To See or not to See … That is the Resolution Question

Three Image Quality Constraints Every Smart Telescope Owner Needs to Know

INTRODUCTION

This blog entry is about astronomical image resolution. I don’t precisely mean resolution in the same sense specified for our cameras, laptops or television displays. A professional photographer’s camera, for example, might have an imaging chip that can produce a 100-megabyte image. Or, the resolution spec of a computer laptop might be 2560-by-1664 native resolution at 224 pixels per inch. These types of resolution specifications describe the total number of pixels used in the devices. This is not the same as the use of the word resolution in an astronomical image.

Astronomical resolution is not the same as total pixel count.

This week there’s going to be a little math at play too. I needed my calculator to work out some of the optical constraints on resolution found in our smart telescopes. I will readily admit that I’m not an astrophotography expert, but fiddling with numbers is how I ultimately understand what’s going on. The numbers are necessary to describe the practical limits on what we can image, and they provide a way to judge different telescopes and their CMOS imaging chips.

I hope at the end of this blog entry you will be equipped to do the math to determine two important quantities:

1. Dawes’ Limit for your scope, which is the theoretical limit of resolution it can achieve;

2. Pixel Sampling for the imaging chip at the heart of your scope, which is the best resolution your system can record.

Finally, I will introduce you to a way to get a handle on a third constraint — the “wild card” we have to deal with — namely, atmospheric seeing.

But first, let’s clarify the dimensional units we’re going to be using. In visual astronomy, we don’t directly measure linear distances like meters, feet, miles or even light years. Rather, we are always measuring angles in the sky. Most people know about degrees -- as in 360 degrees in a circle -- but not everyone is familiar with the smallest unit we will be using — the arc second.

ARC SECONDS — OUR BASIC ANGULAR UNITS

Our final results will be expressed in arc seconds. This is not a unit of time, even though “seconds” is in the term. An arc second is really the practical, basic unit for astronomical imaging for Earth-based telescopes. An arc second is 1/60th of an arc minute. An arc minute is 1/60 of a degree. So, one arc second is then (1/60)^2 = 1/3,600th of a degree. This equals 0.027% of a degree. That’s a very small angle indeed!

With 20/20 vision the human eye can perceive — resolve — angles down to about an arc minute. For example, a basketball, seen from 100 meters away, appears about an arc minute in angular width. Most of us could see the ball. We need telescopes, however, to measure angles in arc seconds because our unaided eye can’t resolve angles that small. A 5-mm high letter on the same basketball would subtend an angle of about an arc second. But, of course our telescopes could magnify this basketball, so we could easily read what was written on it. When we are using our smart scopes to their fullest, we routinely run into performance limits at just these tiny angles measured in arc seconds.

It’s useful to review first the angular sizes of some of our astronomical targets in arc seconds. The universe is a really big place, and the physical sizes therein vary greatly. An object’s angular size seen from Earth then depends on both the physical size and the distance to the object. This is why the range in angular sizes — what we perceive in telescopes — can be quite wide. Here are some approximate angular sizes as seen from our place in the Solar System.

Note that from here on, I’ll use the quotation mark “ to indicate arc seconds and the single apostrophe ‘ to indicate arc minutes.

> Sun and Moon —> each about 30’ across

> Mars —> 3.5” to 25”

> Saturn —> 14.5” to 20”

> Pluto —> 0.06” to 0.11”

> Alpha Centauri -> 0.007” (Even the Hubble Space Telescope can’t resolve this.)

Here’s a picture of the Moon with Mars in conjunction, so we can see how wide a range in angle these two close Solar System neighbors can demonstrate.

Composite image of the Moon and Mars to scale when they were aligned last December. Captured by Sean Walker of Sky and Telescope Magazine.

