Astrometry 101 - How long of exposure can my image frames be?

There is no hard and fast rule other than long enough to get  reasonable SNR on reference stars and your target. Sets of images can be can be easily aligned (process called image registration) and stacked in a number of ways such as add, median combine, averaging. The goal of astrometry is position measurement, not photometry. With photometry, the higher the SNR, the better the result, with 50 highly desired. But such SNRs are not so practical for faint objects. For astrometry, the SNR of the target should be at least 5. You should be able to detect movement of the target with at least two observations to ensure the target is not a star, if performing comet or asteroid astrometry.

A general rule of thumb for exposure length is the time it takes for the moving object to move one pixel.

To determine this you need to know how fast an object is moving across the sky.

pixel scale  = (206.265) * (pixel size in microns) / (focal length in mm)

If your pixel scale is 2.5 arc sec/pixel and your target is moving at 0.01 arc sec/sec, then you can expose for up to (2.5 arc sec/pixel) / (0.01 arc sec/sec) or 250 seconds for one pixel movement.

Depending on light pollution in your area, your images may be background limited. Light pollution causes the same effect  in your images as you see as the sun comes up in the morning, the sky brightens and stars fade and lose SNR. Bright moonlight causes the same stellar extinction phenomenon as the sky background dominates the stellar light signal ad they disappear.

The large aperture survey telescopes may only expose for 30 seconds in order to reach a certain magnitude desired such as magnitude 21. A smaller aperture scope such as a 0.3 meter or 12 inch scope will be limited by the sky or object movement and pixel scale.

To go for fainter comets and asteroids, you can do a number of things:

• use a focal reducer to get closer to optimal pixel scale.
• stack images
• bin camera pixels to increase effective pixel size
• get a larger aperture scope 

To find how fast an object is moving across the sky, sky motion, you can use a number of resources.

Minor Planet and Comet Ephemeris Service: http://www.cfa.harvard.edu/iau/MPEph/MPEph.html

You can select to display motions as:  "/sec   "/min   "/hr   °/day

You can also select to separate R.A. and Decl. sky motions instead of total sky motion to squeeze a little more exposure time depending on whether RA (or x) or Decl. (or y) is more limiting.

A planetarium program such as TheSky6Pro displays sky motion in the info dialog of solar system bodies including the sun, moon, planets, comets, and minor planets.

JPL has an ephemeris service or HORIZONS Web-Interface: http://ssd.jpl.nasa.gov/horizons.cgi

Generally the closer to earth or sun an object is, the faster its sky motion and brightness.

A faint target can also be moving more quickly and you may need a larger image stack or shorter exposures.

Observing known comets and asteroids makes this decision process easier to predict and plan.

There is the possibility that a new or unknown object may show up in your images, with main belt asteroids being the most likely. By using an average sky motion for these objects, you can limit your exposure times based on that, so that an object will not produce a long streak in an image, but a measurable centroid. The larger aperture scopes have a distinct advantage over smaller scopes here because of their immense light gathering power not limited by object motion but by how deep or faint they want to survey. The general idea is to gather three or more images (or sets of images if stacking)  separated by 10 to 15 minutes each so that any moving objects will readily show up when blinking images or when using software detection. Distant objects such as Pluto, Centaurs, Kuiper Belt Objects (KBOs), or TransNeptunian objects (TNOs) may require time separation of an hour or more to detect them. Large aperture survey telescopes have the advantage of stopping down their focal length and flattening the image field so as to image larger portions of the sky and more efficiently survey the sky.

Reference:

IAU Minor Planet Center Guide to Minor Body Astrometry:

http://www.cfa.harvard.edu//iau/info/Astrometry.html

Astrometry 101 - Pixel Scale

Every astrometry observer and measurer needs to know the pixel scale of their images. This is the angular width and height of a pixel. Pixel scale tells how much of the sky is covered by one pixel.  Many CCD cameras have square pixels (equal height and width). If the camera pixels are not square, then both dimensions musts be considered. Normally width is denoted by x and height y.

To calculate pixel scale, one needs telescope focal length and pixel width or height in microns.

pixel scale or sampling in arcseconds = (206.265) * (pixel size in microns) / (focal length in mm)

or

focal length = (206.265) * (pixel size in microns) /  pixel scale

The constant 206.265 converts microns to mm and radians to arc seconds.

A micrometer or micron is one millionth of a meter, or equivalently one thousandth of a millimeter (mm)

[360 deg / (2 pi radians)] x (3600 arc sec / deg) x ( 1 mm / 1000 microns) = 206.26480624709635515647335733078 ~= 206.265

or just about 206 as many use as rule of thumb.

This is a straightforward calculation if you know the two input parameters.

Focal ratio is the ratio of the focal length to telescope aperture

Focal ratio = focal length / aperture

For example, if focal length = 3048 mm and  aperture = 30.48 mm, Focal ratio = 10 or telescope is at f /10.
If you have 9 micron square CCD pixels, then your pixel scale is  206.265 x 9 / 3048  or 0.6 arc sec per pixel.

This has meaning when compared to your "seeing" measured in arc secs of FHWM (Full Width Half Maximum) of the stars in your images and is referred to as "sampling". You have near optimal sampling when your pixel scale is about half of the value of your seeing. If your pixel scale is less than optimal, the sampling is termed "under sampled". If your pixel scale is larger than optimal, the sampling is termed "over sampled".

Focal reducers reduce the effective focal length of your imaging setup and increase your pixel scale.
Binning your CCD camera imaging chip 2x2 (versus high resolution 1x1) doubles the size of each pixel and increases your pixel scale.

Barlow lenses increase the effective focal length of your imaging setup and decrease your pixel scale.

Astrometrica calculates these parameters for you if you can interpret the values provided after doing a plate solution. The default values of its configuration settings must be initially modified by a new user for their imaging setup. Thus, one must be able to calculate or estimate a focal length as an input parameter for this program.

This is  a copy of several lines in a recent Astrometrica.log file from my own astrometry:

The column labeled FHWM reports the seeing of the stars in the image and shows that sthe seeing is about 5 to 6 arc seconds. The pixel size is calculated at 2.58 arc sec or about half the seeing. I was using 2x2 binning on a SBIG ST-8XME CCD camera. There is a f/6.3 focal reducer a filter wheel and an SBIG AO-7 unit in the imaging setup, but due to the spacing of the components resulting in an effective focal ratio of near 4.7. This also gives a nice size field of view to capture more stars which is better for accurate astrometry. The SNR of the stars measured for astrometry is also given. It is important to have sufficient SNR for an accurate centroid calculation of stars and the object to be measured. The Fit RMS is the most important columns because it shows the bottom line of the astrometric calculation and the goal is to have fractional arc second accuracy is all measurements..

Pixel scale is used to determine how long of an exposure one can take of a moving object. That will be the subject of another topic.

The larger your pixel scale the longer you can expose your target per image.

The larger your pixel scale, you provide allow more photons from your target to fall on a pixel which improves the SNR of your target, improves the calculation of the target centroid in the image, improves astrometric precision, and results in smaller residuals.(only up to optimal)

Too high a pixel scale beyond optimal however, is not good for astrometry because stars will be undersampled and are not spread out over one or more pixels. It would be like trying to accurately measure the position of a fly with a ruler with only inch or centimeter tick marks. and is of no astrometric value no matter how reference stars you have imaged..

Reference:

IAU Minor Planet Center Guide to Minor Body Astrometry:

http://www.cfa.harvard.edu//iau/info/Astrometry.html