Lateral offset from a centered star as measured at the focal plane

[Benschu]

TL;DR Summary

At the focal plane, what is the distance from the centerline of an object laterally displaced from a star at local stellar distances?

Reference: https://www.physicsforums.com/forums/astronomy-and-astrophysics.71/post-thread

Hello, I need a bit of help. My age and fading competence are showing (no complaints, just facing up to it). I have an optical SETI observatory in Panama with a 20" Newtonian and a piggybacked 14" Cassy.
I wish to mask a portion of stellar Airy disks with an E-W wire on a small photometer aperture. This has been tested successfully with a 1 mil wire in a 15 mil photometer aperture for 20-minute observations, i.e., tracking is good. Clearly, the 1 mil wire is too large, but it's a start. I plan to electrochemically etch the wire to the needed diameter.

I need to know the relationship for image offset from the optical centerline at the focal plane with respect to the lateral distance from a centered star. As for instance, at the focal plane, what is the distance from the centerline of an object 1 au laterally displaced from a star at 200 ly? I suspect that location is mostly overshadowed by the Airy disk, but there may yet be a useful compromise.

Backup info: 20" f4 Newtonian with a ~2-meter focal length.

The Boquete Optical SETI Observatory has been running continuously since 2009 having observed nearly 10,000 stars in search of pulsed laser signals. Glad to supply more info for anyone interested. A bit of help on this project will be greatly appreciated. Ben Schuetz

 

[Drakkith]

I'm sure there's an easier way, or a calculator out there that makes this stuff even easier, but here's the method I used based on some online calculators I already use a lot.

Try this CCD calculator. Just input your focal length and camera pixel size and it will give you the angular resolution of each pixel. And since you know the size of the pixels, you can then easily calculate the lateral distance for a given angular distance.

For example, inputting 2032 mm as the focal length and 10 um as the pixel size gives an angular resolution of 1.02 arcsec per pixel. Thus, two objects separated by 1.02 arcsec will be 10 um apart at the focal plane for your 20" scope.

To find the angular distance, I use this calculator. Since you're using AU's as your separation distance unit, convert LY to AU and then put that in for the value of r in the calc. Converting 200 LY to AU gives me 1.265e+7 AU. Then put 1 in for g. Hitting 'calculate' then yields an angular diameter of 4.5293E-6 degrees. Multiplying by 60 to get arcmin, and then by 60 again to get arcseconds yields about 0.0163 arcsec.

Since 1.02 arcsec = 10 um, that's $\frac{10}{1.02}=\frac{X}{0.0163}$.
Cross multiply to get $0.163=1.02X$.
Divide to find X: $X=0.159 um$

So two objects at a distance of 200 LY separated by 1 AU will be located 0.159 um apart at the focal plane, which is far smaller than the airy disk diameter of about 5.4 um.

Let's hope my math is all correct.

 

[Benschu]

Thank you so much for your reply and assessment. I've just had a moment to skim your response.
Although I'm using a PMT with high-speed data collection and not a CCD, the info you provided
should enable me to move forward with the project. Yes, as expected the 1 AU separation is much smaller than the Airy disk. Masking may still be useful to improve the SNR if only a partial mask is used. That may be tricky considering how very small the wire must be. I also need to consider alternative methods.
Thanks again Drakkith.

 

[Drakkith]

Although I'm using a PMT with high-speed data collection and not a CCD, the info you provided
should enable me to move forward with the project.

Glad to hear it. I only used the 'CCD Calculator' because I already knew it would let me find the lateral distance based on the focal length of the scope via the pixel size and resolution. I'm sure there's a calculator or formula out there that directly gives you it, but I didn't have one handy at the time.

Yes, as expected the 1 AU separation is much smaller than the Airy disk. Masking may still be useful to improve the SNR if only a partial mask is used. That may be tricky considering how very small the wire must be. I also need to consider alternative methods.

I once tried to image a circumstellar disk with an 8" scope by using a sharpie to put a dot in the middle of clear filter in my filter wheel. Results were... 'inconclusive'.