Object
From chi and h
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There are two main properties of objects of astrophotography that tell us how to take images of them. First is their apparent size, in particular whether they are unresolved and appear point-like, or whether they are resolved and fill an area in the image plane. Second is their brightness. For faint objects we will worry about collecting enough photons to even detect the objects, while for bright objects the main worry is not to overexpose. How we express the brightness of an object depends on whether it is resolved or not.
The brightness of objects
Magnitudes
Since ancient times, the brightness of stars has been expressed in "magnitudes". The brightest stars are "first magnitude" and the faintest stars visible to the naked eye are "sixth magnitude". This magnitude scale goes back to Ptolemy (137 AD), but Hipparchus, around 130 BC, used a similar scale in his star catalogue [1].
Much later, the magnitude scale was put on a sound mathematical footing. Since the human eye responds logarithmically to light, the magnitude is also a logarithmic measure of the number of photons or the energy of the light received. A difference of five magnitudes corresponds exactly to a factor 100 in energy. From a modern scientific perspective, an annoying feature is that the magnitude scale runs backwards: bigger numbers indicate fainter stars.
Eventually, astronomers started to ask not only "how bright?" but also "what colour?". A white star - one of spectral type A0 - has the same magnitude in any colour filter used, but a red star will have a smaller (brighter) magnitude in a red filter than in a green or blue filter. When a magnitude is stated, the colour filter to which it applies should also be stated. The most commonly used magnitudes are:
- visual: as seen by the human eye without filter,
- photographic: as recorded by blue-sensitive monochromatic negative film,
- V: as seen through the V filter of the UBV filter sequence (wavelength 550 nm, bandwidth 90 nm [2]).
Although V is not the same as "visual" and neither is the same as the green filter in a consumer camera, they are roughly the same.
Source flux
In the 20th century radio astronomers took the chance of ignoring the traditional logarithmic upwards-down magnitudes and expressed their measurements in units from the international system of physical units (SI units), such as metre, second, Watt, and Hertz. They also thought in the frequency of radio waves rather than the wavelength of light. Today, professional astronomers are no longer segregated by the part of the electromagnetic spectrum they observe, and so we are now able to look up the conversion from V magnitude to SI units [3]:
- S = 3540 · 10-0.4 V Jy
- 1 Jy = 10-26 W / (Hz m2)
The unit of the source flux is called "Jansky" after the pioneer radio astronomer Carl Jansky. The source flux is the energy (Watt) contained in the radiation received over a certain bandwidth (Hertz) and in a certain collecting area (square metre).
Solid angle
If the object in question is not resolved, then its flux is all we need to judge its brightness. However, if the object is resolved, then we are more interested in the surface brightness, i.e. how much light per unit area on the celestial sphere (per "solid angle") there is. This is where the logarithmic, reversed magnitude scale becomes a real headache. It is far simpler to think in Jansky fluxes, and simply to divide them by the "solid angle" subtended by the source.
Although we tend to measure angles in degrees, arc minutes, or arc seconds, the natural unit for angles is actually the "radian". If you draw a circular arc of the same length as the radius then the angle is one radian (1 rad).
- 1 rad = 180° / π = 57.29578° = 3437.747' = 206264.8"
This can be extended from one-dimensional arcs on the unit sphere to two-dimensional areas on the unit sphere. If you mark an area of 100 square centimetres on a football of 10 cm radius (i.e. the area is the square of the radius) then the marked area, viewed from the centre of the ball, has a "solid angle" of "one steradian" (1 sr).
- 1 sr = 1 rad2
Both radian and steradian are dimensionless, which often leads to confusion as giving the unit "rad" or "sr" is somewhat optional. The whole sphere has a solid angle of 4π, and 2π in the sky above the horizon. (Hence, there's more than one pie in the sky.)
We will usually encounter solid angles only with respect to the image pixels. These are quite small, so that the nightmare of projecting a rectangular field of view onto a spherical sky is not a problem. We can simply calculate the solid angle of a pixel as its size squared divided by the focal length squared.
Surface brightness
The surface brightness of an object is simply its total flux S divided by its solid angle A.
- B = S / A
The unit would be Jy/sr. However, with a steradian being a rather big area on the sky it is convenient to put in a factor of a million and use MJy/sr. It is useful here to carry out an example calculation. Say, Jupiter has a brightness of -2.8 magnitudes and an apparent equatorial radius of 24.4". The polar radius is smaller at 22.8". The area of an ellipse is π times the product of the two radii.
