^ecknlcai ^lote ^o. 209

CALCULATION OF SUNRISE AND SUNSET

TIMES AT IONOSPHERIC HEIGHTS ALONG

A GREAT CIRCLE PATH

A. H. Brady and D. D. Crombie

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

'Cechmcal ^ote 209

Issued November 8, 1964

CALCULATION OF SUNRISE AND SUNSET

TIMES AT IONOSPHERIC HEIGHTS ALONG

A GREAT CIRCLE PATH

A. H. Brady and D. D. Crombie

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington,, D.C. 20402

Price: 20 cents

CONTENTS

Page

1. Introduction 1

2. Determination of Sunrise and Sunset Times 2

3. Determination of Coordinates of Points Along the Path. . . k

h. A Digital Computer Program to Determine the Amount of Path

Illumination 5

4.1 Form of Computer Output 5

h .2 The Computer Program 6

5. Some Comparisons of Path Illumination with Diurnal Variations

of VLF Phase 7

6. Discussion 8

7. Acknowledgment 9

8. References 10

Figures 11

in

Calculation of Sunrise and Sunset Times at Ionospheric

Heights along a Great Circle Path

A. H. Brady and D. D. Crombie

Formulae for determining

the coordinates of points along

a specified great circle path

and for determining the times of

sunset and sunrise at these points

are given. A computer program for

making the calculations and showing

which portions of a great circle

path are illuminated is outlined.

Some illustrations of the dependence

of VLF phase delay on path illumination are also given.

1. Introduction

There is, at present, considerable interest in the diurnal variation in the phase of a VLF signal at great distances and in its dependence upon the distribution of daylight and darkness along the propagation path [Crombie et al. , 1958], For this reason, a simple and

straightforward numerical procedure has been devised for the computation

of the times of sunrise and sunset at a specified height along a great

circle path. The computation then yields the fraction of the path which

is illuminated at any time. It is the authors' experience that much

time can be wasted "both in deciding which spherical trigonometrical

formulae should be used and in programming a computer to make the calculations correctly. Thus, although this report contains nothing mathematically new or sophisticated, it is hoped that the collection of

formulae and the outline of the computer program may be useful to others

who may have to make such calculations for the first time, or have large

numbers of such calculations to make.

Following a discussion of how sunrise or sunset is defined above

the surface of the earth, expressions will be given for calculating the

time at which the sun reaches a given zenith angle, for a specified

longitude and latitude and specified declination of the sun. In

section 3> formulae are given for finding the latitude and longitude of

n + 1 uniformly spaced points along a great circle path (less than l80°

long) whose end points are specified. These values can then be inserted in the earlier formulae and the sunset and sunrise times obtained.

Section k discusses the problem of using a computer for making these

calculations. Section 5 gives some examples of the calculations and

compares them with observed diurnal phase variations. Accuracy adequate

for the intended purpose is obtained with a considerable simplification

by assuming that (a) the sun can be treated as a point source, (b)
atmospheric refraction can be neglected, and (c) the earth is a sphere.

Some consequences of the resulting approximations are discussed in

section 6.

2. Determination of Sunrise and Sunset Times

Sunrise or sunset is usually thought of as occurring when the sun rises or sets, respectively, above or below the horizon, and the time

at which this occurs depends upon the elevation of the observer above

the surrounding country. At a point on the surface of a smooth, spherical, airless earth, the sun's rays would be horizontal when it sank

below the horizon at sunset; i.e., its zenith angle x would be 90° • For

points above the surface, the zenith angle at sunrise or sunset increases

as the distance of the observer above the surface increases.

In speaking of sunrise or sunset in the ionosphere, we are of

course interested in those components of the sun's radiation which affect

the ionization, and most of these cannot penetrate the lower atmosphere.

For example, the ozone layer screens the near ultra-violet light from

heights below about 30 km [Mitra, 1952; Reid, 1961] . In general, therefore, at a given point above the screening height (s) for a particular

wavelength, sunrise and sunset for this wavelength occur when the sun's

rays, arriving at the point, are horizontal at the screening point.

From the geometry illustrated in figure 1, it is easily shown that the

value of x at a height H, at which the sun's rays become tangential to a concentric sphere of radius A + S, is given by

2 -x-Wi-C-Kf-) W

Since it is mainly values of x between 90° and 100° that are of

interest, and, since A is much greater than both S and H, equation 1 can be reduced to the approximate form

x-« 90 + 1.01 V H - S

which is useful for hand calculations. This gives x in degrees when

H and S are expressed in kilometers. A plot of this relation is given

in figure 1 along with the sunrise-sunset geometry.

Equation 1 shows that there is no unique combination of S and H

which will yield a particular value of x (when H > 0). Thus rather than

using particular values of H and S it is better, in general, to use the

appropriate value of x« Nevertheless it is sometimes convenient to

quote the value of H which corresponds , for a particular S , to a given

value of x« Where this is done here, the screening height, S = 30 km?

has been used.
With x fixed "by H and S, application of the law of cosines to the

spherical triangle "whose vertices are the north pole, the sun, and the

zenith gives the local hour angle (LHA) at sunset and sunrise in the

following form [Mitra, 1952]

cos (LHA.) = cos x

- sin ^ sin 6

cos cp-p. cos 6

(2)

where <P is the latitude of the point on the earth and 6 is the declinaU

(l) tion of the sun at noon on the day in question . (North declinations

and latitudes are positive; South declinations and latitudes are negative). The local hour angle of the sun is defined as the angular distance from the meridian of the point in question to the meridian of the

sub-solar point.

If | cos LHA. | > 1, LHA is imaginary and sunrise and sunset do not

occur at the point under consideration. In such a case, account must be

taken of the local season to determine whether the point is in darkness

or sunlight.

To obtain the Greenwich Mean Time (or Universal Time) for sunrise

and sunset, it is necessary to include the Equation of time, E. This is

not printed in the Air Almanac but can be calculated from the value of

the Greenwich hour angle of the sun at Greenwich mean noon, which is

given in the Almanac. (Alternatively, E can be obtained from the

American Ephemeris.) Thus,

E = - GHA

15°" hours if GHA < l80c

and

360° - GHA . j=c hours if GHA > l80c

where E = mean time - apparent time.

The Universal Time of sunrise (TR) and sunset (TS) in hours are then

given by,

Footnote (l) . Use of the value of declination at noon, instead of at

sunrise (or sunset) may introduce small errors in the calculated time

of sunrise (or sunset) at very high latitudes near the equinoxes. These

errors may be reduced by using an iterative process.
TS l = 12 + E - 9„ (In hours) ± LHA (in hours) (3)

TRJ D

where is the longitude of the point expressed in hours (taking East

to be positive and West to be negative), and the algebraic sign of LHA

is chosen to correspond to sunrise (-) or sunset (+). Equation 3 may

yield values for TR and TS which are either negative or greater than

2k hours. To avoid these difficulties, it is suggested that one simply

add 2h hours to the result as a matter of routine, and then compute this

result (modulo 2h) to obtain proper clock values for TR and TS.

3. Determination of Coordinates of Points Along the Path

To determine the illumination status along the path it is necessary

to have the values of the geographic coordinates at any distance D

along the path. Figure 2 shows the path coordinate geometry as used in

this determination, with N corresponding to the north pole, E to the east

end of the path, and W to the west end. 9TM and are the respective

longitudes, and cp^ and ca

7 are the respective latitudes. It is

assumed that the earth is spherical.

First, the arc length EW of the path is computed using the law of

cosines for a spherical triangle:

cos EW = sin cp^ sin cpE+ cos cp^ cos cpE

cos (e E

- 0^) . 00

Also, again using the law of cosines, the angle E is determined:

sin cp^ - sin cpE

. cos EW

cos E = — (5) cos ox-, • sin EW

^E

The path (EW) is divided into n segments so that the angular distance

to the end of the k segment from the east end is

ED = k. M . (6)

Again the law of cosines is used to determine the latitude %
sin cp-p = cos ED . sin cp^ + sin ED . cos cp-p . cos E . (7)

Since -90° ^ y < 90 , cos cp will simply "be given by V 1 - sin^ cp .

(8)

The longitude 9^ is also determined by the law of cosines. Thus,

cos ED - sin cp n

. sin cp cos (a. - ej = ^ 5_ (9) E D' CO s coD

. cos cp E

and

3 D

= 9E

- Arccos [cos (^ - 0^] (10)

where the principal value of the arccosine has been designated, since

( eE

- eD

) <i8o°.

k. A Digital Computer Program to Determine

the amount of Path Illumination

k,l Form of Computer Output

In the calculations, using a digital computer, it was found con- venient to print out the illumination status of each point on the path

graphically, as a function of the time of day. Before outlining the

computer program, a brief account will be given of the form of the display,

Figure 3 shows a sample of the output for June 21 on the Rugby

(England) to Boulder (Colo.) path at an altitude of 70 km and for a screening height of 30 km.

The illumination status of each point on the path was calculated at

15 minute intervals, and time is shown in the left-hand column. The path

was divided into 100 segments, numbered from East to West along the path

and left to right across the top of the page. If a segment is dark, it

is shown by "*"". The percentage of the length of path illumination is

listed down the right-hand edge corresponding to the universal time opposite. It was computed by dividing the number of illuminated segments

by the total number of segments.

At the bottom of the figure, the actual sunrise and sunset times are listed for each segment. This is done to furnish more detailed information if it is required. A sunrise occurring at 0.00 hours with sunset

at 2^.00 hours is used to indicate that the segment is always illuminated.
(The sun does not set!) A sunset at 0.00 hours with sunrise at 2i+.00

hours would be used to indicate a segment which remains in darkness.

(The sun does not risel)

Figure 3 shows illumination variations which may occur on relatively

high latitude paths at ionospheric heights during summer. During the

period when the path, at ground level, is in darkness, there is a short

period during which the ionosphere is illuminated by sunlight from over

the pole. Figure 3 shows that this occurs, on the Boulder -Rugby path, at

approximately 0300 UT on June 21, and that at this time the whole path is

sunlit at ionospheric heights.

k.2 The Computer Program

The following input data are necessary for the computer program:

h

1. Solar declination and Greenwich hour angle at 12 UT

for each day for which the calculation is to be made

(from Air Almanac ) .

2. Screening height and height of ionosphere in kilometers.

3. Coordinates of the eastern and western ends of the path.

k. Number (n) of segments into which the path is to be

divided. (The practical limitation on this number is determined by the number of characters across a page of

printed output from the digital computer to be used.)

The program then proceeds in the following steps

1. Read in the declination ( 6 ) and Greenwich hour angle

(CHA) for each day, convert to radians, and store.

2. Read and store the screening height (S) and ionospheric

height (h) in kilometers.

3. Read the coordinates (cp , 9p, co., 9¥) of the end points

of the path to be computed and the number of segments (n)

into which it is to be divided.

h. Convert the coordinates from degrees to radian values.

At this point one may make whatever checks on the coordinates are necessary, such as for antipodal end points. The

case where the path lies on a meridian should be treated

separately to avoid difficulties with "round-off" error

in the arguments of the inverse trigonometric functions

used in the calculations. (For example, a cosine might

become 1.000001 when computed from a formula.)
5. Compute the angular distance (EW) between path end

points, using equation k. Compute and store the values of

sin EW and cos EW.

6. Compute cos E from equation 5«

7. For each of the n+1 points along the path compute and

store 9~, sin cp , and cos cp using equations 6, 7> 8, and

10.

8. Compute the sunrise and sunset times for the given

height at each point along the path, and store these times

in an array.

a) Compute cos LHA from equation 2.

b) If J cos LHA J > 1, sunrise and sunset do not occur. There, if (6. cp^) < 0, it is winter and the point is

dark and so set the time of sunrise = 2^.00 hours and

sunset =0.00 hours and proceed to the next segment.

If (6. cp^) > 0, it is summer and the point is always

sunlit and so set the time of sunrise =0.00 hours and

sunset = 2^.00 hours and proceed to the next segment.

c) If I cos LHA I ^ 1, then compute LHA and convert to

hours. Compute the times of sunrise and sunset from

equation 3« Add 2k hours to each value and compute the

results (Modulo 2k). This gives positive values for

the times of sunrise and sunset.

9. Determine number of sunlit segments and divide by n. Then print a line with the status of each segment represented

by the appropriate symbol. Repeat until the time 2^-At is

reached.

10. Print out the sunrise and sunset times from 8, and then

repeat steps 8 and 9 for each day and (or) layer height

desired. (The new layer heights may be incremented values

of the original value of H.)

5. Some Comparisons of Path Illumination

with Diurnal Variations of VLF Phase

The phase of a frequency-stabilized very low frequency (VLF) signal

shows a diurnal variation having a nearly trapezoidal shape for paths at

temperate or lower latitudes. Pierce [1955] and Crombie et al. [1958]

have previously pointed out the dependence of the major features of this

variation upon the times of sunrise and sunset at the transmitter and

receiver ends of the path.
An example illustrating this dependence is given in figure k which shows

approximately the variation with time of the length of path in darkness

at 80 km, (x = 97°) and at the ground level (solid lines) during September

and December 1962 for the path between the Canal Zone and Frankfurt,

Germany. The corresponding phase variation for the 13 kc/s signal

propagating from the Canal Zone to Frankfurt are shown by the dashed

lines. The correspondence between the phase and illumination curves is

quite marked.

At high latitudes the diurnal phase delay may show variations which

are not shown by the diurnal variation in the length of the illuminated

path at ground level. Such a situation is shown in figure 5 for the GBR

(Rugby, England) to Boulder, Colo., path during June 1962. Again the

solid lines (obtained in part from figure 3) show the variation in the

length of path in darkness at ground level, 70 km and at 90 km (\ = 0,

96. 5° and 97.5°) approximately. The dashed line is the mean phase delay

for the same period. Of particular interest is the dip in the phase

curve at 0300 UT, which agrees with the increase in illumination shown at

the higher levels in the atmosphere, but not at ground level.

6. Discussion

Some simplifications of the actual problem have been made here in

order to keep the computation simple and because a more sophisticated

treatment appeared unnecessary for the work with which these calculations

was associated. These simplifications are:

a) the sun can be treated as a point source,

b) atmospheric refraction is neglected,

c) the earth is spherical.

For some purposes it may be necessary to consider the effect of

these simplifications. Often, for example, [Mitra,1952] the fact that

the sun is not a point source can be included by increasing the angle of

X given by equation (l) and used in equation (2) by l6', the apparent

semidiameter of the sun. In the present context this would be tantamount

to assuming that full illumination occurred while any portion of the sun

is visible and is not necessarily a better assumption than that used in

this paper. Again, although screening by the atmosphere has been considered, vertical refraction by the atmosphere has not. At the surface of

the earth and for visible light refraction can be included by further increasing the value of x used in equation (2) by 3^'» In both these cases neither the effective diameter of the sun nor the correction for refraction is known exactly. In fact it is hoped that the experimental observations may eventually lead to a better understanding of these values.

It should be pointed out, however, that the errors resulting from

neglect of these quantities, or from the use of incorrect values for

8
them may be appreciable at high latitudes during winter. Since sunrise

and sunset in this paper are defined as the times at which the center of

the sun passes a given zenith angle these times may be considerably in error when the zenith distance of the sun is changing slowly and is near

90°. For example, neglect of refraction and the sun's diameter at certain

places near the Arctic Circle leads to the result that the sun should not

rise during mid-winter whereas in fact several hours of sunshine each dav

are experienced. It should be noted that this is entirely due to the error

of about 50' in determining the zenith distance of the sun and from neglect

of states intermediate between full daylight and darkness. It is possible

that errors of this type decrease as the screening height increases

[Reid, I961]. Indeed, as noted above, it is expected that comparison of

the length of the path illuminated and the corresponding phase change may

yield more information on this subject.

The final simplifying assumption listed above is that the earth is

spherical. This is only approximately true since the polar and equatorial radii differ by some 22 km. This source of error could be overcome if n-'cess?ry by using the geocentric latitude (the angle subtended

at the earth's center by the arc between the place in question and the

equator) instead of the geographic latitude (the angle between the normal

to the earth's surface at the place in question and the equatorial plane).

It can be shown [Barlow and Bryan, 19kk] that the difference between these

angles has a maximum value at a latitude of ^5° and that the difference is

only about 12'. Errors resulting from this cause have been neglected since

at a latitude of U5 the time of ground sunrise or sunset typically changes

at a rate of about 3 minutes for each 1° change in latitude.

7. Acknowledgment

Valuable discussion with J. H. Crary and advice by H. H. Howe on the

preparation of this paper are gratefully acknowledged. The work was done

as a part of a research program supported by the AdvancedResearch Projects

Agency, Washington, D. C, under Order 183.
8. References

Barlow, C. W. C, and G. H. Bryan (19M+), Elementary Mathematical

Astronomy, 5th Edition, 80 (University Tutorial Press Ltd., London).

Crombie, D. D., A. H. Allan, and M. Newman (May 1958)? Phase variations

of the l6 kc/s transmission from Rugby as received in New Zealand,

Proc. IEE, Part B, 301- 30^

.

Mitra, S. K. (1952), The upper atmosphere (The Asiatic Society, Calcutta,

2nd Edition)

.

