Measuring the Parallax of a Near Star with Modest Equipment and Modest Talent

Measuring the Parallax of a Near Star with Modest Equipment and Modest Talent

by Craig Hlady

THE PROBLEM STATEMENT

Is it possible to measure the distance to the nearest stars using by measuring their parallax with a set of affordable astrogear?  In the summer of 2016 I set on a journey to find out.

When I first considered the question, it seemed a very difficult if not impossible task.  Parallax measurements of even the nearest stars are measured in milliarcseconds, whereas with my equipment my resolution was approximately 1 arcsecond/pixel.  So therefore the challenge was to detect the shift in position of a star between measurements of, at the very best, a fraction of a pixel.  I wasn’t sure if that was feasible.  To make the issue even more challenging, I thought, was that the average seeing in my neck of the woods tends to be around 2 arcseconds. Would detecting so slight a shift in the apparent position of a nearby star be possible with equipment available to amateurs on a budget with all its inherent limitations?  Or is the measurement of stellar parallax only the purview of professionals?

CHOOSING THE LUCKY STAR

The first challenge was to find a candidate star I could set my sights on.  Using this list of nearest stars from Wikipedia, I searched for stars that were within 10 light years and located northerly enough that I could ensure capturing an image both in winter and summer.  At my latitude of 45°N, Alpha Centauri at -60° declination was obviously not a candidate.  Barnard’s Star, at a distance of about 6 light years and a declination of +4°, was a better choice in terms of its location in the sky but I worried about its visibility in summer and winter, and besides, at a magnitude of 9.53, I judged it on the faint side.  I had similar objections when considering Wolf 359.

Next on the list was a star I had never heard of, Lalande 21185:  distance 8.3 light years, magnitude 7.5, located in the constellation Ursa Major at a declination +36°.  Bingo!  It turns out that the M-class star Lalande 21185, aka HD 95735, aka HIP 54035, does possess a modicum of notoriety.  It is one of only a handful of stars primarily referred to by its Lalande catalog number, but more importantly in 1945 it became one of the first stars claimed to have a planet detected orbiting it.  The claim was finally refuted in 1974, but in a twist of fate, more modern measurements do in fact suggest this star possesses a planet of approximately 3 Earth masses with an orbital period of 13 days.  Regardless of the presence of exoplanets, real or imagined, Lalande 21185 was nearly ideal for my purposes.

However, with a published parallax of only 392 milliarcseconds, my initial thought was that even this sixth (or seventh?)-closest star to Earth was definitely going to pose a challenge.  I took comfort in the fact, though, that the first reported parallax measurement (511 milliarcseconds) was performed in 1857 – surely I could at least equal that achievement with 160 years of progress and access to modern digital photography? (Information from the Wikipedia page for this star.)

The actual data collection effort at first seemed very simple.  I would take a photograph, repeat in six months, measure the apparent shift of Lalande 21185 in position relative to its neighbor stars between the two photos, impress myself with some fancy trig, and done, right? 

Not so fast!  First of all, at the small change in position relative to my capabilities to detect, I was aware that one measurement would most likely not necessarily be enough to be conclusive.  In order to get repeat measurements to see if truly the parallax of Lalande 21185 would rise above measurement noise and be in fact within the capabilities of my equipment, I prepared myself for a multi-year project.

In addition, with further research, I discovered another wrinkle.  Lalande 21185, moving at 4.8 arcseconds per year, exhibits the ninth-highest proper motion of all stars in the sky.  To put that in further context: the proper motion of the more famous Barnard’s Star is slightly more than twice that of Lalande 21185 at 10.4 arcseconds per year.  With this high level of proper motion, I was confident that two measurements taken six months apart would certainly pick up change in Lalande 21185’s position compared to background stars.  Indeed, the problem that became apparent to me was how to distinguish the displacement of the star due to parallax from the displacement due to the star’s proper motion.

Regardless of the additional difficulties posed by the high proper motion of Lalande 21185, it remained my most promising candidate.

THE METHOD

First I suppose I should explain parallax. Again referring to Wikipedia (hey I’m a semi-regular donor to them so I want to get my money’s worth), parallax is the “...difference in apparent position of an object viewed along two different lines of sight...”.  If you extend your arm and put a finger out, then blink one eye and then your other while looking at your finger, the way your finger appears to move compared to the wall (or whatever the background is) is a very simple example of parallax.

