Proper Motion and Calculating Angular Distance

This week’s biggest questions solved

Heroes and Highlights:

Tony Dunn’s proper motion simulator: the inspiration for this page, a beautiful simulation of proper motion’s impact on familiar constellations

One of two websites to include the full equation for calculating angular distance between star by hand. I had to do a little digging through the source code but it was all there.

in-the-sky.org: I wanted this page to be able to extend to any collection of stars or constellations and that required having the proper motion and the angle of proper motion for any given star which turned out to be difficult to find. This site includes proper motion, angle of proper motion, right ascension, and declination for all stars I looked up.

For examples below I use the proper motion and proper motion angle for Alkaid within the Big Dipper (in-the-sky.org details)

Try it Yourself:

To try out these techniques for any collections of stars or constellations I’d recommend the following steps:

1. Find each star on in-the-sky.org and record its proper motion to determine how far it will travel over time and its right ascension/declination to determine the direction it will move as well as how far it is from other stars (more commonly referred to as angular distance)
2. Determine the angular distance between at least two stars in the constellation to get a reference for how far a degree is
3. Calculate the distance that a star will travel for a given time period (in this guidebook page I used 100,000 years since it is large enough to get visible shearing for stars even with small proper motions) in degrees and use the reference guide create to graph the final results
4. Plot the change on the constellation with the given angle of the proper motion (pos angle) and the reference for a degree calculated from angular distance

How to determine the Angular Distance between two Stars with Right Ascension and Declination?

The sky can be treated as a celestial sphere, therefore the distance between any two stars will be a curved angle along the edge of the sphere. This requires angular distance rather than a simple linear distance.

$$\text{Declination as degree.arcminutearcsecond (DD.MMSS)}$$ $$\text{Declination of star 1 = dS1}$$ $$\text{Declination of star 2 = dS2}$$ $$\text{Right Ascension as hours.minutesecond (HH.MMSS)}$$ $$\text{Right Ascension of star 1 = raS1}$$ $$\text{Right Ascension of star 2 = raS2}$$

To calculate angular distance, convert to decimal degrees from degrees (for declination) or hours (for right ascension).

$$\text{Declination as decimal degrees} = \text{(+ or -)} (\text{degrees} + \frac{\text{arcminutes}}{60} + \frac{\text{arcsconds}}{3600})$$ $$\text{Right Ascension as decimal degrees} = (HH + \frac{MM}{60} + \frac{SS}{3600})$$

$$\text{Then, declination and right ascension should be converted from degrees to radians,}$$ $$\text{ degrees} * \frac{\pi}{180} = \text{radians}$$

$$\text{Finally,}$$ $$cos(A) = sin(\text{dS1})sin(\text{dS2}) + cos(\text{dS1})cos(\text{dS2})*cos(15*(\text{raS1} - \text{raS2}))$$$$\text{A} = arcos(A)$$ The math here can also be found on the History Survival Guide Github repo where you only need the two stars’ right ascension and declination to have it return the angular distance.

The purpose of this step allows us to have a scale for far a ‘degree’ is so when the movement for each star is calculate we can use the new scale to determine how far a degree is in reference to other stars. This is useful since you won’t need any additional tools to plot the final locations of the star as you will see below.

How Long it Takes for a Star with X Proper Motion to Move 1 Degree?

Milliarcseconds will be used in this example since that is typically how the proper motion of a star is reported

$$\text{Remember: 1 degree = }3600 \text{ arcseconds} = 3,600,000 \text{ milliarcseconds}$$

$$\frac{x\text{ proper motion}(\mu)\text{ of a star (milliarcseconds)}}{1\text{ year}} = \frac{1 \text{ degree}}{3,600,000\text{ milliarcseconds}}$$ $$\text{Therefore, solve for }z$$ $$\frac{y\text{ degree}}{1 \text{ year}} = \frac{1 \text{ degree}}{z \text{ years}}$$

$$\text{Example: Alkaid has a proper motion of 122.1 milliarcseconds/year}$$ $$\frac{122.1\text{ proper motion}(\mu)\text{ of a star (milliarcseconds)}}{1\text{ year}} = \frac{1 \text{ degree}}{3,600,000\text{ milliarcseconds}}$$ $$=0.000034 \text{ degrees/year}$$ $$\frac{0.000034 \text{ degrees}}{1 \text{ year}} = \frac{1 \text{ degree}}{z \text{ years}}$$ $$z = \frac{1}{0.000034}$$ $$z = 29,484.03 \text{ years}$$ $$\text{So, it takes Alkaid roughly 30,000 years to move 1 degree}$$

How many Degrees will a Star Move in 100,000 years?

$$\frac{x \text{ milliarcseconds}}{3,600,000 \text{ milliarcseconds}} * 100,000 \text{ years}$$

$$\text{Example: Alkaid has a proper motion of 122.1 milliarcseconds/year}$$ $$\frac{122.1 \text{ milliarcseconds}}{3,600,000 \text{ milliarcseconds}} * 100,000 \text{ years}$$ $$0.00003392 * 100,000 = 3.39167 \text{ degrees in 100,000 years}$$

Plot new position of stars with the angle/direction of proper motion

From above we have calculated that Alkaid moves 3.39167 degrees in 100,000 years. Alkaid moves at an angle of 263.0 degrees (for more information about Alkaid)

Past: Future: 