# Midpoint Declinations

May 2014

In my work as a software engineer working on astrology programs, I recently looked into the issue of Midpoint Declinations. A customer disagreed with our program's calculation for the midpoint of Jupiter and Pluto. On March 1, 2014, for example, Jupiter had a declination of about 23°, while Pluto had a declination of -20°. Our software gave the midpoint declination as 6°. The customer asked, shouldn't it simply be (23 + -20)/2, which would be 1.5°? To confuse things further, after hanging around in the 6° region for a while, in late April the midpoint declination suddenly shifted to -4°. What's going on here?

The trouble is in how Declination is defined; it's the distance above or below the celestial equator. The celestial equator is very different from the plane of the Zodiac, because of the tilt of the Earth's axis. These two circles in the sky meet in two points, at zero Aries and zero Libra, but are 23 degrees apart around 0 Capricorn and 0 Cancer. This is why the declination varies wildly as planets move around the zodiac, even while their Latitudes, the distance above or below the zodiac itself, remain very small.

So if we were looking for the "midpoint of the declinations", we might say that the declinations are two numbers, and we would simply want to average them. But if we're looking for the "declination of the Midpoint", we need to first ask where the midpoint lies on the celestial sphere, and then get the declination of that point.

Since the planets all hew close to the plane of the Zodiac, it's reasonable to average their longitude and latitude to get the midpoint. (There's a more subtle definition based on the Great Circle connecting the two, but the two definitions produce very similar results.) In our example of March 1, 2014, we get Jupiter with a longitude of 100° 29' and a latitude of 0° 14', while Pluto has longitude 283° 05' and latitude 2° 41'. This gives a Midpoint at Longitude 11° 47', Latitude 1° 27'. (We flip the longitude by 180 degrees to put it in Aries because Pluto and Jupiter are more than 180 degrees apart.)

What's the declination of this point? Declination is generally given from Longitude, Latitude, and the Earth's axial tilt angle by the formula

Declination = Arcsin(cos(tilt)*sin(Lat) + sin(Tilt)*cos(Lat)*sin(Long)

This gives us a declination for the midpoint of right about 6°.

But any formula should be checked against intuition - does it make sense that the midpoint would have this declination? Imagine the night sky, and the plane of the Zodiac, where all the planets move. Jupiter and Pluto are nearly at opposition, but we can imagine Jupiter nearly setting as Pluto has just risen in the east. Follow the zodiac along the sky from one to the other; the midpoint is right up there at 11° 47' Aries. The declination of the zodiac itself at this degree of aries is about 4° 40'. Add on a latitude of about 1° 27', and we're pretty close to 6°. Note that this is very different from the straight average of the two declinations, which again would be about 1.5°.

So what happens in late April? On April 21, Jupiter reaches opposition with Pluto, so the midpoint between them flips around to the other side of the sky, from mid-Aries to mid-Libra. On April 25, for instance, the Jupiter/Pluto midpoint is at 13° 51' Libra, or a longitude of 193.85°, while the latitude has barely changed at 1° 28'. The zodiac has a declination of -5° 28' at this latitude; the declination of the midpoint itself is

Arcsin(cos(23.4375)*sin(1.467) + sin(23.4375)*cos(1.467)*sin(193.85)) = -4° 07'.

Of course, all of this can be done by hand and calculator, but this is one of the reasons why software is so useful for astrology. I've been working with Solar Fire for years, and highly recommend it. Here are sites to learn more: