Algorithm series (19) Using astronomical method to calculate Sun Moon (Crescent)

The synodic of the Chinese lunar calendar is the basis of lunar calendars, and synodic is strictly to the sun and Moon, the day of the beginning of the month as the first, so the moon and sun time calculation is the key to make the lunar calendar. This paper will introduce the theory of ELP-2000/82 Moon operation and how to calculate the time of sun and moon with Elp-2000/82 Moon Operation theory.

To calculate the moon and sun time, first of all to the sun and moon, the astronomical phenomenon of the mathematical definition. Synodic is the lunar phase period observed on Earth, the average length is about 29.53059 days, and the Star Moon (astronomical month) is the moon around the earth a week of time, length of about 27.32166 days. The lunar phase cycle is about two days longer than the star's, because the moon rotates around the earth for a week while the Earth rotates around the sun for a certain angle, so the lunar phase cycle is related not only to the moon's operation but also to the Sun's operation. The Sun, Moon and earth three close to a straight line, the moon is not illuminated the face of the earth, so the earth can not see the moon, at this time is called the New Moon. Fig. (1) is the schematic diagram of the astronomical phenomenon of moon and sun.

Figure (1) schematic diagram of the astronomical phenomena of the sun and moon

The moon around the Sun White road surface and the earth around the sun's ecliptic surface has a maximum of about 5 ° angle, so in most cases, the sun and moon when not strictly in the same line, but will also occur in the same line of the situation, this time will occur solar eclipse. Fig. (1-b) shows the three possible positions of the moon on the side plane of the sun and Moon, when the moon is in position 2 o'clock the eclipse occurs. by figure (1), the mathematical definition of sun and Moon is the time when the Earth's core is 0.

To calculate the sun and moon, we need to know how to calculate the yellow meridian of the Earth's core and the Earth's geocentric view. The third chapter of the Calendar Generation algorithm series, "using astronomical methods to compute Shiber", has introduced how to calculate the vsop82/87 of the sun by using the theory of the planet of the Earth, and this paper will continue to describe how to calculate the yellow Meridian of the moon with the elp-2000/82 Moon theory. ELP-2000/82 Moon theory is M. Chapront-touze and J. Chapront, a semi-analytical theory of the moon's position, presented in 1983, like other semi-analytical theories, the elp-2000/82 theory also contains a set of computational methods and corresponding iterative cycle terms. The theory consists of 37,862 periodic terms, 20,560 of which are used to calculate the longitude of the moon, 7,684 to calculate the latitude of the moon, and 9,618 to calculate the Earth-moon distance. However, many of these periodic items are very small values, such as some of the calculation of latitude and longitude of the results of the gain only 0.00001 corner of the second, and some of the monthly distance cycle of the results of the gain of the distance is only 0.02 meters, for the precision of the calendar calculation, can be ignored.

There are many improvements or simplifications based on the theory of Elp-2000/82 Moon, the 45th chapter of the astronomical algorithm introduces an improved algorithm, whose periodic term parameters are converted from the periodic term parameters of the elp-2000/82 theory, ignoring the small periodic term. The accuracy of the lunar yellow Meridian calculated by this method is only 10 ", the Moon Huang Wei Precision is only 4", but only 60 cycle terms are calculated, the speed is very fast, this paper uses this modified elp-2000/82 theory to calculate the Earth's core of the moon. The periodic term of this method is divided into three parts, which are respectively used to calculate the lunar Huang Wei, the lunar distance and the Earth-moon distances, and the contents of the three-part cycle term, which consist of four calculation of the angular coefficients and a sine (or cosine) amplitude. Calculate lunar Huangching and Earth-moon distances using sine expression summation: a * sin (θ), calculates the lunar Huang Wei with cosine expression summation: A * cos (θ), where the calculation formula for the angular θ is:

θ= A * D + b * m + c * m ' + d * F (4.1-type)

4.1-type four-angular coefficient A, B, C and D are given by each iteration cycle term, the sun and moon angle D, the solar flat near ground angle m, the moon flat near the ground angle M ' and the lunar nodal point distance f are calculated by 4.2-4.5 type respectively:

D = 297.8502042 + 445267.1115168 * T-0.0016300 * T2

+ t3/545868-t4/113065000 (4.2-type)
M = 357.5291092 + 35999.0502909 * T-0.0001536 * T2

+ t3/24490000 (4.3-type)
M ' = 134.9634114 + 477198.8676313 * T + 0.0089970 * T2

+ t3/69699-t4/14712000 (4.4-type)
F = 93.2720993 + 483202.0175273 * T-0.0034029 * T2

-t3/3526000 + t4/863310000 (4.5-type)

The above results are measured in degrees, where T is the number of centuries, and T is calculated by the 4.6 formula:

T = (JDE-2451545.0)/36525.0 (4.6-type)

Taking the calculation of the second term of the cycle of the lunar Yellow Sutra as an example, the second data is as follows: the angular coefficient a = 2,b = 0,c = -1,d = 0, amplitude a = 1274027, and yellow is computed with a sine expression, the I2 calculation is as follows.

I2 = 1274027 * sin (2d–m ') (4.7-type)