Pierce, J. A. (1955)? The diurnal carrier phase variations of a l6 kc/s

transatlantic signal, Proc. IRE k3, 58^-588.

Reid, G. C. (1961) , A study of the enhanced ionization produced by

solar protons during a polar cap absorption event, J. Geophys. Res.

66, 4071-^085.

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SEGMENT- 10 20 30 40 So An »*

30-»

115- - - 75. HO- - - 73. 1*5 * - 72. 200- - - 71. 215 • " - 70. ?3Q...... •••••••••••••••• - - 69. 24S-««»a«a»»»»«««a«aa«s»»i - - 69.

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UT 1 I 1 I I 1 I 1 I I i

SOLAR ILLUMINATION OF THE LOWER IONOSPHERE

OVER A GREAT CIRCLE PATH

(SEE LEGEND AT TOP)

SUNRISE AND SUNSET TIMES (HOURS. HOURS

ALONG PATH. DISTANCE IN KILOMETERS.

NUMBER 1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 DISTANCE 74 149 223 297 371 446 520 594 668 743 817 891 966 1040 1114 1188 1263 1337 1411

SUNRI SE 2.75 2.75 2.75 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76 2.76

SUNSET 21.35 21.47 21.59 21.71 21.84 21.96 22.09 22.22 22.36 22.49 22.63 22.77 22.91 23.06 23.21 23.36 23.51 23.66 23.82 23.98

NUMBER 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 DISTANCE 1485 1560 1634 1708 17B2 1857 1931 2005 2080 2154 2228 2302 2377 2451 252 5 2599 2674 2 748 2822 2B97

SUNRISE 2.76 2.75 2.75 2.74 2.74 2.73 2.71 2.70 2.68 2.64 2.58 0. 0. 0. 0. 0. 0. 0. 0. 0.

SUNSET 0.15 0.31 0.48 0.65 0.83 1.01 1.19 1.38 1.58 1.79 2.02 24.00 24.00 24.00 24.00 24.00 24.00 24.00 24.00 24.00

NUMBER 40 41 42 43 44 45 46 47 4B 49 50 51 52 53 54 55 56 57 58 59 DISTANCE 2971 3045 3119 3194 3268 3342 3416 3491 3565 3639 3714 3768 3B62 3936 4011 4085 4159 4233 4308 4382

SUNRISE 0. 0. 0. 3.72 3.97 4. IB 4.3B 4.57 4.75 4.93 5.10 5.28 5.45 5.61 5.78 5.94 6.09 6.25 6.40 6.55

SUNSET 24.00 24.00 24.00 3.21 3.14 3.11 3.08 3.06 3.05 3.04 3.04 3.03 3.0 3 3.02 3.02 3.02 3.02 3.01 3.01 3.01

NUMBER 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 DISTANCE 4456 4530 4605 4679 4753 482B 4902 4976 5050 5125 5199 5273 5347 5422 5496 5570 5645 5719 5793 5867

SUNRISE 6.70 6.85 6.99 7.13 7.2 7 7.41 7.54 7.67 7.80 7.93 8.05 8.17 8.29 8.41 8.53 8.64 8.75 8.86 B.97 9.07

SUNSET 3.01 3.01 3.01 3.01 3.01 3.02 3.02 3.02 3.02 3.02 3.02 3.02 3.03 3.03 3.03 3.03 3.03 3.03 3.04 3.0*

NUMBER 80 81 82 83 84 85 86 87 88 69 90 91 92 93 94 95 96 97 98 99 DISTANCE 5942 6016 6090 6164 6239 6 313 63BT 6462 6536 6610 6684 6759 6633 6907 6961 7056 7130 7204 7278 7353

SUNRISE 9.17 9.27 9.37 9.47 9.57 9.66 9.75 9.8* 9.93 10.02 10.11 10.19 10.28 10.36 10.44 10.52 10.59 10.67 10.75 10.82

SUNSET 3.04 3.04 3.04 3.05 3.05 3.05 3.05 3.06 3.06 3.06 3.06 3.07 3.07 3.07 3.07 3.08 3.06 3.08 3.08 3.09

SUNRISE 10.89

SUNSET 3.09

Figure 3. Sample computer output for the Rugby-Boulder path

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Figure 5- Comparison of phase delay expressed as a percentage of its total diurnal change, with

variation of amount of path in darkness at

90 km, 70 km, and at ground level (solid

lines). (Screening ht. =30 km.)

15 G PO 845 - 233