Before photographing an image of the deep sky or a planet, some astronomical calculations are necessary to assess the equipment to be set up to photograph the object. For example, we can ask whether the entire object fits within the field of view of the CCD camera. In addition, before buying equipment dedicated to astrophotography, it is very interesting to know and master these calculations, because they will allow you to make a better choice. Here is a list of interesting calculations to know before undertaking the acquisition of an image. **The dimension of a deep sky object**

Astronomy books and planetariums present the dimensions of deep sky objects and planets in degrees, minutes and seconds of arc. Here is what these dimensions represent:

Circumference = 360 degrees (360^{o})

1 degree (1^{o}) divides into sixty arc minutes (60 ′) and one minute (1 ′) contains sixty arc seconds (60 ″)

1^{o} = 60 ′ = 3600 ″**Field of view of the imaging camera in arc minutes**

Formula Cv = S x 3438 / f

Cv = Field of view in minutes of arc

S = Dimension of one side of the die in mm

f = focal length of the telescope in mm

Other formula S = 205 x P / FL

S = Field of view covered by one pixel in arc second

P = Physical dimension of a pixel in microns

FL = Focal length of the telescope in mm

It remains to multiply by the number of pixels in width and height. To convert to minutes of arc divide by 60.

When we know the field of view of the camera in minutes of arc, we can compare it with the dimension of the object to be photographed in minutes of arc. Then we will know if the object is in the field of view of the camera.**Value of a micron in millimeter**

1 micron = 0,001 mm

Very useful for performing various calculations. For example, if we want to know the width of the matrix in millimeters. If we know the number of pixels in width of the matrix, for example 3 pixels and each pixel is equal to 326 microns, here is the width of the matrix in mm:

3326 pixels x 5,4 microns x 0,001 mm = 17,96 mm**Focal aperture and exposure time**

Hobbyists who practice photography know that the exposure time depends on the focal aperture. A large focal aperture favors a shorter exposure time. In the field of deep sky imagery, the focal aperture is therefore an important element to consider since the tracking (or autoguiding) of the object to be photographed is greatly facilitated with a shorter exposure time.

The focal aperture is represented by a factor f / followed by a number. For example f / 5, f / 10. The smaller the number, the larger the focal aperture. To compare the exposure time of one focal aperture against another, take the focal aperture number and square it. Here is an example :

To compare an exposure time of 20 minutes with a focal aperture of f / 10, here is the exposure time equivalent to the focal f / 5:

f / 10: 10 * 10 = 100

f / 5: 5 * 5 = 25

100 / 25 = 4

The exposure time at f / 5 will be 4 times less for an exposure time of 5 minutes instead of 20 minutes at f / 10.**Telescope focal length and focal length reducer**

With the same setup, a telescope with a shorter focal length will give a larger field of view. So to increase the field of view (or decrease the magnification) of a telescope we can use a focal reducer. The focal reducer also makes it possible to increase the focal aperture of the telescope. Here are examples with a 2000mm focal length telescope open to f / 10.

50% focal length reducer: the focal length of the telescope will increase to 1000 mm and the focal length to f / 5

63% focal length reducer: the focal length of the telescope will increase to 1260 mm and the focal length to f / 6.3

It is easier to follow a deep sky object with a telescope of shorter focal length. So favor a focal length of 1000mm or less with the majority of telescope mounts. Only very precise (and therefore very expensive) mounts allow good object tracking with a telescope with a focal length of 2000 mm and more.**The resolving power of the telescope**

Formula: Ps = 120 / diameter of the telescope in millimeters

Ps = resolving power in arc seconds

The telescope's resolving power refers to the telescope's ability to distinguish two contiguous objects (eg double stars).**The sampling of a pixel in arc second**

Formula: (205 * dimension of a pixel in microns) / focal length of the telescope in mm

The result of the formula is expressed in seconds of arc. It takes 2 pixels to resolve the image. For example, for a sampling of a pixel of 2 ″ of arc, the theoretical possible resolution (or resolving power) of the camera is 4 ″ of arc. So the sampling can be compared to the resolution of the image and to the resolving power of the telescope expressed in seconds of arc. Here are examples of calculations that will help understand the importance of knowing the sampling of a camera pixel. This table represents an analysis that I performed from my current equipment.

EDGE HD 800 or Orion 80 ED and Atik 383L-M telescopePlanetary: ZWO ASI120MM | 2x Barlowand ASI120MM | Without reduction.focal length | Reducerfocal f / 6.3 | GlassesOrion 80 ED |

Telescope focal length | 4064 | 2032 | 1280 | 480 |

Telescope diameter in mm | 203,2 | 203,2 | 203,2 | 80 |

Telescope focal length | 20 | 10 | 6,3 | 6 |

Dimension of a pixel in microns | 3,75 | 5,4 | 5,4 | 5,4 |

Camera field of view in arc minutes | 4,04 x 3,03 | 30,2 x 22,74 | 47,94 x 36,09 | 127,84 x 96,25 |

Sampling of one pixel in sec. bow | 0,19 | 0,54 | 0,86 | 2,31 |

- Separating power of the camera | 0,38 | 1,09 | 1,73 | 4,61 |

Bin 2 × 2 mode: One pixel sampling in sec. bow | n/a | 1,09 | 1,73 | 4,61 |

- Separating power of the camera | n/a | 2,18 | 3,46 | 9,23 |

Scope of separation of the telescope in arc second | 0,59 | 0,59 | 0,59 | 1,50 |

Periodic error mount (PEC and autoguiding active) | n/a | 5 | 5 | 5 |

- Celestron CGEM (in arc seconds) - Tracking quality: Very good |

**Sampling for the deep sky**

Here is the recommended sampling for photographing distant deep sky objects (requiring a medium field of view or for the deep sky) and those closer to us (requiring a large field of view from the camera). To divide these two categories of deep sky imagery, we will base ourselves on the focal length of the telescope (more than 700 mm for the deep sky and 700 mm and less for objects requiring a large field of view).*Images for deep sky objects requiring a medium field of view or for deep sky*

- The focal length of the telescope is over 700m
- Camera sampling should be 1 ″ to 3 ″ arc per pixel
- The tracking (or autoguiding) accuracy of the equatorial mount should be 6 ″ of arc and less.

In the majority of nights, the air turbulence allows a resolution of 2 ″ to 3,5 ″ of arc. So the ideal (and minimum) camera sampling is 1 ″ of arc which will give a possible resolution (or resolving power) of the camera of 2 ″ of arc. It compares to air turbulence under very good viewing conditions. In deep sky imagery, do not go down to a sampling lower than 1 ″ of arc, because the theoretical resolution of the camera will be lower than the air turbulence (2 ″ of arc) which will not bring no benefit. Sampling up to 3 ″ of arc (possible resolution of 6 ″ of arc) is considered very satisfactory for the deep sky,

In the examples shown in the table above, using the Edge HD 800 telescope without a focal reducer (f / 10) and with the f / 6,3 focal reducer both provides camera oversampling. Indeed, the sampling is 0,54 ″ of arc at f / 10 and 0,86 ″ of arc at f / 6,3. They are both less than the 1 ″ arc minimum. We will see below that we can solve this problem by using the **Bin Mode**.

In order to appreciate the maximum value of 3 ″ of arc for this category of images (medium field and deep sky), here is the nebula of the veil NGC6960:

*Click on picture to enlarge*

The image was taken in a site without light pollution with high-end equipment with the following characteristics:

- The camera sampling is 3,38 ″ of arc (very close to the maximum value of 3 ″ of arc)
- The focal length of the telescope is 1095 mm
- 2 × 2 Bin Mode was used for the luminance image (to adjust sampling to 3,38 ″ of arc)
- Air turbulence was 2 ″ of arc and less (3/5)
- Sky transparency was above average (4/5)
- Mount tracking accuracy is less than 2 ″ of arc

With these ideal conditions, the highest value (or the weakest link) is the camera sampling of 3,38 ″ arc. Look at the finesse (precision) of the details revealed. This image clearly shows that we can go up to a sampling of 3 ″ of arc for the deep sky and the medium field of view.*Images for deep sky objects requiring a large field of view*

- The focal length of the telescope is 700mm or less.
- Camera sampling can be more than 3 ″ arc per pixel, without exceeding 5 ″ arc per pixel.
- The tracking (or autoguiding) accuracy of the equatorial mount can be more than 6 ″ of arc.

In order to appreciate the recommended values for this category of images (large field of view), here is the “Les Pleiades” nebula and cluster M45:

*Click on picture to enlarge*

The image was taken in a site without light pollution with high-end equipment with the following characteristics:

- Camera sampling is 3,5 ″ arc
- The focal length of the telescope is 530 mm
- Bin Mode 1 × 1 was used for the luminance image
- Air turbulence was 2 ″ of arc and less (3/5)
- Sky transparency was excellent (5/5)
- Mount tracking accuracy is less than 2 ″ of arc

With these ideal conditions, the highest value (or the weakest link) is the camera sampling of 3,5 ″ arc. Look at the finesse (precision) of the details revealed. This image clearly shows that the recommended values can be used for this image category.**Bin Mode**

Bin Mode consists of grouping pixels to resolve the image. For example, Bin 2 × 2 will group 4 pixels to resolve the image (Bin 1 × 1 = 1 pixel, Bin 3 × 3 = 9 pixels). The image will be 4 times smaller, but the exposure time will be reduced by the same factor of 4, which is the advantage of Bin Mode.

Bin Mode is useful for balancing the resolution of the image with the optics used. For example, if the camera sampling is 0,5 ″ of arc in Bin 1 × 1, it will change to 1 ″ of arc in Bin 2 × 2 (2 pixels high and 2 pixels wide). The camera resolution will therefore be 2 ″ of arc, which is equal to air turbulence under very good viewing conditions.

In the examples presented in the table above for the Edge HD 800 telescope at the focal length f / 10 and f / 6,3 (with focal reducer), the use of the Bin 2 × 2 allows to obtain a sampling of the 10 ″ arc f / 1,09 and 6,3 ″ arc f / 1,73 camera. The Bin 2 × 2 will therefore allow the camera sampling to be adjusted to the recommendation for the deep sky (sampling between 1 ″ and 3 ″ of arc). So here are two great examples of using Bin mode.

For the Orion 80 ED scope which is used for the large field of view, the camera sampling in Bin 2 × 2 is 4,61 ″ of arc. We will also favor this Mode for this large field of view, because we can exceed 3 ″ of arc as sampling (without exceeding 5 ″ of arc) for telescopes with focal length 700 mm and less (in this case, the telescope has a focal length of 480mm). The image will be very well resolved for this large field of view. It will not appear blurry.

The fact of using the Bin 2 × 2 will make it possible to obtain four times more light than the Bin 1 × 1 or to decrease the exposure time by a factor of 4. The tracking of the object in autoguiding will also be facilitated. with the Bin 2 × 2, because the exposure time will be much less.

It should be noted that the Bin 2 × 2 mode which makes it possible to acquire four times more signal than the Bin 1 × 1 is not accessible for cameras with CMOS sensor. For more information, click on this link.**The tracking precision of the frame**

The most important element for deep sky imaging (long exposure time) is the tracking quality of the autoguider mount. It is therefore essential to know this precision.

You can now assess the tracking quality of your frame yourself using the software PHD Guiding 2. You can display the guidance graph in real time by activating the menu **Display | Display Graph**. I also recommend using the software PHD2 Log Viewer to evaluate in detail the quality of guidance of your mount. Before starting autoguiding, the star image log must be activated in PHD Guiding 2 (**Tools | Activate the star image log**) in order to make the autoguiding data accessible to the PHD2 Log Viewer software. Here is an example of the analysis I performed for my CGEM mount:*CGEM mount autoguiding analysis using the PHD2 Log Viewer software. Click on picture to enlarge.*

Looking at the graph, the maximum deviation in right ascension and declination is 5 ″ of arc (+/- 2,5 ″ of arc) for the total duration of the worm which is eight minutes. The RMS value (the average of the deviations) is +/- 1,84 ″ of arc or 3,68 ″ of arc in total for the entire imaging session (see the total RMS value at the bottom of the graph) . This represents a very good performance of the frame. There is also several other analytical information about the autoguiding session. Note that maximum peaks should not be considered, as they occurred at the start of the autoguider before it stabilized. I always wait for autoguiding to stabilize before starting an imaging session.

If you haven't purchased your mount yet, search the web for "periodic error *name of the frame* ". There is a good chance that an enthusiast will provide their assessment of the mount they own that matches the one you intend to buy.

Now looking at the table above in both columns **Without focal reducer** et **F / 6,3 focal reducer.** Although we have made sure to use the correct sampling of the camera according to the above parameters, the resolution of the image will be equal to the tracking accuracy of the mount which is 5 ″ of arc and not the resolution of the camera which is respectively 2,18 ″ of arc (without focal reducer and with Bin 2 × 2) and 3,46 ″ of arc (with focal reducer and Bin 2 × 2). Indeed, we cannot go down to the bottom of the tracking quality of the frame in autoguiding. So that's all the importance that must be given to the tracking precision of the frame.

