M OON,

2002

D22

NOTES AND FORMULAE

Use of the polynomial coe cients for the lunar coordinates

On pages D23 D45 for each day of the year, the apparent right ascension and declination of

the Moon are represented by economised polynomials of the fth degree, and the horizontal parallax

is represented by an economised polynomial of the fourth degree. The formulae to be evaluated are of the

form:

a

0

+

a

1

p

+

a

2

p

2

+

a

3

p

3

+

a

4

p

4

+

a

5

p

5

where

a

5

is zero for the parallax.

The time-interval from 0

h

TT is expressed as a fraction of a day to form the interpolation factor

p

, where

0

p

1, and the polynomial is evaluated directly, or by re-expressing it in the nested form:

a

5

p

+

a

4

p

+

a

3

p

+

a

2

p

+

a

1

p

+

a

0

to avoid the separate formation of the powers of

p

. Alternatively this nested form for and may be written

as:

b

n

+1

=

b

n

p

+

a

5

,n

;

for

n

= 1 to 5

;

where

b

1

=

a

5

and

b

6

is the required value. For the parallax

a

5

is zero, so that:

b

n

+1

=

b

n

p

+

a

4

,n

;

for

n

= 1 to 4

;

where

b

1

=

a

4

and

b

5

is the required value.

The polynomial coe cients are expressed in decimals of a degree, even for , and the signs are given on

the right-hand sides of the coe cients to facilitate their use with small calculators. Subtract 360 from if

it exceeds 360 . In order to obtain the full precision of the polynomial ephemeris the interpolating factor

p

must be evaluated to 8 decimal places 10

,

3

s; estimates of the precision of unrounded interpolated values

are:

RA

Dec

HP

0

s

.0003

0

00

. 003

0

00

. 0003

Particular care must be taken to ensure that the coe cients are entered with the correct signs.

Example

. To calculate the apparent right ascension the declination and the horizontal parallax

for

the Moon on 2002 January 21

d

13

h

23

m

48

s

.32 UT, using an assumed value of

T

= 67

s

.

TDT = 13

h

24

m

55

s

.32, hence

p

= 0

558 973 61

right ascension

declination

horizontal parallax

b

1

,

0

000 1458

,

0

000 1624

,

0

000 009 43

b

2

+0

000 2716

,

0

001 0980

,

0

000 012 74

b

3

+0

040 5245

,

0

026 3080

+0

001 420 64

b

4

+0

178 2591

,

0

059 0432

+0

007 976 83

b

5

+11

102 2159

+4

827 3134

= +0

914 899 82

b

6

= 28

799 4888

= +7

127 7010

= 1

h

55

m

11

s

.877

= +7 07

0

39

00

. 72

= 54

0

53

00

. 639

Moon Polynomial Coe cients

Supplement to The Astronomical Almanac 2002

c HMNAO