Orion’s Waistline

One night, while gazing at Orion’s Belt, I wondered what if it was actually a literal belt? Three stars make up Orion’s Belt. They’re definitely not collinear. These stars form the vertices of a gigantic triangle. With some imagination, the perimeter of this triangle is analogous to Orion’s stellar waistline. So how do I calculate it?

First, let’s get acquainted with the stars that make up Orion’s Belt.

The left star in Orion’s Belt is Alnitak, also known as Zeta Orionis. It was named by the Arabs. One translation is “string of pearls”. Different cultures saw those three stars very differently. Some see it as a belt, others see it as jewelry. Alnitak is actually a triple star system where one star is orbiting a binary star system. Talk about third wheels.

The center star in Orion’s Belt is Alnilam, also known as Epsilon Orionis. Its name continues the string of pearls theme. That’s much more romantic than just a belt. This one is a supergiant 500,000 times brighter than the Sun, bound to flame out sooner than the flanking stars.

The right star in Orion’s Belt is Mintaka, also known as Delta Orionis. This time, the star is named after the Arabic word for belt. It is a double star with an orbital period of 5.73 days. They must be dizzy.

So, how far apart are these stars?

Each and every star perceptible to our eyes has been catalogued. Two measurements that help us find the stars in the sky are right ascension and declination, similar to Earth’s longitude and latitude. Right ascension (RA) is the horizontal angle away from the Sun’s location on the March equinox. It is usually measured in hours, minutes, and seconds. Declination (DEC) is the vertical angle away from the celestial equator. These two measurements allow us to pinpoint where stars shine. However, they say nothing of how far away the stars are.

The third measurement, distance, came much later with the advent of radio technology.

These three measurements form the basis of spherical coordinates. With some trigonometry, spherical coordinates are converted to Cartesian coordinates. That is, a star defined by (right ascension, declination, distance) can be transformed into (x, y, z). One important thing to note is the units of right ascension. It’s measured in hours, minutes, and seconds. This is from the fact that right ascension is measured in the time it takes Earth to rotate from the reference (the March equinox). We’ve got 24 hours in a day. 360 degrees divided by 24 is 15 degrees. So one hour of right ascension is equal to 15 degrees. 60 minutes in a hour means a minute is 15/60 = 0.25 degrees. Similarly, a second is 0.25/60 = 0.00416… degrees.

As for declination, it’s measured in degrees, arcminutes, and arcseconds. An arcminute is 1/60th of a degree, and an arcsecond is 1/60th of an arcminute. With these conversions in mind, here’s the above table adjusted for plain old degrees:

Now we just need to transform these spherical coordinates to Cartesian coordinates. We apply the following equations where r is distance, \phi is the horizontal angle, and \theta is the vertical angle:

\begin{aligned}
x&=r\sin{\phi}\cos{\theta}\\
y&=r\sin{\phi}\sin{\theta}\\
z&=r\cos{\phi}
\end{aligned}

Plugging and chugging gives us this table of the stars’ Cartesian coordinates:

Now that we have the 3D Cartesian coordinates of Alnitak, Alnilam, and Mintaka, we apply the 3D Distance Formula (shown below) for each pair of stars.

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

Alnitak and Alnilam are 741 light years apart. Alnilam and Mintaka are 801 light years apart. Mintaka and Alnitak are 84 light years apart. Adding these up gives us the perimeter: 1626 light years.

That’s one huge belt. Orion probably shops at the Big & Tall.