Calculation of lunar eclipses

Author: Andres Mejia Valencia

Submitted to: Computing section of A.L.P.O. / “The Digital Lens”, Editor Mike McClure
Keywords: Astronomical calculations, eclipses, Moon, ephemerides
Date: May 24th, 1998
_________________________________________________________________________________

Definitely, lunar eclipses do not cause such a great popular stir as their solar counterparts, however
they are interesting astronomical phenomena for which their calculation is within the reach of any
amateur astronomer involved with the mathematics of the heavens. We shall discuss a derivation of the
method to calculate a lunar eclipse and apply it to a real life example for the partial eclipse of july 28 of
1999.

Basically, a lunar eclipse happens when the moon passes within the shadow cast by the earth in space,
forming in line with the Sun and earth, as it ocurrs every full moon. However, we do not see a lunar
eclipse every month because the lunar orbit is inclined about 5º 09’ with respect to the ecliptic, which is
the plane that contains the sun and the earth. The conditions necessary for a lunar eclipse must satisfy
that the moon is full and on or very near the nodes of its orbit, where it crosses the ecliptic.

The shadow geometry is explained in figure 1, where it can be seen that the penumbral shadow DXYC
and the umbral shadow XYV, being cast by an approximately spherical earth are really a couple of
cones each one with the vertex on opposite sides. If these cones at cut perpendicular to the plane of the
figure at the moon distance, we shall see two circles of shadows which are represented in figure 2. The
instants of time in which the moon comes into contact with these circles define the different phases of
the eclipse, and the different types of eclipse depend on how deeply or near the moon comes to the
common center of the shadows.

figure 1

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If the moon enters completely inside the umbral shadow cone (interior circle in figure 2) the type of the
eclipse is total. If the moon enters the umbra, but the disc does not come completely inside then the
eclipse is partial. Also if the moon enters totally or partially the penumbra without actually touching the
umbra we have a penumbral eclipse.

figure 2

We shall use the following notation:

Description Sun Moon
Horizontal equatorial parallax P P1
Semidiameter S S1
Geocentric distance (in units of the equatorial radius) r r1

Referring to figure 3, we can see that the umbral shadow cone at moon´s distance is MN and the
umbral angular radius , with respect to T is NTM. If XVT is v, then XNT =  + v.

figure 3

But XNT is the angle subtended at the moon by the equatorial radius, that is the horizontal equatorial
parallax of the moon, thus XNT = P1, and P1=  + v.

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Also, STA =XAT + v. As STA is the angle subtended at the earth by the radius of the sun it is its
semidiameter S. Is obvious that XAT is the parallax of the sun P, thus S = P + v.

From the simple definitions given and for an observer at the center of the earth, we have that the radius
of the umbra is  = P + P1 – S. In a similar way the penumbral radius is found to be  = P + P1 + S.

In practice, it has been determined that the earth´s atmosphere has the effect of increasing the shadow
cones in about 1/50th, which has been verified over the years by timing the contacts of the shadows
with the lunar craters. Also, as the earth is not a perfect sphere, the shadows are affected somehow by
the flattening of the earth. This can be considered by correcting the moon´s parallax by reducing it to a
mean radius of the earth at 45º latitude. So, the adopted formulas for the radius of the shadows are:

 = 1.02( P + 0.99834P1 − S ) (1)

 = 1.02( P + 0.99834P1 + S ) (2)

We will not study in detail the derivation of the necessary geometrical conditions for a lunar eclipse to
happen, as it will be enough to have the following definitions, where  is the moon´s latitude:

Condition Conclusion
 > 1º 36’ 38’’ No eclipse in the penumbra
1º 26’ 19’’ <  < 1º 36’ 38’’ Possible eclipse in the penumbra
1º 03’ 46’’ <  < 1º 26’ 19’’ Eclipse in the penumbra. No umbral eclipse
0º 53’ 26’’ <  < 1º 03’ 46’’ Eclipse in the penumbra. Possible eclipse in the umbra
 < 0º 53’ 26’’ Eclipse in the umbra

Now, the actual derivation of our method follows. Let us suppose that the geometrical conditions for a
lunar eclipse are fulfilled. We refer to figure 4 and imagine that T represents an observer at the center
of the earth and that C is the center of the umbral shadow. Likewise M represents the center of the
moon at the instant of contact with the umbra, with P being the celestial pole.