Most of our deep sky objects are much bigger than everything we observe in the Solar System, except the Sun and Moon. Here are some deep space examples:

> M57 aka the Ring Nebula -> 65” or just over 1’

> M13 aka Great Hercules Cluster -> 20’

> M42 aka Great Orion Nebula -> 1 degree by 1.5 degree

> M31 Andromeda Galaxy -> 3 degrees by 1 degree

To bring these angular sizes full circle, here’s an image of how big the Moon would appear if its path in the sky enabled it to pass directly in front of M31, the Great Andromeda Galaxy.

The Moon is in reality (2x10^6 x 3.15^7)/1.3 ~ 48,000,000,000,000 closer!

The first take away from this preamble on units is that it helps us understand why our smart scopes excel with the large, angular sized Messier Objects and aren’t so great with planetary detail. There’s a huge angular size scale difference and planetary detail requires extreme astronomical resolution that our scopes can’t really achieve.

Understanding the challenge of creating images down to arc second resolutions, helps explain three factors astro-photographers need to worry about when seeking high resolution images.

1. DAWES’ LIMIT

The first limit we run into is something called Dawes’ Limit. It’s named for William Rutter Dawes, a 19th Century English astronomer. His formula expresses the maximum resolving power of any optical instrument, telescope or even microscope. It applies, for example, when trying to “split” double stars in the sky. Stars closer in angle than the telescope’s Dawes’ Limit will appear as one.

It turns out that Dawes’ theoretical resolution is due to the wave nature of light and only depends on the diameter (D) of the optical objective. The objective is either the big lens at the front of your scope or its large primary mirror if it’s a reflector. (Dawes’ Limit does vary with wavelength of light, but the equation quoted below is for an average visible light wavelength.)

Dawes’ Limit, R, is the resolution expressed in arc seconds when D is expressed in millimeters.

R=116/D

For our smart scopes, the following table lists what we get for R. I’ve also included a couple popular sizes for amateur reflecting telescopes for your comparison too.

My takeaway from this first exercise is that I anticipate most of the scopes are engineered to be able to work at the 1 to 2 arc second range. Two of the most affordable scopes — DWARF II and Hestia — can, nevertheless, theoretically work down to less than 5 arc seconds, which is not bad for deep sky objects that often span tens of arc minutes or even degrees.

No matter what imaging chip a smart telescope is using or even if it were orbiting above the atmosphere in outer space, it can never do better than the numbers in the table above. The Dawes’ Limit is our ultimate resolution limit.

So, what about our imaging chips? Can they take advantage of these resolutions? To answer these questions, we have to understand the next limiting factor that depends on pixel sizes.

2. PIXEL SAMPLING

Pixel sampling (S) is the term used to represent the angle of the sky each pixel in our imaging chips can detect in arc seconds. It only depends on two quantities: the telescope’s focal length = L in mm; and the individual pixel size = P, in millionths of a meter or microns.

S =206.3*P/L

Here’s a table of S computed for the smart scopes.

The first take away from this table comes from comparing the last two columns. We see that the sampling sizes are very close to the Dawes’ Limits for each scope (except for maybe the DWARF II). As I think about that it makes perfect sense. Why would you make the pixel limit any better? And, why not take advantage of the pixels?

In reality, I suspect the optical designers of these smart telescopes worked backwards from the pixel sizes available from Sony, the chip manufacturer. If we wonder why they all have 50 to 100 mm in diameter objectives at f/4 focal ratios in relatively affordable and portable designs, then blame Sony… ;-)

There might be some super photo experts who will parse these differences in the table and discuss oversampling(pixels basically too small for Dawes’ Limit) or undersampling (pixels much larger than Dawes’ Limit)*. But I’m not so sure it makes any difference at this level of analysis since the last two columns are so close. Our scopes are good and convenient imaging devices, but they’re not going to reach state-of-the-art astronomical photos.

It’s important to note that field of view (FOV) of our scopes may be important to us, but it doesn’t figure directly into these calculations at all.

FOV, however, figures into comparisons of the Vespera models. The upcoming Pro model basically has the identical Dawes’ limit as the standard Vespera. But the Pro’s FOV is about 80% greater and its pixel sampling, because of a slightly longer focal length and pixel size, is almost 1/2 smaller. Nevertheless, the theoretical resolution limit is no different between the models because the apertures of the objective lenses are the same.

