Wednesday, August 25, 2021

learned cross-hair method

Back on 25 July, I attempted to use the method described by John McCue, with only a cross-hair eyepiece, to measure the separation and position angle of a double star.

At the time I could not get it to work. It seemed to me that something was missing. It was a nagging feeling as I left the web page.

Early today, I had a go using Beish's method, where he demonstrated using a custom bi-filar eyepiece with a micrometer. It got me thinking...

I returned to the British Astronomical Association web page with McCue's notes and diagrams. Built a spreadsheet (again, must have discarded the other) and hammered out the numbers.

Some light bulbs lit this time. I grokked the triangle formed by the initial cross-hair alignment and the second alignment. Trig at work. I started to get some good numbers.

I carefully read every-single-word. Spotted the remark about converting the seconds of time into seconds of angle by multiplying by 15. I did not recall seeing that before.

And then I tried working his sample values. Worked his numbers backwards. Things did not seem to align. And again, hit a roadblock. Again, it felt like something was missing. Or is McCue operating at a different level? Does "sin θ" mean something that I don't know, never learned, never understood?

I did some algebra and ended up with an equation which would solve the separation. And there it was! It worked! I had a value that matched his. Bit more brain-bending formulae and I got the separation! Holy Universe. It actually worked.

Tested it with three random doubles and was impressed. Pretty close on Albireo and HD 206224. A fair result with 94 Aquarii.

So let me try to explain the process and the maths is an obvious and easy way. McCue explanations, I feel, leave a lot to be desired.

  1. align cross-hair to EW or parallel to RA and drift a star across the field
  2. roughly estimate the position angle considering N and W in the field
  3. get time (t1) between primary and secondary stars drifting across the NS line, e.g. 1.93 seconds
  4. put the primary star at the centre
  5. align cross-hair through both stars
  6. get time (t2) of the secondary star drifting across the NS and EW lines, e.g. 3.04 seconds
  7. get the apparent/current declination (d) of the star, e.g. +28.0
  8. calculate the separation
  9. calculate the position angle

The formulae:

formulae for calculating sep and PA

Hopefully I'm using the correct nomenclature here. To be clear, the sin-1 indicates the use of arcsine.

You should get a separation of 32.1" and a position angle of 53°.

Now, McCue explains that the initial PA can impact the final PA calculation. Again, I suggest you roughly estimate it at step 4 above. If the PA is between the degrees:

  • 0 and 90, the primary will lead, and you do not need to modify the calculated value
  • 90 and 180, the primary will lead, but use a final formula: 180 - your calculated PA
  • 180 and 270, the secondary will lead, but use a final formula: 180 + your calculated PA
  • 270 and 360, the secondary will lead, but use a final formula 360 - your calculated PA

McCue also cautions that if the PA is near zero or 180, timing is very difficult.


  • you need an equatorial mount where you can toggle sidereal tracking on and off, if that's not abundantly clear
  • you'll want to work at a long focal length for greater resolution
  • a stopwatch (app) with a lap timer will be very handy
  • the alignment drifting and the timing runs should be done a few times, perhaps a dozen, to yield an average
  • if you use an electronic spreadsheet such as Excel or Google Sheets, don't forget to convert to and from radians, as required

So, there you have it.

I really wanted to work through this, understand it. It is a proof of concept. It shows that if an observer really desires to measure doubles visually and has a cross-hair eyepiece, they are good to go. 

With the dearth of astrometric eyepieces now, there are few options for the visual observer trying to save a few bucks...


If you don't want to reinvent the wheel, ask me for my Excel workbook.


Edited on 25 Aug to move "rough estimate" of PA earlier in the sequence. It's much more obvious after step 1.

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