- A = π · 22.8" · 24.4" = 4.11 · 10-8 sr
- S = 3540 · 101.12 Jy = 46700 Jy
- B = S / A = 1.14 · 106 MJy/sr
Strictly speaking, this is the average surface brightness. Objects tend to have brightness variations over their extent: Jupiter has dark bands, the Sun and planets have limb darkening, the Moon has dark maria and bright highlands. This becomes very serious for deep sky objects: a galaxy or bright nebula is very much brighter in its core than on its periphery. Dark nebulae are defined by being darker than the surrounding sky, hence the concept of surface brightness is no use for them.
Camera calibration
The object sends light - measured as magnitude V or flux S - through our optics into our camera. The raw image we download to the computer consists of pixels, each with a brightness value in what we might call RIU - raw image units. There is proportionality between the RIU and the flux S, but what are the effects of exposure time, aperture, etc.?
Consider a single image pixel, one at the centre of a stellar image. Like any other pixel, on the sky it has a solid angle
- A = (δ/f)2 sr
If all the light from the star falls into this single pixel, it sees a surface brightness
- B = S / A
If the pixels are small enough for the light from the star to be distributed over a small neighbourhood of pixels, then it will form something similar to a two-dimensional Gauss curve of surface brightness. If the Gaussian has a full width at half maximum of w (in radian), then the peak brightness is
- B0 = 0.88 S / w2
The surface brightness B tells us how much energy per unit of time, unit of bandwidth, unit of aperture area, and unit of solid angle, the object sends our way. In addition, we know that the light is made up of photons, each with an energy related to the frequency of the light. Multiplying the surface brightness B with the bandwidth Δν, pixel size A on the sky, aperture area, and exposure time t, and dividing by the photon energy hν gives us the number of photons coming into the image pixel during the exposure. These photons are then converted into electrons, subject to a quantum efficiency of less than one.
- nγ = B πΔν/(4hν) (δ D/f)2 t
- ne = B πΔν/(4hν) QE (δ D/f)2 t
The camera amplifies the voltage or currant generated by the electrons and then converts the amplified signal from analogue to digital units. The resulting units are commonly called ADU (analogue-digital units) and the conversion factor from ADU to electrons is called the ADC factor (analogue-digital conversion factor). Our ISO setting is part of the ADC factor. The camera processing will in general scale the results again before we get an actual raw image downloaded to the computer, with Bayer matrix interpolated or binned, and possibly with some colour balance correction applied. Let's call the effective scaling factor CP (for "camera processing"):
- ADU = B πΔν/(4hν) (QE/ADC) (δ D/f)2 t
- RIU = B πΔν/(4hν) (CP QE/ADC) (δ D/f)2 t
Some of the factors in the equation above are of interest to us, because we have control over them, such as the exposure time, ISO setting and the f ratio. The rest are in themselves of no interest to us. With a view to determining one constant per camera empirically, we re-write
- RIU = d ISO B (D/f)2 t
- RIU: raw image units
- d: camera efficiency
- ISO: ISO setting
- B: surface brightness
- D: aperture
- f: focal length
- t: exposure time
Once we have determined d for each camera by taking an image of a star of known brightness, we can predict roughly how bright any object will appear given the camera and exposure parameters. This will work as rough prediction only, as there are significant and unpredictable effects like atmospheric extinction loss of light in the optics. For proper photometry, one has to calibrate each image from known stars in the image. In addition, there is an assumption of linearity in all this. If the camera processing includes a gamma correction - as is the case for JPG from compact or dSLR - then this method of camera calibration cannot be used. Calibrating your camera's raw images in this way can still be useful for JPG photography, because raw images and JPG will saturate simultaneously - a well-balanced raw image exposure is also a well-balanced JPG exposure.
Example
Only compact and dSLR have ISO settings. For webcams and CCD cameras we can determine the combined factor (d·ISO) instead. If we use different gain settings, we have to calibrate separately for each setting. By comparing this with d and ISO for a compact or dSLR, we can determine a formal ISO value. Here is the example for my Canon EOS 400D dSLR and my Philips ToUcam Pro webcam with the gain set to 81 in qastrocam [4]. The dSLR and webcam have been measured to have
- ddSLR = 0.065 sr/(MJy s)
- dwebcam · ISOwebcam = 8.4 sr/(MJy s)
Two cameras at the same ISO setting and with the same camera-processing factor CP should have about the same d, because the ISO setting is a standard designed to give the same image brightness for the same object brightness, irrespective of the camera details. However, the webcam digitiser has 8 bit, while the dSLR raw data end up as 16-bit RIU numbers. Hence there is a factor 256 (two to the power of 16 minus 8 bit) in RIU at the "same image brightness". To compare the webcam with the dSLR we would use d·ISO = 2160 and find
- ISOwebcam ~ 33000
- dwebcam = 0.00025 sr/(MJy s)
Brightness of various objects
The table below lists the magnitude, flux, and surface brightness of various objects of astrophotography. The size is roughly the diameter of the object in arc seconds. If the object is resolved by your optics and camera, simply use the surface brightness to estimate the exposure parameters needed to accomplish a suitable RIU value. For unresolved objects use the flux and either the solid angle of one pixel or the width of the point spread function (see above).
