Astrophysics
Orbit Events
There are a number of events that occur in the orbits of the Earth, Moon, and other planets. Here we define the events and define what the event is.
Orbital Periods
Defining a unique number to the period of the orbit of a body is complicated by the fact that there are several different definitions of orbital periods.
Calendar years are used to define a particular orbital period. The calendar new year doesn’t correspond to any astronomical events. For event calculations a good choice of the orbit is the anomalistic period. This starts at perihelion that currently occurs in the first week of January for Earth.
Moon orbits are assigned lunation numbers. The Brown Lunation Number defines lunation 1 to be the first New Moon of 1923 which occured on 17 January. Again, we will define the start of a lunar orbit as perigee. The BLN number will be associated with New Moons.
Astronomical events are associated with a body such as the Earth or the Moon. Some events, such as eclipses, require the positions of several bodies. The concept of an orbits is useful for calculations. An orbit will be defined as the interval betwenn two consecutive periapses. A Julian day is in a particular orbit if it is greater than or equal to the Julian day of the start periapis and stricly less than the Julian day of the next periapsis.
Apses
The apses of an orbit are the closest and furthest points in an orbit. The closest point is perihelion for planets and perigee for the Moon. The furthest is Aphelion for planets and apogee for the Moon.
The times of these events has to be determined by iteration to find the closest and furthest distances.
In the case of the Earth, the times of the apses vary by several days from year to year. The reason for this is that it is the Earth-Moon Barycentre (EMB), not the Earth, that orbits the Sun. When the EMB is at perihelion, the Moon can be closer to the Sun than the Earth. In this case the Earth will be at perihelion a few days later.
Apsis Determination
Given a Julian day number \(j\) we need to find the Julian day number \(j_p\) of the periapsis before it. This must satisfy the condition \(min(j_p - j) \ge 0\). The function \(\rho(j)\) returns the radius of the body from the body it orbits.
Search for the periapsis day interval.
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Set \(j_m = j\) the Julian day number.
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Set \(j_s = j_m - 0.5\) half a day before.
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Set \(j_e = j_m + 0.5\) half a day after.
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Calculate the radii \(\rho_s = \rho(j_s)\), \(\rho_m = \rho(j_m)\), and \(\rho_e = \rho(j_e)\).
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if \(\rho_s > \rho_m < \rho_e\) exit as we have an interval containing the periapsis.
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Set \(j_e = j_m\), \(j_m = j_s\), \(j_s = j_s - 1\).
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Got to step 4.
We now know that a periapsis occurs in the interval \((j_s, j_e)\). Perform a golden section search on the using the radius function to find the periapsis Julian day number \(j_p\).
There is one case where this returns the wrong result. That if is the day interval contains the Julian day number and the periapsis and the periapsis is after the day number \(j < j_p\). In this case we need to find the previous periapsis. Repeat the process using \(j = j - 1\).
Solstices and Equinoxes
The times and dates of the equinoxes and solstices are when the ecliptic longitude of the Sun from the Earth is a multiple of \(90^\circ\).
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Vernal Equinox \(\lambda = 0^\circ\).
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June Solstice \(\lambda = 90^\circ\).
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September Equinox \(\lambda = 180^\circ\).
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December Solstice \(\lambda = 270^\circ\).