The moon sometimes looks bigger. Is it closer to us?
First calculate the perigee and apogee times, and then compare the corresponding distances.
The distance from the moon to the earth will change. But "perigee" is not equal to "full moon", and not every perigee is equally close. The original book calculates the time of extreme distance and the horizontal parallax respectively, and then converts the parallax to the distance from the center of the earth.
Guess first
When the moon reaches perigee, is it necessarily a full moon?
Show answer and explanation
uncertain — The near-far cycle is different from the moon phase cycle. The smallest distance does not guarantee that the half of the sun's illuminated side is facing us.
Then connect experience and science
Distinguish between near and far and roundness and lack
Changes in distance affect the apparent size of the moon's disk; the distance between the disk and the moon is mainly determined by the direction of illumination from the sun, moon and earth, and their periods are not the same.
Perigee is not equal to the full moon
Near and far locations each use their own corrections.
The average periapsis month gives a rough date. Perigee and apogee use different periodic terms and cannot be replaced by a single sine wave.
Table 50.A: 60 items for perigee and 32 items for apogee
Then calculate the horizontal parallax at that moment
Table 50.B gives equatorial horizontal parallax. The greater the parallax, the closer the moon is to the earth.
sin π = Earth’s equatorial radius / Earth’s center distance
Different rounds will be different
Switching between the front and back rounds, you can see that the distance and interval are not fixed. This is a period approximation, not a modern numerical ephemeris.
Time correction and distance calculation are verified separately
Perigee and apogee use different time correction tables; horizontal parallax uses its own cosine table. Finally, divide the earth's equatorial radius by sin π to find the distance; the arc seconds of π must first be converted into the angle used in the trigonometric function.
give it a try
Compare the perigee and apogee of the same year; switch the next time and see if the distances and intervals are exactly the same. The disk only compares the relative size, which does not mean that the moon will be full on that day.
No need to memorize symbols
- k
- Perigee integer, apogee half-integer number
- π
- Equatorial horizontal parallax, the interface displays angle minutes
- Δ
- Distance from the center of the earth in kilometers
- JDE / TT
- Dynamics consecutive days/time scale
Algorithm, accuracy and scope of application
Achieve 50.1 average time, 60 near-Earth/32 far-Earth time terms of Table 50.A, 46 near-Earth/17 far-Earth parallax terms of Table 50.B, and retain T correction. After π is rotated, the distance to the center of the earth is obtained by 6378.14/sinπ; it is not estimated directly from the pixel length of the image. Scanned table 50.A (printed 359/PDF367) has been returned to check the 32 apogee coefficients item by item, and independently re-enter the sum: k=−148.5 in example 50.a is corrected to −0.4648325814 days, JDE2447 442.354300357, consistent with AA+; the example of printing 357 pages/PDF365 is revised to −0.4654, JDE2447442.3537, the difference between the two is 51.87 seconds, which is not regarded as consistent by relaxing the tolerance. The page retains the table formula results; the AA+ file history annotation also records the same 2447442.3543003569 value. The remaining intermediate angles D/M/F are consistent with the examples in the original book, and the parallax is consistent with the rounding digits in the book. Displays TT/JDE, does not automatically change the local calendar, and does not include moon phases or moonrise.
my understanding check
What is the normal apparent size of the Moon of the same actual size when it is closer to us?
bigger — The same objects are closer together and the viewing angle is wider.
Is the distance from the center of the earth exactly the same for every perigee?
no the same — Factors such as the sun produce periodic disturbances, and all perigees cannot be predicted using just one fixed ellipse.
If we write the perigee time as TT, will we also get the local moonrise time?
no — Perigee is the extreme value of distance, and moonrise also depends on the location of observation, the Earth's rotation, and horizon conditions.
The moon's distance, waxing and waning, and local rising and setting are different issues; first determine which event you are calculating.
Jean Meeus · Astronomical Algorithms, 2nd ed. · No. 50 chapter