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I'm currently building a web app and am getting the RA's and Dec's of solar system objects based on date and time chosen by the user. Then using the gmst for 0 hours of the date, the app calculates rise and set time of the body. Now when observers time changes the rise and set time also changes which should not be. I'm looking for a way to get the rate of change of RA and Dec of these bodies so that I can always use RA and Dec of a specific time (say 0 hours).

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To a rough approximation: "not much".

The sun and the moon increase their ecliptic longitude by 360 degrees per year and month respectively. That is 0.041 degrees and 0.55 degrees per hour. That could cause an error of a couple of minutes in your sunrise calculator and tens of minutes in your moonrise calculator.

Since the ecliptic is tilted by only 23 degrees to the equator, the RA increases by about the same amount per hour. Their declension won't vary by much, but sometimes it can be increasing and sometimes decreasing (as ecliptic crosses from northern to southern hemispheres)

The other planets have more complex motions as a result of the combination of their orbital motion and the Earth's. Their ecliptic longitude varies by values on the order of 0.04 degrees or less, but it might be either positive or negative. Planets can appear to move retrograde. Venus and particularly Mercury move quickly. The outer planets move much more slowly.

If you want accuracy, you need to do a proper ephemeris at the time chosen.

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