Brightess of various objects. object V S (Jy) size (") B (MJy/sr) ref. bright blue sky (e.g. halo) 30,000,000 [7] cloudy sky (e.g. rainbow) 8,000,000 sky at civil twilight 6,000 [9] sky at nautical twilight 40 [9] ISS (space station) -2.0 20,000 20 24,000,000 Iridium flare -4.0 140,000 Moon, 10% illuminated -7.4 3,000,000 1800 400,000 [5] Moon, 25% illuminated -8.7 10,000,000 1800 600,000 [5] Moon, 50% illuminated -9.9 30,000,000 1800 800,000 [5] Moon, 75% illuminated -10.9 80,000,000 1800 1,500,000 [5] Moon, 100% illuminated -12.6 400,000,000 1800 5,000,000 [5] Moon, earthshine 450 [6] lunar eclipse, penumbra 1,500,000 [6] lunar eclipse, bright umbra 1,500 [6] lunar eclipse, dark umbra 150 [6] Sun -26.7 170,000,000,000,000 1800 2,000,000,000,000 [5] Sun, Hα, 0.1 nm bandwidth 200,000,000 solar eclipse, prominences 1,500,000 [6] solar eclipse, inner corona 100,000 [6] solar eclipse, outer corona 30,000 [6] sky during solar eclipse 1,000 [7] horizon during solar eclipse 20,000 [8] Mercury, with 1 mag extinction 0.0 3500 10 1,500,000 Venus -4.5 200,000 50 15,000,000 Mars -2.5 35,000 20 4,000,000 Jupiter -2.9 50,000 50 900,000 Galilean moons 900,000 [7] Saturn 0.0 3500 20 250,000 Uranus 5.5 20 4 60,000 Neptune 7.8 3 2.3 20,000 comets 5.0 35 300 15.0 (1) Ceres 7.0 6 (134340) Pluto 13.7 0.01 M42, M27, M57, etc. 30.0 [6] Horsehead, California, etc. 1.0 [6] Milky Way 9.0 [8] galaxies, bright cores 8.0 [6] galaxies, outer regions 0.5 [6] 3C 273 (brightest quasar) 12.8 0.03 city sky 12.0 [6] town sky 3.0 [6] country sky 0.7 [6] dark country sky 0.3 [6] desert/mountain sky 0.1 [6]
Where numbers from [6,7,8] were used, they have not been taken directly, but multiplied with 0.75 to make the table self-consistent.
References
- Richard Miles (2007). "A light history of photometry: from Hipparchus to the Hubble Space Telescope". J. Br. Astron. Assoc., 117. p. 172.
- "Photometric system". Wikipedia. http://en.wikipedia.org/wiki/Photometric_system.
- UKIRT (1998). "Fluxes from a zero mag source and conversion of magnitudes to Janskys and F-lambda". http://www.jach.hawaii.edu/UKIRT/astronomy/calib/phot_cal/conver.html.
- Franck Sicard (2007). Qastrocam. http://3demi.net/astro/qastrocam/doc/. Newer CVS repository at http://sourceforge.net/projects/qastrocam/.
- Horst Meyerdierks (2009). Sputnik 3.1. http://www.chiandh.me.uk/soft/.
- Michael A. Covington (2000). Astrophotography exposure calculator. http://www.covingtoninnovations.com/astro/astrosoft.html
- Rainer Beck, Andreas Hänel (1982). "Grundlagen der Astrofotografie". 10 Jahre Volkssternwarte Bonn - 1972-1982 - Festschrift. p.45. Volkssternwarte Bonn.
- Alois Lohoff (1982). "Die richtige Belichtungszeit - aber welche?". Sternzeit, 8. p.41.
- Horst Meyerdierks (2009). "Noctilucent cloud". http://www.chiandh.me.uk/p/Noctilucent_cloud.