The amount of apparent movement of a star against the background then is a fairly simple exercise in trigonometry, with the Earth’s position six months apart creating the base of a very long, skinny isosceles triangle of distance two A.U., or approximately 0.00003 light years.   (Did I mention it was a long, skinny triangle?)  The helpful thing I personally learned in this exercise is that published parallax measurements are given using the annual parallax definition, which is the difference in apparent position as seen from the Earth and the Sun, i.e., the base of the triangle is 1 A.U..  This was a fortunate realization for me, since my measurements taken six months apart meant that I was looking for the change in a star’s apparent position by twice the published amount – which made the task I had set myself upon twice as easy as I initially thought!  I was now looking to measure a shift in position along the order of a full pixel rather than just a half-pixel, a welcome realization.

Armed with a refreshed knowledge of high school geometry, I worked out a method of data-collection and measurement I thought would work.  If I were to take measurements at the same time each year (say summer), the result would capture the proper motion of Lalande 21185.  If I would also take a mid-year measurement in the winter, I would not only capture the proper motion of the star in six months, but also the parallax.  If I were to take photos each summer and winter approximately six months apart, the star would exhibit a zig-zag motion as it moved across the sky.  The summer measurements and winter measurements, taken separately, would result in (in a perfect world with no error in measurement) two parallel lines, the distance between the lines being the parallax.  The figure shows the resulting stairs-like measurements I hoped to capture.

The axes are arbitrary – Lalande 21185’s proper motion is relative to the stars surrounding it, so my thought was I would measure Lalande 21185’s distance from key surrounding stars in each photograph, figure out a convenient axis, do some tedious trigonometry, and then lay out Lalande 21185’s path as in the graph.

Would it work?  There was only one way to find out, and that was to get started!

THE EQUIPMENT

Having headlined this article partly by saying ‘with modest equipment’, at this point it is time to reveal just how modest.  Let me start by stating that I am the individual you have been warned not to be in countless Cloudy Nights posts: the guy who is so cheap buying equipment he ends up spending more in the end.  First off, the mount – an AP Mach1, my second mount after my old Celestron CG5 just couldn’t keep up with my desires anymore.  OK so this mount is perhaps not so modest, but considering that I was taking photographs only on the order of 15 – 30 seconds, the need for extreme tracking accuracy wasn’t very high so I don’t think anyone who wants to try this needs an exceptional mount.  Perhaps even the CG5 could have sufficed although that might have been pushing it.  Second, the telescope – a Celestron C8N – an eight-inch, f/5, 1000-mm focal-length Newtonian.  This is as budget as it gets, probably a $300 new telescope.  To mitigate coma I also had a Baader Multi-Purpose Coma Corrector (MPCC).  The camera I started out with was my trusty old Canon 450D, but a few years in I switched to a Nikon D5500.  Changing equipment mid-way through an experiment is not necessarily recommended, but I thought the higher resolution and lower noise characteristics of the Nikon vs. the Canon was preferable to consistency.  As it turns out, for some reason I swapped back again at one point which probably didn’t help improve any accuracy of data collection but what is done is done.

However, the ultimate point I want to make is, I performed these measurements with equipment well within the reach of an astrophotographer with less than an unlimited bank account.

If you are curious about the resolution of my setup, here is what I calculated based on my telescope’s focal length and pixel sizes from each camera:

To reiterate, with the published value of Lalande 21185’s parallax measurement of almost 400 milliarcseconds, I was therefor looking for a shift in position of twice that amount or 800 milliarcseconds, or essentially a shift of one pixel in photographs taken six months apart.  Would a longer focal-length telescope have helped, or the use of a camera with smaller pixels?  Probably, but the issue was moot for me, since I didn’t have a higher focal length telescope anyway.  I was worried, though, that the inevitable errors in individual measurements creeping in from limits of seeing, tracking inaccuracies from measurement to measurement, focusing issues, etc., would combine to swamp the very small shift I was attempting to sift out from all the noise.

