# Calculating the position of stars at a given point in time

Is there any particular methods that help us calculate the position of a star at a certain point in time? This question is what I am trying to achieve, but for me, I do not want to use the data because the data that we can access to calculate the position is limited to only a few years. I want a method which uses the current position, speed, velocity of stars and calculate its position for many years back (like in 11000 BCE to 13000 BCE). I am trying to use astropy and I am really new to this, so I just need a push in the right direction; helpful resources will serve a great deal as well, thank you.

Edited: There are a few stars' positions that are recorded and mentioned in one of our history books. SO what I am trying to do is collect all the recorded stars' positions and try to map them in real-time at a particular year and see if the recorded data exists or not.

Now out of this data, the one that stands out as puzzling as is was the movements of Alcor and Mizar in the sky.

If we observe these 2 stars from earth, for a naked eye it would seem like Mizar is always in front and Alcor is always in following Mizar.

But in one of our history books, there is a recorded event where Mizar follows Alcor (it looks like it is following, I know they just revolve around each other), so what I am trying, is to see if there was any particular period where this would be possible.

This is the starting problem there are many observations like this so it would be really helpful if I could get any info on achieving this.

• Roughly how many years is "many years back"? 10? 100? 1 000? 10 000? Oct 20, 2021 at 13:08
• Should have been clear, many years in the sense back to 11000 BCE to 13000 BCE. Oct 20, 2021 at 14:24
• Can you be clear what you mean by "positions". Oct 20, 2021 at 16:41

The projected separation of Mizar and Alcor is 12 arcminutes. If they are both at a distance of 25 pc (suggested by the Hipparcos satellite), then this equates to a physical separation of at least 18,000 au. Using Kepler's third law we can then estimate an orbital period of at least 800,000 years (assuming a total system mass of around $$9M_\odot$$. Thus over a period of time of 13,000 years or so I think you can safely neglect the acceleration caused by the interaction of Mizar and Alcor.

Similarly over the course of 13,000 years, the distance to the stars will only change by about 0.0013 pc per km/s of radial velocity. The radial velocities listed in SIMBAD are $$-6.3 \pm 0.4$$ and $$-8.9 \pm 0.9$$ km/s for Mizar and Alcor respectively, and proper motions in RA of $$119.0 \pm 1.5$$ and $$120.21 \pm 0.12$$ mas/year and in Declination of $$-26.0 \pm 1.7$$ and $$-16.04 \pm 0.14$$ mas/year. Assuming the velocity tangential to the Sun remains constant then these low radial velocities will not change the measured proper motions very significantly over that time and certainly not by more than the current uncertainties in the relative distance to the two stars or their proper motions.

Thus all you need to do then to work out (relative) positions on the sky is extrapolate the current positions and proper motions backwards for the requisite amount of time assuming that they they do not change. How accurate these predictions are will depend on how accurate the proper motions are.

Roughly speaking (and you should do this properly using the astropy apply_space_motion routines) the change in Alcor's position relative to Mizar is about $$1.2 \pm 1.5$$ mas/year in RA and $$10.0 \pm 1.7$$ mas/year in Dec. One simply multiplies these numbers (and their uncertainties) by -10 to get the change of position of Alcor relative to Mizar 10,000 years ago in arcseconds. Given that the current separation is 12 arcminutes, then there wouldn't really be much change in their relative positions visible to the naked eye (I think they would get about 1.6 arcminutes closer together in Declination).

The answer probably depends on what you want, what is 'long time' for you. Astropy offers a method to handle coordinates and proper motion: https://docs.astropy.org/en/stable/coordinates/apply_space_motion.html.

However, unless you have actual positional information for times you are interested in, you have to forward or backward integrate from what you know and basically have to do what you always have to do in such case: obtain the 3D positional information and 3D velocity information and then integrate forward or backward in time.

In practice you probably will get the proper motion of stars in right ascension and declination and the 3rd velocity coordinate along the line of sight is hard to come by and might not be available at all; similarily and additionally the distance might be the least well-constrained part of the stellar coordinates which adds uncertainty to the conversion of proper motion to 3D velocity. You might in many cases get acceptable results, if you ignore the radial component in both spatial and velocity coordinate and keep your integration time small compared to the galactic year (revolution time of the sun around the galactic center, ~220 Myr).

In order to simulate the view of the night sky from Earth 100s or 1000s of years ago or in the future you will also want to simulate the precession of the Earth's rotational axis which will change the apparent position of the stars in the sky by changing the point on the celestial sphere the sky seems to revolve around.

• What I am trying to do is simulating the night sky view from earth, yes, I understand the earth's postion also becomes a factor here, this is what I wanted to know, how to introduce this, or does astropy handles this....not sure....! I have update the initial question as well, added further details, about what I am trying to do. Oct 20, 2021 at 14:35

I was able to see this using skygazer 4.5M, quite frankly, I am new to astronomy and inspired by other's (mainly Nilesh Oak) work I started doing this.

He had already proved this and I wanted to see if everything that he said is true. writing code to simulate this in astropy is just to much work and I am not that familiar with the package so, for now I cannot do anything about it(but I have not quit on this yet)

For this to work we need to consider the change in pole star, from Polaris to Vega which happens every 13000 years(half of full precession of equinoxes).

Now again, since we are looking at the sky with naked eye and it is from earth, we can only say how it might have looked from earth at specific latitude and longitude, so if we consider all this and movement of stars and the movement of earth itself, it will look like the star Alcor is crossing the meridian first before Mizar, really fascinating observation.

If you are interested, here is the link to the video where Nilesh Oak, talks about it:

Mizar is known as Vasishtha and Alcor as Arundhati in the video.

Both of the answers has points that helped me I will give up votes for both and accept mine as answer to close the question.