So, How to Build an Astrolabe: Mater – Back Plate

This week, build an astrolabe’s various calendars engraved on the back of the astrolabe

Mater: Back Plate

The back of the astrolabe is where the core of the measurements are made. By measuring the position of a known star, the back of the astrolabe can be used to determine not only time, but dates. Each mark is engraved on the back of the mater—the main body of the astrolabe.

The back of the mater is a host to any number of useful measurement circles. Among the most important are ways to determine the altitude, equal hours, and calendars. The ordering of each circle is irrelevant, but for ease of use, the outermost rings are typically for measuring altitude.

Altitude (10°):

For the purpose of this astrolabe, the outermost ring splits the astrolabe into 360°, with tick marks for each 10°. This is one of the simplest rings, but is very useful in determining the rough position of stars in the night sky or the Sun.

360°:

For more fine-grain observations, it is useful to also include a reference which splits the circle into 1 degree segments, 1 for each of the 360 degrees.

Equal Hours:

Like the front limb, equal hours split a circle into 24 equal parts, each 15° wide. The equal hours are numbered 1-24 (or two sets of 1-12).

Zodiac Calendar:

The zodiac houses are a type of astronomical calendar. While typically associated with astrology, the zodiac divides the ecliptic—the apparent path that the Sun takes across the sky during the year—into 12 equal parts. Each segment is named for a prominent constellation visible at that time of the year.

Each “sign” is 30° wide, and the calendar begins with Aries during the Vernal Equinox at the beginning of spring (in the northern hemisphere). This is aligned along the horizontal at 0°.

Spherical projection projects from the south pole, so the signs are read in a reverse—or, counter-clockwise—order.

Gregorian Calendar and Days:

There is a vast array of calendars used by different cultures and religions throughout history, but the Gregorian calendar is the most widely used civil calendar. The Gregorian calendar replaced the Julian calendar in 1582 at the order of Pope Gregory XIII (which it is humbly named after).

The Gregorian calendar can be constructed with two different methods: either as an eccentric or concentric calendar. A concentric calendar is more difficult, as the spacing between each of the days varies based on the Sun’s rate of longitude change throughout the year. Meanwhile, the eccentric calendar assumes the Earth as the center of the astrolabe and the Sun moving around it along the ring at a constant speed, so the length of the days, and by extension, the width of the segments will be equal.

Because the calendar depends on astronomical dates, the calendar will represent a specific year. However, the resulting error in using it for other years is minor, and, for the sake of simplicity, astrolabes typically omit leap years, so the calendar is 365 days long.

The eccentric calendar is the most common form used on astrolabes. An eccentric calendar will be mostly aligned with the zodiac calendar. However, because Aries represents the beginning of the Vernal Equinox, which can run anywhere from March 19th to March 21st we first need to determine where to place the first day of the year: January 1st.

An eccentric calendar assumes that the Sun moves at a constant speed through the year, however, the specific day of an astronomical event like the equinox or solstice will not always land on the same day in the Gregorian calendar. This is due to the Sun’s true anomaly.

To calculate the true anomaly, first, we need to understand the Earth’s eccentricity. Orbital eccentricity is how much the orbit of Earth deviates from a perfect circle. The eccentricity of a perfectly circular orbit is 0, while for a fully elliptical (parabolic) orbit it is 1. Earth’s orbit is a bit elliptical, its eccentricity is about 0.0167. However, it does change over time. This difference can be calculated for a specific year. First, we will calculate the time in reference to the year 2000.

T = (year - 2000) / 100

So, if the current year is 1969 then:

T = -0.31

Which can be used to roughly calculate the eccentricity for a given year:

Eccentricity = 0.01670862 - (0.000042037 * T) - (0.0000001236 * (T2)) + (0.00000000004 * (T3))

Eccentricity = 0.016722

The change over time is small, and, thankfully, and over time the effect will be minimal.

The aphelion is when the Earth is furthest from the Sun and the perihelion is when the Earth is closest to the Sun. So, the position of the aphelion and perihelion will determine when the astronomical year begins and ends.

Perihelion = 102.937348 + (1.7195269 * T) + (0.00045962 * (T2)) + (0.000000499 * (T3))

Aphelion = Perihelion + 180°

So, for the year 1969:

Perihelion = 102.404339°

Aphelion = 282.404339°

However, because the center of the astrolabe represents the Earth, the calendar is based on a geocentric view of the solar system.

So, the position (longitude) of the aphelion and perihelion would be 180° from the heliocentric longitude of Earth, so the actual values are flipped:

Aphelion = 102.404339°

Perihelion = 282.404339°

The angles are measured from the Vernal Equinox (between Aries and Pisces) at 0°.

Because the Earth’s orbit is not a perfect circle, the reference point to measure angles on the calendar will be some offset point off center from the center of the astrolabe.

offset = 2 *  eccentricity * radius_of_plate
x_delta = offset * cos(perihelion)
y_delta = offset * sin(perihelion)

To simplify the math, we can assume the radius of the astrolabe is 1.

offset = 0.033443
X offset = -0.007184
Y offset  = 0.032663

The offset will also change over time.

The line of apsides is the line connecting the two points—aphelion and perihelion—together which will cross the new offset.

The mean anomaly (M₀) on the first day of January begins at midnight on December 31, which can be called January 0. This represents the position of the Sun in its elliptical orbit on January 0.

Because the position of the perihelion is astronomical and depends on the Earth’s orbit, it does not always fall on the same day in the Gregorian calendar. Instead, it can occur anywhere in the first week of January. The mean anomaly will determine how far apart the perihelion and January 0 are.

M0 = 357.52910 + (35999.0503 * T) - (0.0001559 * (T2)) - (0.00000048 * (T3))

So, the mean anomaly will fall somewhere within a few degrees of 360°. For 1969, the mean anomaly is:

M0 = 357.823492°

So, about three degrees away, the perihelion falls on roughly January 3, 1969. So, the beginning of the month will begin 3 days (~3 degrees) before the perihelion. This will mark the beginning of January.

365 Days (Every Five Days):

It takes the Earth roughly 365.25 days to completely orbit the sun. On calendars, this is rounded to 365 days with a leap day every four years to account for the remainder. One way to view this is with evenly spaced marks every five days—each 4.93° wide, beginning on January 1st.

365 Days (Every Day):

For greater resolution, the individual days are marked within the five day segments, each roughly a degree or 0.9863° (where 360/365) apart.

Assuming the year is 365 days long, then the following months can be found by counting the day marks for each subsequent month, in counter-clockwise order.

This will align the months of the year in the Gregorian and zodiac calendar

Each 90° mark will represent the rough position of the equinox and solstices.

For those in a hurry, some of these values can be assumed. So, in a trade-off for fewer steps and reduced accuracy, you can assume that there is no offset of the astrolabe. Often, the perihelion is on January 3rd and the Vernal Equinox is on March 20th, so the starting position of January 0 is ~357°.