THE DATA

On August 6, 2016, I was present at the Oregon Star Party (OSP), which took place within the Ochoco National Forest in Oregon.  This is a wonderful dark sky site, although alas in recent years susceptible to smoke from forest fires which seem to happening more and more often.  (And has now been canceled multiple years in a row due first to COVID, then forest fire fears, then untimely major forest service road repair blocking access.  I wonder if it will ever be the great event it was again.)  According to my notes from that night, “...except for some distant clouds which stayed SE all night, plus intermittent lightning in the far southern horizon, [the] night was as clear as I’ve ever seen.”  I started out my night as the sky was still darkening by taking 13 sixty-second subs of my target before swapping scopes and moving on to more pretty-picture-oriented activities.  The result of this first effort is below, annotated in PixInsight with star names from the Tycho catalog and also with declination and right ascension grid lines.

Ta-da!  There Lalande 21185 sits proudly in the middle left of the photograph, shining happily away at a blinding magnitude 7.5, at 11h 03m 20s and +35° 58’ 12” (actually those are Epoch J2000 coordinates, not its current position!)

Hopefully at this point you are not feeling too underwhelmed.  Admittedly this is a very dull star field with nary a faint fuzzy in sight, but really, how much drama do you expect out of an M-class dwarf star?  I do have a sense of wonder about this star, actually, since although it is one of our nearest neighbors, and the brightest red dwarf in the northern hemisphere, it is yet invisible to the naked eye.  Just think about the vast number of stars like this, completely unseen to us poor Terran observers.

The star of interest I have relabeled in larger font than the original for legibility.  You might wonder why it isn’t centered in the photo.  Good question!  For this first photograph, I was shooting ‘blind’ – after I had aligned my scope as best I could, I slewed to the published coordinates of the star, then took the picture.  I had no way of knowing at the time whether or not I had achieved enough accuracy in my pointing to capture Lalande 21185 in my field of view.  Luckily for me my skill level was at least up to getting it somewhat in the interior of the frame.  The identification of other stars in the photograph was important because I anticipated that in subsequent photographs I would measure Lalande’s change in position with respect to selected background stars.

And, that was pretty much it for six months!  I just had to wait for the Earth to languidly amble along in its orbit for six months before capturing another image of the same part of the sky.  And then I did it again. And again.

The dates that I took my photographs are as follows.  As I stated before, I used two different cameras, but kept the same telescope and mount throughout.

As you can see, I did vary my exposure times over the years.  Again, maybe a mistake on my part, but as I progressed, I gained the philosophy that shorter subs were probably preferable to longer ones, to minimize any errors and elongation due to poor tracking, or having poor seeing or the wind causing my telescope to vibrate causing the star to spread out and make the estimation of its center less accurate.  In retrospect I could have experimented with this more at the beginning and stuck to a more constant approach.  On the other hand, on each occasion I tried to match my approach to the sky conditions of the time and it was a very much learn-as-you-go activity.

THE TECHNIQUE

In each photograph, I identified not only my target but five reference stars spread around Lalande 21185.  After four years of just taking photographs, I finally got serious about sitting down and examining the data.  The first thing I wanted to do was measure my selected reference stars’ distance from each other from photograph to photograph, to make sure that          (a) none of them were moving with respect to each other; and

            (b) that the distance measurements were steady enough to use as good reference points from which to calculate Lalande 21185’s motion.

Using the five reference stars I mentioned that more or less surrounded Lalande 21185, I crunched the numbers.  To my vast disappointment, after I measured the star-star distance in each of the seven photographs I had taken to then, I determined the standard deviation in the background star distances from each other was on the order of two pixels.

Two pixels!  Oh the humanity!  Given that I was attempting to measure a shift in Lalande 21185’s position on the order of ONE PIXEL, I felt that this was the end of the road.  This initial result indicated that the errors in measurement from all the various sources had indeed added up to make my stated goal impossible, as I had feared when I first began this quixotic study.  I am glad it wasn’t the case, otherwise the ending to this article would have been rather anti-climatic to say the least.

What saved the day was that after some gnashing of teeth it dawned on me that what I was really trying to do was manually register each picture to each other.  That is, I was attempting to make each reference star static from photo to photo so I could properly isolate Lalande 21185’s motion for measurement.  Well good news, image processing software such as PixInsight do exactly this!  Usually image integration is done for the purposes of ‘stacking’ photographs to improve image quality, but in this instance what I was looking for was a technique to allow me to examine each photograph with all the background stars held in constant positions so I could accurately track Lalande 21185’s ever-shifting position.

PixInsight’s StarAlignment module did just the trick.  All the photos except one were successfully registered, thus ‘freezing’ all the stars in the same location in each photograph, except of course for the one traveler star of my interest.  (The image that was not successfully registered was taken on 8/6/21, and that particular occasion, clouds prevented me from taking my shots until the star was very close to the horizon, so much so that some tree shadows impinged upon some of the photographs and most likely some atmospheric diffraction was taking place.  So I felt I had enough reason to legitimately discard this data point.)

After picking my best-quality image as the master and registering all the other photos to it, all the stars aligned, so to speak.  The next step was to record, as precisely as I could, Lalande 21185’s position in each photograph.  Again PixInsight came to the rescue.  The module that I used was the DynamicPSF process, PSF standing for Point Spread Function.  In a typical PixInsight workflow towards making a ‘pretty picture’, this process is normally used as a precursor to deconvolution efforts as part of trying to sharpen pictures.  The DynamicPSF tool is designed to fit the star to fitting function to describe the star, including its FWHM (full-width half-max), eccentricity amount and angle, etc.

In my case, however, the only data produced by the DynamicPSF tool I that was interested in was the coordinates of the centroid of the star.  Luckily for me, this value is expressed to the hundredth of a pixel by the tool.  Plenty of precision for my needs, but as I entered the data, I wondered how much of that precision translated into actual accuracy?  Most of the fitting functions the DynamicPSF tool gave for Lalande 21185’s brightness curve were Gaussian fits, an indication of a star brightness profile that is relatively soft and rounded compared to, say, a Moffat fit.  This star profile therefore wasn’t the most conducive to determining the most accurate value of the star’s true centroid position.  However, that was the data I had.

THE RESULTS

The graph below sums up my efforts over the past six years.

The data point at the 0,0 position represents the starting position of Lalande 21185 on August 6, 2016.  This initial reference position was, according to Stellarium (my personal favorite planetarium software), located at 11h04m15s/+35°51’25”.  (Compare this to the J2000 coordinates of 11h03m10a/+35°56’52” for an idea of how far this star moved in sixteen years due to its high proper motion.) Each consecutive data point to the right of the initial 0,0 data point represents the position of Lalande 21185 each six months thereafter (with the data point from August 2021 missing).  The blue squares represent my summer (August) measurements and the orange diamonds my winter (January) measurements.  Again I note, the axes are arbitrary, merely the x- and y- axes established by the particular angular position of my camera when I took the reference photograph.  So the movement of Lalande 21185 is the hypotenuse of the movements along the two axes.  Each consecutive data point progressing rightward from the initial point represents six months in time, with the last data point at approximately 24,16 represents the final measured relative position of Lalande 21185 on August 13, 2022.

The most salient feature of this graph for the purpose of my study is that indeed the individual data points progress in a step ladder shape as I anticipated they would.  To my delight there is a clear delineation between the summer and winter data points, indicating that my data was good enough to pick up the parallax present from the 2 AUs of Earth’s position change from summer to winter.  The lines of regression of the winter and summer data (W_reg and S_reg) are nearly parallel, and furthermore, the coefficients of determination are extremely high (R²=0.9966 for the winter data, 0.9940 for the summer data), indicating high quality of data.

But how to actually get some measurements of the parallax from the graph?  If the lines were exactly parallel to each other it would be an easy exercise, but this wasn’t quite the case.  I decided to choose the winter regression line as my reference line, due to its slightly higher coefficient of determination, and then for each summer data point calculate the parallax between the point and the winter reference line.  Aa simple average would then represent the best estimate of parallax I could get from six years of observation.

So, the moment of truth, the vindication of the better part of a decade of effort, the opportunity for me to brag to my friends at my local astronomy club (a shout out is due here to the Rose City Astronomers), came down to the answer to a simple math question:  what is the average of 717, 118, 478, 359, 446, and 335?

