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The Rate of Movement of Celestial Objects


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nrothschild Neil is an expert in several areas, including camera support Registered since 25th Jul 2004
Mon 10-Sep-12 09:53 PM | edited Sat 13-Apr-13 10:41 AM by nrothschild

The single most important number for the astrophotographer is the rate of movement of the celestial objects. Interestingly, many beginning astrophotographers vastly overestimate the exposure speed required, thus blaming soft images on the movement of the moon rather than technique or support. Therefore it is important to know these numbers in order to help interpret any problems with your images and to plan exposures.

If you are at 1/200s and the image is soft it wasn't the moon's fault- it was your fault

The table below includes some interesting numbers for most of the current and previous camera sensors. The executive summary is highlighted in yellow

The celestial objects all rotate around a 360 degree sphere in roughly 24 hours. Converted to arc-seconds the rate of movement is ~15 arc-seconds of a degree per second of time.

From that you need to determine the field of view at any given focal length and work in the width of the sensor's pixels, in terms of arc-seconds of distance against the apparent celestial sphere. I've done all the ugly math for you.

For any given sensor, I like to think in terms of rate of movement, in terms of pixels per second, per 100mm focal length. From that number it is easy to multiply by 1/100th the actual focal length to arrive at the rate of movement. I can do that in my head while out in the field.

Some rules of thumb have been developed to further simplify things but they necessarily assume a certain level of tolerance for blur. And that depends on your output size and personal expectations. My formula lets you pick your own personal threshold of pain

The rates indicated in the table assume declination = zero, which is the fastest possible rate. The 2nd table indicates an adjustment factor ("K") that is simply the cosine of the declination in degrees. The declination of 90 degrees is the North Pole, where the rate of movement is zero. In the southern hemisphere the declination of -90 degrees points to the South Polar Region.

To determine a precise rate of movement, compute your rate at declination = 0 on the first table. Determine the declination of your subject via astronomical maps or ephemera and find "K" closest to that declination. Multiply the result by "K" in the second table.

For the moon and planets, ignore the K adjustment...

The sun travels a precise path in the sky each year, plus or minus about 23.5 degrees declination. In the northern hemisphere it is at +23.5 degrees on the summer solstice and at -23.5 degrees at the winter solstice. You can see from the chart that even at its extremes the rate is only reduced by about 10 percent, which is insignificant for our exposure computations.

The orbits of the moon and planets are all slightly tilted in relation to the sun so they traverse slightly greater extremes. The moon's orbit is tilted about 5 degrees relative to the sun's orbit and therefore varies roughly +/- 28.5 degrees declination each month. As you can see from the table, K is still about 88% so again, not much help to the fixed mount astrophotographer.

That 2nd table is helpful for planning constellation shots that might be much closer to the poles.

Click on image to view larger version

The row labeled "Arc Seconds/Pixel/100mm (FOV of one pixel)" can also be very helpful. In this case divide the number by 1/100th your focal length. For example, my D300 at 850mm results in a "Field of View" of 11.3/8.5 = 1.33 arc-seconds per pixel.

On Jan 4, 2011 Jupiter had an apparent diameter of 46.9 arc-seconds when I shot it with my D300 and 500/4 + TC17E-II working 850mm. Just for laughs . Exposure was 1/50s f/11 ISO 200.

According to the calculation Jupiter would span 46.9/1.31 = 35.8 pixels. The image below (a pixel-peeping 100% pixels) measures 32x31 pixels. It is not clear to me that the outer perimeter of Jupiter was fully imaged - I kept the exposure low to bring out the equatorial bands as best possible. Had I more fully exposed the image I would have expected something much closer to the computed 35.8 pixels.

Click on image to view larger version

More focal length!!!

The same number can be used to compute the diameter of the moon, in terms of pixels, at any given focal length. The moon averages about 30 arc-minutes in diameter (1800 arc-seconds) although it varies by 10% or so due to the eccentricity of its elliptical orbit.

As an example of a rate of movement computation, for my D300 working 1000mm, I multiply 1.313 * 10 = 13.1 pixels per second. At certain times of the month (at extremes of declination) it could be as little as ~11.5 pixels per second.