figure 4

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If (1, 1) and (0, 0) represent the apparent equatorial coordinates of the center of the moon and C
respectively and we define CM as  and PCM as Q, we have by means of basic definitions of spherical
trigonometry that:

sinsinQ = cos 1sin (1 −  0 )

sin cos Q = sin1 cos  0 − cos 1sin 0 cos(1 −  0 )

Now, as  and (1 - 0) are small angles we can simplify these equations by adopting these
suppositions:
sin = 
sin ( 1 −  0 ) = 1 −  0
cos(1 −  0 ) = 1

Therefore:
sinQ = ( 1 −  0 ) cos 1
 cos Q = sin1 cos  0 − cos 1sin 0
 cos Q = sin (1 −  0 )
 cos Q = 1 −  0

So, with sufficient accuracy we have:

sinQ = (1 −  0 ) cos 1 (3)
 cos Q = 1 −  0 (4)

It should be easy to see that the coordinates of the center of the shadows differ by 180º exactly with
those of the sun, thus the right ascension of C is equal to that of the sun increased in 12 hours and its
declination has the same magnitude but contrary sign to that of the sun.

If the rectangular coordinates of the moon are defined as:

x = (1 −  0 ) cos 1 = sinQ (5)
y = 1 −  0 =  cos Q (6)

Then, if we choose a reference hour T0, for instance the closest integer hour to the time of opposition in
right ascension of the sun and the moon, with x0, y0 being the rectangular coordinates of the moon at
time T0, we have:

x = x 0 + x' t (7)
y = y0 + y' t (8)

Where x’, y’ are the hourly variations of the rectangular coordinates of the moon, found numerically
with the coordinates x, y calculated at one hour intervals, and t is the time measured since the reference
time T0.

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Merging equations 5 and 6 with equations 7 and 8, we have:

sinQ = x0 + x' t
(9)
 cos Q = y 0 + y ' t

Let us define the following quantities:

x0 = msinM
(10)
y 0 = m cos M

x' = nsinN
(11)
y ' = n cos N

The values m, M, n and N can be found directly with the numerical known values of x0, y0, x’ and y’.

With equations 9, 10 and 11 we have:

sinQ = msinM + ntsinN

(12)
 cos Q = m cos M + nt cos N

If we square and add equations 12, we find:

 2 sin 2 Q +  2 cos 2 Q =
m 2 sin 2 M + 2mntsinMsinN + n 2 t 2 sin 2 N + m 2 cos 2 M + 2mnt cos M cos N + n 2 t 2 cos 2 N

 2 (sin 2 Q + cos 2 Q) =
m 2 (sin 2 M + cos 2 M ) + 2mnt (sinMsinN + cos M cos N ) + n 2 t 2 (sin 2 N + cos 2 N )

 2 = m 2 + 2mn cos( M − N )t + n 2 t 2

Thus,

n 2 t 2 + 2mn cos(M − N )t + (m 2 − 2 ) = 0 (13)

Equation 13 is a quadratic expression on t that represents the time on which the moon is at a particular
position defined by its rectangular coordinates x and y. Thus, this equation allows us to define the
instants of the different phases of a lunar eclipse by defining certain conditions for the position of the
moon.

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The solution of equation 13 can be easily achieved by using the well-known formula for finding the
two roots of a second degree equation, like this:

− 2mn cos( M − N )  (4m 2 n 2 cos 2 ( M − N ) − 4n 2 (m 2 −  2 )

t=
2n 2
1
m cos( M − N )  m 2 cos 2 ( M − N ) − m 2 +  2 2
t=−   
n  n2 
1
m cos( M − N )  m 2 (cos 2 ( M − N ) − 1) +  2 2
t=−   
n  n2 
m cos( M − N )   2 − m 2 sen 2 ( M − N ) 
t=−   
n  n2 

If we define:

m cos( M − N )
A=−
n (14)
B = m sen ( M − N )
2 2

Then we finally have the solution for t as:

1
 2 − B  2
t = A   2
 (15)
 n 

Equation 15 allows us to find the times, with respect to the reference time T0, of all phases of the
eclipse, having in mind that the positive sign refers to the first contacts with the shadows and the
negative sign refer to the last ones.

According to our previous definitions we have the following values for , to be introduced in equation
15 in order to find the time of the phases:

Contacts with the penumbra:

 = 1.02(P + 0.99834P1 + S ) + S1 (16)

Contacts with the umbra:

 = 1.02(P + 0.99834P1 − S ) + S1 (17)

As customary, in both equations 16 and 17, we see that the moon´s semidiameter has been added to the
radius of the penumbra and umbra, to refer the times of contact to the moon´s limb and not to its center.

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For total lunar eclipses the value of  is  = 1.02(P + 0.99834P1 − S ) − S1 , which correspond to the
internal contacts of the moon´s limb with the umbra.

The time of maximum eclipse is defined to be the time at which the moon comes closest with the axis
of the shadows. This time of maximum eclipse is given by T0 + t’, where:

m cos(M − N )
t' = − =A (18)
n

However, as x and y represent the rectangular coordinates of the center of the moon with respect to the
center of the shadows, we can define d = x 2 + y 2 as the distance of the moon in a given moment. At
maximum eclipse is obvious that the distance d must attain its minimum numerical value, so we can
minimize this equation to obtain the corresponding time, like this:

d = x2 + y2

( ) (2 xx )
1
dd 1 2 −
= x + y2 2 '
+ 2 yy ' = 0
dt 2
xx ' + yy ' = 0

Introducing equations 7 and 8, we have:

( x0 + x ' t ) x ' + ( y 0 + y ' t ) y ' = 0
x0 x ' + x '2 t + y 0 y ' + y '2 t = 0
t ( x '2 + y '2 ) + ( x0 x ' + y 0 y ' ) = 0

Thus obtaining:

x0 x ' + y 0 y '
t=− (19)
x '2 + y '2

In equation 19, the variations of x and y are taken also for the reference time T0. This equation is
equivalent to equation 18, and both give us the time of maximum eclipse, or time of closest approach
between the center of the moon´s disc and the axis of the shadows of the penumbra and umbra, with
reference to time T0.

The magnitude of the eclipse is a quantity that help us to define the extent that the moon will enter into
the shadows. In a partial eclipse, the magnitude is the fraction of the moon´s diameter that is covered or
obscured by the umbral shadow at the time of maximum eclipse.

In a partial eclipse, the magnitude is given by:

 − m sen( M − N )
Mag p = (20)
2 S1

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Where msin(M - N) is to be taken as a positive quantity and  is the mean value used for the first and
last contact with the umbra.

Alternatively, the magnitude can be calculated with:

 − d + s'
Mag p = (21)
2s '

Where d is the distance at maximum eclipse,  is the umbral radius and s’ is the moon´s semidiameter.

Lastly, all that remains to be calculated are the position angles of the contacts of the moon with the
shadows of the penumbra and the umbra. These position angles are referred to the celestial pole, and
not to the local direction of the zenith. Again, referring to figure 4 we see that the drawing represents
the fourth contact (last contact with the umbra) and that the great circle that joins the point of contact of
the moon disc with the shadow of the umbra forms and angle PMC with the meridian PM through the
center of the moon. Because the position angles are measured from the celestial north toward celestial
east, the position angle of contact  is given by:

 = 360º − PMC
 = 180º +CMR

Because the angle CM is small, the angle PCM is approximately iqual to CMR, thus the position angles
is  = 180º +Q . Angle Q is determined with equations 12, inserting the appropiate value of t for every
contact.

Numerical application of the method: Our purpose is to apply the method described in this
article with the partial lunar eclipse of july 28 of 1999

The basic information needed for the calculation of any eclipse is an accurate ephemeris of the sun and
moon. We know that on july 28 will occur a lunar eclipse, so we calculate the following ephemeris:

Ephemeris for the sun and moon on July 28th 1999

Sun Moon
Time TDT   P S   P S
10:00:00 8h 29m 00.3s 19° 02' 18'' 0.00241 0.2625 20h 25m 21.7s -18° 25' 27'' 0.92793 0.2528
11:00:00 8h 29m 10.1s 19° 01' 44'' 0.00241 0.2625 20h 27m 33.0s -18° 20' 48'' 0.92826 0.2529
12:00:00 8h 29m 19.9s 19° 01' 09'' 0.00241 0.2625 20h 29m 44.4s -18° 16' 02'' 0.92855 0.2530
13:00:00 8h 29m 29.7s 19° 00' 34'' 0.00241 0.2625 20h 31m 55.7s -18° 11' 11'' 0.92893 0.2531
14:00:00 8h 29m 39.5s 19° 00' 00'' 0.00241 0.2625 20h 34m 07.0s -18° 06' 14'' 0.92927 0.2532
Table 1

The notation of the quantities is the same as described in the article.

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First of all, we must find the instant of opposition in right ascension between the sun and the moon,
instant at which the function is  + 12h - 1 = 0. From the tabular data is easy to see that this instant
will occur around 12:00:00 TDT. By using any of the well-known methods of inverse interpolation we
find this instant to be 11h 47 55s TDT.

At this time, we have the following apparent coordinates for the sun and the moon:

Sun Moon
  Dist. (A.U.)   Dist. (km.)
8h 29m 17.9s 19° 01' 16'' 1.015439 20h 29m 17.9s -18° 17' 00'' 393585

Note how the right ascension of the sun and the moon differ exactly by 12 hours. Also, at 11h 47 55s
TDT the geocentric latitude of the moon is +0º 42’ 57.0’’ well within the limits of the condition for an
umbral lunar eclipse to occur, as mentioned before in the article. Recalling our method, as the nearest
integer hour to the opposition is 12:00:00 TDT, we define this time as our reference hour T0.

The fundamental information for the calculation of the eclipse is described in table 2:

Shadow Moon Coordinates and variations

Time TDT  ()  ()  ()  () x y x' y'
10:00:00 307.25125 -19.03833 306.34042 -18.42417 -0.86414 +0.61416 +0.48013 +0.06806
11:00:00 307.29208 -19.02889 306.88750 -18.34667 -0.38401 +0.68222 +0.48095 +0.06973
12:00:00 307.33292 -19.01917 307.43500 -18.26722 +0.09694 +0.75195 +0.48101 +0.07110
13:00:00 307.37375 -19.00944 307.98208 -18.18639 +0.57794 +0.82305 +0.48147 +0.07306
14:00:00 307.41458 -19.00000 308.52917 -18.10389 +1.05941 +0.89611
Table 2

Note that all quantities, including the right ascensions, are now expressed in degrees (remember that 1
hour is equivalent to 15 degrees). Also, note that we have introduced the coordinates of the shadow as
the right ascension of the sun increased in 12 hours (or 180º) and the declination with the opposite sign
to that of the sun, just as described in the text of the article. The values of the rectangular coordinates of
the moon are calculated with equations 5 and 6 as follows:

At 10:00:00 TDT:
x = ( − )cos = (306.34042 – 307.25125)cos –18.42417º = -0.86414º
y = ( − ) = (-18.42417º - -19.03833º) = +0.61416º

The hourly variations of x and y are calculated as follows:

At 10:00:00 TDT:
x’= -0.38401 – (-0.86414) = +0.48013
y’= +0.68222 – (+0.61416) = +0.06806

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We proceed in a similar way to calculate the rest of table 2.

The reference coordinates (for our reference time T0=12:00:00 TDT) are:

x0 = 0.09694
y0 = 0.75195

We find with equations 10, the values M and m:

x 0.09694
tanM = 0 =
y 0 0.75195
M = 7.34595º
x0 0.09694
m= = = 0.75817º
sen M sen 7.34595º

From table 2, we have that the hourly variation at reference time T0 are:

x’0 = 0.48101
y’0 = 0.07110

With equations 11 we find N and n:

x' 0.48101
tanN = 0' =
y 0 0.07110
N = 81.59177º
x0' 0.48101
n= = = 0.48624º
sen N sen 81.59177º

Once we have found these basic values, by equations 14 and 15, we define our fundamental equation
like this:

m cos( M − N ) 0.75817º cos(7.34595º −81.59177º )

A=− =− = −0.423353
n 0.48624º
B = m 2 sen 2 ( M − N ) = 0.75817 2 sen 2 (7.34595º −81.59177º ) = +0.532447

Equation 15 finally is formed as:

1
 −B
2 2
t = A   2

 n 
1
  2 − 0.532447  2
t = −0.423353   
 0.23643 

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The values of , for the various events are:

Ingress/Egress of the penumbra:

 = 1.02(P + 0.99834P1 + S ) + S1 = 1.02(0.00241 + 0.99834 x 0.92855 + 0.2625) + 0.2530 = 1.46876º

Ingress/Egress of the umbra:

 = 1.02(P + 0.99834P1 − S ) + S1 = 1.02(0.00241 + 0.99834 x 0.92855 − 0.2625) + 0.2530 = 0.93326º

Note how the values P, P1, S and S1 are taken from table 1 for the reference time T0

With our fundamental equation formed and the values of  properly determined, we are ready to
calculates the times of the differente phases of our eclipse, with the following comments:

• We use the negative sign of our fundamental equation for the ingress events, and the positive sign
for the egress events.

