 1.16.

Enter
Data

Year:

Month:

Day of the Month:

Latitude:

Longitude:

Observer's height above horizon (meters):

Observer's location:

MMT Observatory, Arizona

Calculations

Intermediate
Variables

Convert to decimal form:

The functions necessary to convert between sexagesimal and
decimal form are defined past the right margin of the page.

The day of the year N:

First, calculate the Julian
Date for the calendar date at 0h UT:

Then calculate the Julian
Date for the calendar date on 0 January, 0h UT for the given year:

Define an initial value for t, the approximate time of the
phenomena:

(for morning phenomena)

(for evening phenomena)

The zenith distances
for the phenomena:

for Sunrise or Sunset

for civil twilight

for nautical twilight

for astronomical twilight

Calculate the
Morning Phenomenon

Convert the time T to a range between 0 and 24h:

Calculate UT of the phenomenon. If UT is greater than 24h , the phenomenon occurs on the following
day, Greenwich time. If UT is negative, the phenomenon occurs on the previous
day, Greenwich time.

Calculate
Evening Phenomenon

Convert the time T to a range between 0 and 24h:

Calculate UT of the phenomenon. If UT is greater than 24h , the phenomenon occurs on the following
day, Greenwich time. If UT is negative, the phenomenon occurs on the previous
day, Greenwich time.

Results

The Morning
Phenomena

(converting to
sexagesimal notation)

Sunrise:

Civil twilight:

Nautical twilight:

Astronomical twilight:

Note: To convert to Eastern Standard Time (for example),
subtract 5 hours from the above times.

The Evening
Phenomena

(converting to
sexagesimal notation)

Sunset:

Civil twilight:

Nautical twilight:

Astronomical twilight:

Note: To convert to Eastern Standard Time (for example),
subtract 5 hours from the above times.

Accuracy

In order to provide the user with an indication of the
accuracy of the above method, we have calculated the times of Sunrise and Sunset
for the year 1993 and compared these times with those times provided by the
Astronomical Almanac. The times were calculated for a latitude of 42 degrees and
at the Greenwich meridian.

The differences in Sunrise DRise = Risealmanac - Risehandbook and Sunset D Set = Setalmanac - Sethandbook in minutes for the beginning of each
of the twelve months of 1993 are:

A plot of the above differences: