# Measurement unit of coordinate systems in astronomy

Nearly all of the coordinate system I have studied in Positional astronomy use the degree system. This makes me curious about why astronomers prefer degrees over radians as we know that Radian is a necessary intrinsic measure of an angle, which is related to the "fact of nature" that the length of a unit circle is 2π.

• I mean we’re talking about people who still measure brightness in a 2300 years old number system.
– pela
Commented Aug 18, 2022 at 20:22
• Except for measuring longitudinal angles in equatorial coordinates, for which one obviously uses "hours" to measure the angle. Commented Aug 18, 2022 at 20:31
• Except when we measure times in radians... UT0 and Earth Rotation Angle Commented Aug 18, 2022 at 23:08
• I would like to know where you routinely encounter the use of radians. The trig functions in programs (Excel, Visual Basic, etc) are the only place that I know where radians are used. Everywhere else that I am familiar with (engineering, physics, drafting, protractors) use degrees. Why would astronomy be any different? Commented Aug 19, 2022 at 16:01
• NASA's ephermedes measure length in astronomical units. We are a strange people. Commented Aug 19, 2022 at 16:55

In science, there are cases in which some weird units are used for historical reasons only and we would all be better off without them. One could argue that the magnitude systems is one of those cases, but I don't think that the use of degrees versus radians falls in this category.

Both radians and degrees have their own advantages and disadvantages, and are most suited for different purposes.

One reason to use radians to measure angles is that the derivative and integrals of trigonometric functions assume the simplest form

$$\sin' x = \cos x$$ $$\int\sin x dx= -\cos x$$

Also, Euler's formula

$$e^{ix} = \cos x +i\sin x$$

is only valid if $$x$$ is measured in radians. As a consequence, radians will be the preferred unit in all the cases where these properties are important, so when dealing with Fourier transforms, partial differential equations (wave equation, heat equation, Schrödinger equation...), rotations, path integrals, so basically Mathematics and Theoretical Physics.

The degree system is instead much more intuitive to visualize, because 360 is a superior highly composite number, which means that it has a lot of divisors for being so small.

How much is 2.1293 radians? Difficult to say, but if I say 122 degrees, it becomes trivial: 120 is one third of 360, so 122 is just a little more than one third. The advantage of having many divisors is that most angles will be close to one of them. In positional astronomy one does not need all the fancy properties of the complex numbers, one just want to locate a star, and the degree system is great at doing just that. For the same reason, degrees are also used in engineering and other fields that deal with actual measurements and with problems that have a proper scale.

How many theoretical physicists, while solving a complex integral, need to wonder how large is an angle of 0.2618 radians? I would argue very few of them.

Conversely, an engineer may well need to consider whether a rotation of 15 degrees is sufficient for its robot arm to accomplish its task. And may also need to know that after 6 rotations it will have reached a right angle, the maximum rotation of the joint.

In the end all boils down to using the unit that is more convenient, not the one is closer to reflect a "fact of nature". Otherwise we would all be using Plank units, wouldn't we?

Have a nice $$4.658 \times 10^{49}$$ Plank time ;P

• use of $\pi$ or better $\tau=2\pi$=turn helps somewhat. It is almost as easy to say "a quarter turn" as "ninety degrees" So how much is $0.34\tau$ ah just a little more than 1/3 of of a turn! Commented Aug 19, 2022 at 18:01
• @JamesK sure, but you are not really using radians at that point, you are using turns. Which is completely reasonable, but makes you loose the nice properties of the trig functions and complex numbers, so you need to carry $2\pi$ factors around. Commented Aug 19, 2022 at 18:06
• but I see what you mean, just using $2\pi$ as a pseudounit to append to numbers. Like $\alpha = 3.47 \times 2\pi$ Commented Aug 19, 2022 at 18:09

Degrees are far easier for humans to work with. If you say something is $$90^\circ$$ from the horizon, people know what you mean instantly. In radians you'd have to say $$\pi/2$$ or 1.57rad, and that would get even worse with less common angles (e.g. $$45^\circ = .785rad$$, and $$30^\circ = .524rad$$). While the invention of the degree system predates history, it's obvious it was designed to be fairly intuitive for humans to recognize common angles.

If you look at any astronomy software, you'll find most functions accept and return radians, or a unit vector, and those are only converted to degrees for display to humans. Many algorithms are designed using degrees or arcseconds, but they are generally converted to radians before returning the value to the caller.

For example see the IAU SOFA Library, most functions all accept and return radians. Another example is the NOVAS library, which also uses radians in most places, but in some places uses unit vectors.