I should then be able to shoot an image at 1/15s with slightly less than 1 pixel of blur. At 1/20s I am at ~2/3 pixel of blur, with some margin of comfort. And of course, somewhere around those speeds and focal lengths the gentlest breeze can become quite problematic.

The first table includes sidereal rates and lunar rates. As you can see, the differences are quite insignificant, on the order of 3%, and can be disregarded.

As proof of concept I offer this image of the 18 day 10 hour moon, shot at 1/20s at 1000mm on a D200. According to the above formula I would expect 12 pixels of movement per second, requiring about a 1/15s exposure for one pixel of blur.

Click on image to view larger version

The story behind that image is that for many years I attempted to put together a complete set of daily phases of the moon, well synchronized. For a variety of reasons this is very difficult to do and requires a lot of luck in terms of weather, libration of the moon, and time and opportunity. After a couple of years I learned that the best way to do this is with a series of long runs of a week or more.

In late August of 2007 I started shooting nightly images (without this plan in mind), the weather held, and I managed to shoot the 16th through 27th days, except for day 17 where I had very marginal weather. But I got 11 of 12 days, all in reasonably good clear weather. That required unheard of clear weather at a bad time of year for that. Late August is generally very humid and cloudy (as it was this year and it only broke yesterday).

As a bonus, that time of the year is perfect for that series, the most optimum days falling roughly in the middle of the series. And further, the late summer/early fall somewhat favors a long run wrapping the 3rd quarter because there are more favorable days (with the moon north of the ecliptic) in that particular cycle than other times of the year. This is due to some complex celestial mechanics involving the positioning of the ecliptic.

The 16th day is a few days after full, and from there the moon steadily wanes into a thin crescent. At the 3rd quarter the moon is due South at sunrise. After the 3rd quarter (~21st day of the lunation) the moon rises quite late, and in the final days it is not nearly due south and highest in the sky at dawn. In fact it has just risen and is quite low.

So essentially each day the moon is lower and lower in the sky at dawn, and it is also a thinner, dimmer, more difficult to render crescent. As the moon moves toward the thin crescent the exposure steadily drops each night.

I believe that at least with my own DX cameras it is very important to shoot the moon at base ISO whenever possible and with the D200 that is ISO 100. The moon is an extraordinarily noisy subject. I was using a TC-301 with my 500/4 Ai-p. That TC is probably about comparable to the TC20E-II… not a great TC. I generally shot the moon at f/11, or down one stop. That TC (and the TC20E-II) could probably use another stop (f/16) but for practical reasons of exposure speed and diffraction I figured f/11 was maybe optimum and certainly easier than f/16.

The exposure of a full moon high in the sky is generally about one stop down from sunny 16 (sometimes called "Luny 11"). It drops quite a bit only a day or two either side of the full moon. The result was an exposure of 1/30s f/11 at ISO 100 only a few days after full.

As the sequence proceeded each night the exposure generally dropped a bit (with an interesting exception documented in my gallery that I cannot explain). Below the 1/30s to 1/20s level I increased ISO to 200, which was as far as I would go.

When I was back to about 1/30s or so I threw in the towel on that TC and switched to a TC14B (700mm), buying me the stop I needed. On the 27th day I dropped to 500mm mainly because I knew from experience that that thin crescent would yield no additional detail above 500mm. Craters are barely discerned.

That is just the briefest explanation possible why most of my own lunar imaging is done around 1/30s or so, or it was when I used the D200. The D300 bought me a full stop by virtue of its higher base ISO.

The 28th day was probably clouded out and I had to return home. The last 5 days or so was shot at a friend's house at the New Jersey shore where I knew I would get the low horizon I needed to complete the series as best I could. And I think that next morning was likely clouded out. In a future post I'll talk about chasing very new and old moons.

All this to explain the thought process I went through to shoot the series and try to explain how I used my tables to push the edge of the envelope. You can see the results of that 12 day marathon here in my Daily Phases gallery. Where the EXIF indicates 800mm that was the closest I could get, using the 500/4P + 1.4x TC14B, with Nikon's limited non-CPU focal length tables (no 700mm option). In all other cases the exif times, dates and exposures are accurate.

I'll probably never get another run of good weather at the right month of the year (with that access to the horizon) quite like that again, and even if I do I'm not sure I want to get up at 3am for a solid week straight to try to do it . It was a tough week

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