(Vespera’s FOV is 1.6 x 0.9 degrees, whereas Vespera Pro's will take in 1.6 x 1.6 degrees. Mosaic mode will be coming to many of these scopes in time. Vaonis already offers it. That really doesn’t increase the optical resolution, but it can yield much larger FOVs. The Pro model will be able to go 2.6x2.6 degrees at full resolution. That’s a large chunk of sky in any case.)

The software used to make mosaics will make a huge difference in extending the view, but it will still not really help with resolution, which is technically limited by the objective lens or mirror and the pixel sizes.

Whew! All that tech talk can be head spinning. But… there’s also one other major random factor we have to deal with. It’s the optics of the Earth’s atmosphere, and it’s called “seeing.”

3. ATMOSPHERIC SEEING

No matter how precise our smart telescope optics and sensors are, they still must contend with the unsteady and random turbulence in the atmosphere above our heads. Even if the Hubble Space Telescope were on Earth’s surface, it too would have to deal with the limits imposed by atmospheric seeing.

Atmospheric Seeing, or just seeing for short, refers to the fact that the atmosphere refracts (i.e. bends) the light from the objects as it travels to our telescopes. This distorts the wavefront of the light, which causes a constant moving and blurring of the image. Twinkling stars might appear beautiful to most people, but to astronomers they signal bad seeing.

A star, for our purposes, is a perfect point source. But its light will be distorted and is sometimes described on a scale of seeing described by the 19th-20 th Century Harvard astronomer, Edward Charles Pickering. (To technically measure the seeing and really use this scale requires standard telescope sizes, star magnitudes, etc. We won’t go that far here.)

The numbers below the images refer to the quality of the seeing with 10 being the ideal and 1 the worst. I usually see images like those in middle of the range here. I think a 4 or 5 here is about 1-2 arc seconds seeing, which is about where our scopes are designed to perform best. Usually, I see patterns that are more chaotic and probably mean the seeing is 3 arc seconds or worse. When they are in that range, it’s the atmosphere that is ultimately in control. I might then get great galaxy images, but the stars will not be as small as they might be under steady seeing.

Here is a great web page from the Canadian weather service on astronomical seeing that I highly recommend. It will also return an estimate of predicted seeing for North America.

https://weather.gc.ca/astro/seeing_e.html

I’ve often thought I would conduct a little smart scope program to measure seeing by observing double stars. I think that’s possible. If you’d like to think about that, here’s a good introduction to some that might be useful from the Milwaukee Astronomical Society: https://milwaukeeastro.org/beginners/double_stars.asp

CONCLUSION

I hope we now have a tool kit to better understand and evaluate the imaging resolution possibilities or our scopes. IMHO, these scopes are pretty well optimized, but the atmosphere holds the wild card. A possible way to work around the seeing is to take one’s scope on the road to places where the air is very steady. Seeing is typically best on mountain tops where the major observatories are located. Smart scopes that only weigh 5 to 10 kg make this more practically possible.

I’ve also heard that the skies in the Bahamas can be very steady too. Now that sounds like an

even smarter place to set up my tripod -- sittin' on the dock of the bay, watching the photons roll in …. ;-)

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Comments and Credits:

*If you want to go further into the subject of sampling, High Point Scientific has a great discussion here: Undersampling and Oversampling in Astrophotography.

>Sir Laurence Olivier as Hamlet. Available on Criterion. A performance for the ages.

> Moon and Mars Image: S&T's Sean Walker captured round Mars hanging below the shadow-filled craters of the lunar limb in this shot, taken post-egress. He used an Astro-Physics 92-mm Stowaway equipped with a 4x Tele Vue Powermate and Player One Uranus-C video camera. Stack of 800 frames recorded at 4:20 UT.

Sean Walker / Sky & Telescope

> Moon over Andromeda… Composite Image Credit & Copyright: Adam Block and Tim Puckett