The answer: 409 milliarcseconds.  409!  Well dang, closer to the accepted value of 392.8 millarcseconds than I dared to dream!  I converted this to light years and what I got was a calculated distance from Sol to Lalande 21185 of 7.98 light years.  A quick double-check against the published value of 8.30 light years brought a huge smile to my face.  Finally, an answer to my question, is it possible with amateur equipment to discern the incredibly small parallaxes associated with stellar distances?

I believe the results show that, with patience, the answer is yes.

Ancillary to this study, but a good double-check as to the accuracy of the data, was the estimate of the annual proper motion of Lalande 21185 calculated by its total observed change of position in the sky for the duration of this study:  in the amount of time between the first and last pictures I took of this star, I calculated it moved 28.86 arcseconds through the sky.  This yields 4.794 arcseconds per year based on my observations, which agrees with published data within 0.1 percent.

The following table summarizes the results of this study, comparing them to published (data from Wikipedia):

CONCLUSIONS & DISCUSSION

As an amateur with no formal training in astronomy, I freely admit this study likely has some flaws in technique or reasoning.  Despite these, I believe I achieved my goal of determining the parallax and thereby the distance to a nearby star with my modest telescopic equipment with some reasonable amount of accuracy.  In fact my calculated value is accurate to about four percent of its true value, thereby besting the accuracy of my arch-rival Friedrich Winnecke’s estimate of 511 milliarcseconds back in 1857.  Take that, Freddy!

Am I done?  As February of 2023 nears, I am debating whether or not to keep on going.  I doubt I will; I have accomplished what I have set out to do.  Perhaps I will spend another six years on a more challenging star like 61 Cygni or Struve 2398, stars of some eleven light years’ distance from us.  On the other hand, I wonder what more I could accomplish by doing so.  The quest I was on, after all, wasn’t to improve upon the existing values of the distance to Lalande 21185, but only to see if it could be done at all by someone with less than professional-level equipment and knowledge.

I am sure I could improve the accuracy of my results with better equipment and better techniques, but in the end all I would end up doing is trying to see how close I could get to Hipparcos.  It was a mainly a personal journey of discovery, albeit one I’d like to advertise and encourage other people to try.

I do wonder how far out some more advanced imager could measure before error inevitably swamps the data as the apparent shift for a target star would get smaller and smaller.  For my own data, to be honest, I am not sure how to properly express its uncertainty.  The standard deviation of my seven parallax measurements is 197 millarcseconds, but here is my difficulty:  on the minus side (say, the average minus one standard deviation) this translates to a calculated distance of 5.4 light years, or 2.6 light years less than the calculated value.  However, on the plus side, the calculated distance for the average plus one standard deviation is 15.4 light years, or 7.4 light years more than the average.  The problem being that the uncertainty in my data in arcseconds on the one side of the average does not translate to the same uncertainty in distance as on the other side.  So is my estimate of Lalande’s distance 8.0 ly + 7.4 – 2.6 at one SD? Or would I say, my estimate of the distance of Lalande 21185 within one standard deviation of certainty is 5.4 – 15.4 ly away? But that would imply my best guess is the average of the two values is 10.4 ly, which is not true.  I suspect there is a better way to calculate uncertainty of measurement in this case.  I will leave the solving of this exercise to the reader!

And finally, I am somewhat in awe how this star, in many ways so humble, is also a reminder of the vastness of the scales of time and space.  Since I first learned of its existence and started following it, it has glided across the sky by about one-sixtieth the angular diameter of a full moon.  Despite being one of the fastest-moving stars in the sky, it would take multiple lifetimes to record it glide even a full moon diameter.  And yet, it is approaching closer and closer to us until it will become, in about 20,000 years, at less than five light years away, the second-closest star system to us before it recedes again.  A mere blink of an eye on the timescale of the universe.

If a picture is worth a thousand words, what are sixteen pictures taken over the span of six years worth?  I close this little attempt at science of mine with a short video of the movement of Lalande 21185.  I note that the slight ‘zig-zag’ motion I have referred to is too small to be picked up by eye (at least mine).  But I hope it does capture some part of your imagination, as it has mine: