[Content moved from http://photographingtransitofvenus.wordpress.com/]

Summary: Estimate the apparent image size of the solar disk and Venus in pixels for certain camera and simulate the ideal observed image.

Disclaimer:In this post, all discussions are based on the apparent image SIZE of the subjects — solar disk and Venus. I have not discussed any issue of resolving fine details, which is very important, and that shall be discussed in another post. Whether you will be able to capture the fine details such as the Granule, Lanes and Bright points on the solar spots, will depend on the ability to resolve fine details called the “spatial resolution”. The spatial resolution that you can achieve depend on the quality of the lens, quality of the sensor, pixel size of the sensor, the focus, presence (or absence) of vibration during the exposure, and lastly but perhaps the most limiting of all in the case of astro-photography is “seeing.” For today’s discussion we are not concerned with any of those, assuming our camera is the perfect camera and there is not air-turbulence.

You may have been wondering whether the focal length of the lens that you have will be “good enough” (see disclaimer) for taking photographs of the transit. The answer to that question really depends on what you want to photograph and the focal-length of the optics that you have at your disposal. While a 300 mm lens attached to a camera with APS-C sensor will suffice (not ideal) for taking pictures of the entire solar disk with Venus (see the simulated images below & some photographs of the Sun that I took during the past few days), you will need focal length beyond 2000 mm if you want to take frame-filling close up Venus in front of a partly visible solar disk.

It is not very hard to calculate the focal length of the optics required for your desired photograph.

It is a common practice in astronomy to specify the apparent sizes of celestial objects in terms of their angular diameter as seen from the Earth. The units used to measure angular diameter are radians, degrees (°), arc minutes (‘) and arc seconds (“) (see this for more details on the units and/or see the appendix for a schematic of the relationship between the units.) Although the average apparent angular diameter of the Solar disk is about 1920″, its diameter will be about 1891″ during the transit. The apparent size of Venus varies between 9.9″ during superior conjunction and 68″ during inferior conjunction. On the day of the transit its apparent size will be about 58″.

Using simple trigonometry, the (image) diameter of the solar disk projected on the sensor is calculated as:

(millimeters) 

where, is the focal-length of the optics in millimeter () and is the apparent angular-diameter of the subject (solar disk and/or Venus) in arc seconds.

The number of diameter-pixels in the image is calculated as:

where, is the pixel pitch (size of individual pixels) in microns ().

These formulae were used to calculate the image size of the Sun and Venus as observed through a camera with APS-C sensor and with a 35mm full-frame sensor for lenses with focal lengths of 300mm, 640mm, 1000mm and 2000mm. The resulting image sizes (in millimeter and in terms of number of pixels) are shown in the tables below, and the associated simulated images are shown following the tables.

Canon Rebel XT [Canon APS-C sensor]:
Pixel pitch = 6.4 
Sensor size = 22.20  x 14.80 

Table 1: Estimated image size of the solar disk and Venus for different focal lengths for Canon Rebel XT.

Focal length	Solar disk diameter (mm)	Pixels in solar disk diameter	Venus disk diameter	Pixels in Venus disk diameter
300 mm	2.7504 mm	430	0.0844 mm	13
640 mm	5.8675 mm	917	0.1800 mm	28
1000 mm	9.1679 mm	1432	0.2812 mm	44
2000 mm	18.3358 mm (cropping)	2865	0.5624 mm	88

Simulated images:

F = 300 mm on Canon Rebel XT.

F = 640 mm on Canon Rebel XT.

F = 1000 mm on Canon Rebel XT.

Canon 5D Mark -III [Full format sensor]:
Pixel pitch = 6.08 
Sensor size = 36.00  x 24.00 

Table 2: Estimated image size of the solar disk and Venus for different focal lengths for Canon 5D Mark-III.

Focal length	Solar disk diameter (mm)	Pixels in solar disk diameter	Venus disk diameter	Pixels in Venus disk diameter
300 mm	2.7504 mm	452	0.0844 mm	14
640 mm	5.8675 mm	965	0.1800 mm	30
1000 mm	9.1679 mm	1508	0.2812 mm	46
2000 mm	18.3358 mm	3016	0.5624 mm	92

Simulated images:

F = 300 mm on Canon 5D Mark-III.	F = 640 mm on Canon 5D Mark-III.	F = 1000 mm on Canon 5D Mark-III.
F = 2000 mm on Canon 5D Mark-III.

How accurate are these calculations?

One must always verify the accuracy of one’s calculations. I compared the calculated image size of the solar disk to the actual size of the solar disk that I observed in the photograph (see below). I took this photograph using my Canon EF 70-300mm f/4.5-5.6 IS USM lens (used at 300mm) attached to my Canon Rebel XT body. The calculated size of the solar disk diameter (Table 1) is 2.75 mm which corresponds to 430 pixels assuming the pixel pitch is 6.4 microns. The actual number of pixels observed in the photograph is about 417 pixels which is very close to the calculated number. There could be several reasons for this (small) error, such as the true focal length of my zoom lens might not be actually 300mm (which is not that uncommon for inexpensive zoom lenses), the pixel pitch may not exactly be 6.4 microns but a smaller number than that, etc.  Nevertheless, I think, we can get a very good idea about the size of the solar disk and Venus that we can obtain using the above formulas. I hope it will be useful to you in planning your photograph for the transit of Venus.

References:

• “400 years of Sunspots,” McDonald Observatory, The University of Texas at Austin, 2011: http://www.as.utexas.edu/mcdonald/scope/poster/sunspots.pdf
• Wikipedia article on Minute of arc: http://en.wikipedia.org/wiki/Minute_of_arc
• “Observing Sunspots,” Jim Scala, 2000: http://www.eastbayastro.org/2000/0700/r0700-7.htm
• “Photographing the transit of Venus,” Andrew Cooper, A Darker View, May 20, 2012. http://darkerview.com/wordpress/?p=4161
• “Transit of Venus,” Steve Owens, Dark Sky Diary, May 23, 2012. http://darkskydiary.wordpress.com/2012/05/23/transit-of-venus/

Appendix

Relationship between the different units for measuring angular diameter:

[This content was originally posted by me at http://photographingtransitofvenus.wordpress.com/, a blog dedicated to the 2012 transit of Venus]