Altitude of the Moon During Lunar Eclipses

Fred Espenak

Calculating the Moon's Altitude and Azimuth

To determine whether a lunar eclipse is visible from a specific geographic location, it is simply a matter of calculating the Moon's altitude and azimuth during each phase of the eclipse. The calculations can be performed on any pocket calculator having trig functions (SIN, COS, TAN). Armed with the latitude and longitude of the location, the lunar eclipse catalog provides all the additional information needed to make the calculations. (For those wishing to avoid the tedium of performing of these calculations, several Microsoft Excel spread sheets are available to automate the calculations for any geographic location and for all lunar eclipses from 1951 to 2050. See: Local Visibility of Lunar Eclipses.)

The altitude 'a' and azimuth 'A' of the Moon during any phase of an eclipse depends on the time and the observer's geographic coordinates. Neglecting the effects of atmospheric refraction and lunar parallax, 'a' and 'A' are calculated as follows:

```             h  = 15 * (GST0 + t - ra ) + l
a  =  ArcSin  [ Sin d Sin f + Cos d Cos h Cos f ]
A  =  ArcTan  [ - (Cos d Sin h) / (Sin d Cos f -  Cos d Cos h Sin f) ]

where:
h  =  Hour Angle of the Moon (in degrees)
a  =  Altitude (in degrees)
A  =  Azimuth (in degrees)
GST0  =  Greenwich Sidereal Time at 00:00 UT
t  =  Universal Time
ra  =  Right Ascension of the Moon (in hours)
d  =  Declination of the Moon (in degrees)
l  =  Observer's Longitude (East +, West -)
f  =  Observer's Latitude (North +, South -)

```

For example, determine whether the Moon will be above the horizon at greatest eclipse during the total lunar eclipse of 2000 Jan 21 as seen from Washington DC. The geographic coordinates of Washington DC are:

```                  Latitude:   f =  38°53´N  =   +38.9°
Longitude:   l = 077°02´W  =  -077.0°```

From the catalog record, we have:

```             Time of Greatest Eclipse:     t =  04:43 = 4.72
Greenwich Sidereal Time at 00:00 UT:  GST0 =   7.992
Right Ascension of the Moon:    ra =   8.173
Declination of the Moon:     d =  19.76 ```

Thus:

```    Hour Angle of the Moon:  h  = 15 * (GST0 + t - ra ) + l
= 15 * (7.992 + 4.72 - 8.173) + -077.0
= 15 * (4.55) -077.0
h  = -9°

Altitude of Moon:   a = ArcSin [Sin d Sin f + Cos d Cos h Cos f]
= ArcSin [Sin(19.8) Sin(38.9) + Cos(19.8) Cos(-9) Cos(38.9)]
= ArcSin [0.339 * 0.628 + 0.941 * 0.988 * 0.778]
= ArcSin [0.213 + 0.723]
= ArcSin [0.936]
= 69°```

With an altitude of 69°, the Moon will indeed be visible at greatest eclipse during the total lunar eclipse of 2000 Jan 21 as seen from Washington DC.

The expression for the Moon's azimuth contains the trigonometric function ArcTan. The ArcTan function results in an angle between -90° and +90°, with an ambiguity of + or - 180°. If the desired calculation has the form A = ArcTan [ x / y], then the ambiguity can be resolved using a simple test: if the denominator y is negative, then add 180° to the final answer.

In our current example the azimuth of the Moon is then:

```
Azimuth of Moon:  A = ArcTan [-(Cos d Sin h)/(Sin d Cos f - Cos d Cos h Sin f)]
= ArcTan [-(Cos(19.8) Sin(-9))/(Sin(19.8) Cos(38.9) - Cos(19.8) Cos(-9) Sin(38.9))]
= ArcTan [-(0.941 * -0.156) / ((0.339 * 0.778) - (0.941 * 0.988 * 0.628))]
= ArcTan [-(-0.147) / ((0.264) - (0.584))]
= ArcTan [ +0.147 / (-0.320)]
= ArcTan [ -0.459 ]
= -24.7°

Since the denominator in ArcTan [ +0.147 / (-0.320)] is negative, we must add 180° to the final answer:

A = -24.7° + 180°
A = 155.3°
```

This places the Moon in the southeast at greatest eclipse during the total lunar eclipse of 2000 Jan 21 as seen from Washington DC.

Acknowledgments

All eclipse calculations are by Fred Espenak, and he assumes full responsibility for their accuracy. Some of the information presented in these tables is based on data originally published in Fifty Year Canon of Lunar Eclipses: 1986 - 2035.

Permission is freely granted to reproduce this data when accompanied by an acknowledgment:

"Eclipse Predictions by Fred Espenak, NASA/GSFC"

2005 July 31