• The time for each instant is found as T=T0 + t

• The times thus found will be expressed in Terrestrial Dynamical Time. If necessary they can be
converted to Universal Time or Local Time. Recalling that T = TDT – UT, and as T will be
about 64 seconds, we have that UT = TDT – 64 seconds.

The different times for the events/phases are:

Event  t Time TDT Time UT Time UT Time EDT

Ingresss to penumbra 1.46876 -3.0449 8.9551 8.9373 8h 56m 3h 56m
Ingress to the umbra 0.93326 -1.6199 10.3801 10.3623 10h 22m 6h 22m
Maximum eclipse -0.4234 11.5766 11.5588 11h 34m 7h 34m
Egress from the umbra 0.93326 0.7732 12.7732 12.7554 12h 45m 8h 45m
Egress from the penumbra 1.46876 2.1981 14.1981 14.1803 14h 11m 10h 11m
Table 3

For instance, the ingress to the penumbra is calculated as follows:

=1.46876º
1 1
  2 − 0.532447  2  1.46876 2 − 0.532447  2
t = −0.423353    = −0.423353 −   = −3.0449h
 0.23643   0.23643 
T = T0 + t = 12h – 3.0449h = 8.9551h TDT
Universal Time = 8.9551h – 64/3600h = 8.9373h UT = 8h 56m UT
Eastern Daylight Time = 8h 56m UT –5h +1h = 4h 56m

The time of maximum eclipse can be calculated by equation 18, like this:

m cos(M − N )
t' = − = A = −0.423353
n

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Time of maximum eclipse = 12h – 0.4234h = 11.5766 TDT

Also, by means of equation 19, and using the rectangular coordinates and their variations taken form
table 2, we find:

x0 x ' + y 0 y ' 0.09694 x 0.48101 + 0.75195 x 0.07110

t ' = − '2 =− = −0.423358
x +y '2
0.481012 + 0.07110 2

Which is basically the same instant obtained before with equation 18.

The magnitude of the eclipse is found by equation 20, like this:

 − msin ( M − N )
Mag p =
2 S1
0.93326 − 0.75817sin (7.34595º −81.59177º )
Mag p = = 0.402
2 x 0.2530

The value of the magnitude of 0.402 means that 40.2% of the moon´s diameter enters into the shadow
of the umbra, thus being a partial eclipse. We leave the exercise of calculating the position angles of
contact with the shadows to the readers.

With all the calculated information, and with the position angles we can make the following diagram
depicting all the phases of our eclipse:

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Finally, just how accurate are our results?. As our ephemeris of the Sun the Moon was an accurate one,
we should expect the results to agree with a precise calculation. Our reference will be NASA´s
reference publication 1216, “Fifty year Canon of lunar eclipses 1986-2035” written by Fred Espenak, a
world renowned authority in the calculation of eclipses. His book, on page 111 gives the following
information for the eclipse in question:

Times of contact:
P1= 8h 56.1m UT (ingress to the penumbra)
U1= 10h 21.9m UT (ingress to the umbra
Maximum eclipse: 11h 33.7m
U4= 12h 45.4m UT (egress from the umbra)
P4= 14h 11.1m UT (egress form the penumbra)

Magnitude in the umbra: 0.4016

So, we can see that our results perfectly agree with those of the the reference.

References
1. William Smart (1977), Textbook on spherical Astronomy (sixth edition), Cambridge University
Press
2. Her Majesty´s Nautical Almanac Office (1961) – reprint 1974, Explanatory Supplement to the
Astronomical Ephemeris and the American Ephemeris and nautical Almanac. (H.M. Stationary
Office, London)
3. Fred Espenak(1989), Fifty Year Canon of Lunar Eclipses 1986 –2035, NASA reference publication
1216
4. Andrés Mejía (1998), Efemérides Astronómicas 1999, (Sociedad Julio Garavito para el estudio de
la Astronomía, Medellín, Colombia)

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