To evaluate this performance, here is an evaluation grid for the quality of monitoring of a self-guided mount:

CUSTOMER EXPERIENCE | Tracking accuracy | Deep sky | Large field |

Excellent | 2-3 ″ arc tracking | x | x |

Very good | 5-6 ″ arc tracking | x | x |

Good | 8-12 ″ arc tracking | +/- | x |

Minimum | 15-20 ″ arc tracking | n/a | +/- |

*Source: for the first two columns, **The New CCD Astronomy**, Ron Wodaski, New Astronomy Press, page 218*

By referring to the examples above, you will be able to adapt these calculations with your equipment (sampling, Bin Mode and mount tracking accuracy). You will then know the maximum configuration to use in terms of image resolution, camera sampling, exposure time and tracking quality (autoguiding) of your mount.

**Camera cooling**

At each cooling of 6^{o} Celsius of the camera matrix, thermal noise (includes dark current and hot pixels) decreases by 50%. It is therefore an important element to consider when choosing a deep sky camera.

Formula: DB = (1-(1/(DT/6)*0,5)*100

- DB = Reduction of thermal noise in percentage
- DT = Temperature decrease in degrees Celsius relative to ambient temperature

*Here is an example : *

Decrease temperature by 20^{o} Celsuis compared to the ambient temperature:

- DB = (1-(1/(20/6)*0,5)*100
- DB = (1-(1/3,33)*0,5)*100
- DB = (1-0,15)*100
- BD = 85%

Also cooling the camera to a fixed temperature, for example -20º Celsius below 0º Celsius, greatly facilitates the production of blacks (Dark).

**Sampling for planet imagery**

For planet imagery, sampling should be less than 1 ″ arc (2 ″ arc resolution). Indeed, to find details in the structure of the surface of the planet, the resolution of the image must be lower than the turbulence of the air! The question that immediately comes to mind is how to get down to a resolution lower than air turbulence? This presents a good challenge for the planetary imaging enthusiast. To maximize the results, choose a night when the turbulence is low. Then take a lot of images (see section Suggested minimum exposure time of this site for more details). We then select the images where the turbulence is very low. Finally, we assemble the chosen images to produce a composite image of the planet that will have a resolution of less than 2 ″ of arc.

In the imaging of planets, the resolving power of the telescope is the most important element. Of course, it must be less than 2 ″ of arc. The resolving power of the telescope depends on a single element which is the diameter of the instrument (see formula above). So you should use a telescope with a diameter of at least 100 mm (4 inches) which will give a resolution of 1,2 ″ arc (120/100). Then you have to balance the sampling of the camera to achieve the resolution of the telescope. The camera sampling should therefore be 0,6 ″ of arc (2 pixels = 1,2 ″ of arc). The element to look for is the focal length of the telescope necessary to be able to bring the sampling of the camera closer to the resolving power of the instrument. We will therefore adapt the camera sampling formula as follows:

Focal length of the telescope in mm = 205 * dimension of a pixel in microns / sampling

Referring to the table above, the necessary focal length of the telescope at camera sampling of 0,295 (50% of 0,59 ″ arc which is the resolving power of the Edge HD 800 telescope) in use with the camera ZWO ASI120MM will be:

205 * 3,75 / 0,295 = 2606 mm

It will therefore take a telescope with a focal length of 2606 mm to balance the sampling of the camera to the resolving power (or resolution) of the telescope. In the table, the telescope has a focal length of 2 mm. To extend the focal length of the telescope close to the reference value, we will therefore use a Barlow of 032x (1,3/2606). As it is not commercially available, use a Barlow 2032x which is closest to the reference value. The resulting focal length of the telescope will be 2mm (4064 x 2032) and the camera resolution 2 ″ of arc which is closest to the telescope's resolution of 0,38 ″ of arc. The effective resolution of the image will then be 0,59 ″ of arc, as it cannot be lower than the resolution of the telescope. See the result of the calculations in the table above.

It should be noted that the tracking quality of the mount is a much less important factor for photographing the planets, as the exposure time is very short (less than a second).

Here are two sample images (the planet Jupiter and Saturn) taken with the above recommendations in mind:

In conclusion to this section devoted to astronomical calculations, by using these few formulas or simple calculations (at least not too complicated!), You will be able to maximize the use of your equipment. I therefore recommend that you master these calculations for your images of the deep sky and the planets. In addition, before buying equipment dedicated to astrophotography, it is very interesting to know and master these calculations, because they will allow you to make a better choice.

Richard Beauregard

Sky Astro - CCD

Revised 2022/08/14