Closer look to photometry through the telescope
|Andromeda galaxy (M31)|
(c) Sergi Verdugo Martínez (astrophoto-sv.com)
A good photo depends basically upon an adequate illumination, that is, the flux of light that comes from the object towards the camera for a certain period of time.
The photometric relation of this Flux that involves 2 surface elements (i.e., a nebula of extension S, and the pupil or a camera CCD of area S') would be dF = L · dS · domega · cos(theta), where L means the object's luminance, omega the falling solid angle differential ( = dS' / r^2, being r the distance between the nebula and the lens of the camera), and cos(theta) applies for the cosine component of the surface, its orientation (Lambertian cosine law).
Luminance is defined from the spectral radiance emitted by the object (see Greenhouse effect post), but weighted to the average sensitivity curve of the human eye to wavelength, which drives to a "brightness perception", the amount of light the eye would perceive from a particular viewpoint.
There are 2 different shifted curves for sensitivity due to both types of eye photo-receptors (cones and rods). Photopic vision affects cones, which are composed of three separated photo pigments to enable color perception; and the scotopic vision implies rods, that are more sensitive to light and less to color. Photopic responses under normal lighting conditions. This curve peaks at 555 nanometers, so the eye is most sensitive to a yellow-green color (oddly human eye has evolved to match Sun's). At low light levels, near to darkness, the eye response fits the scotopic curve, and peaks at 507 nm, closer to blue-violet.
Thus, the incoming flux of light entering the pupil, or the lens of a telescope, of diameter D would be dF = L · dS · domega · cos(theta) = L · dS · [ pi·(D/2)^2 ]/r^2 · cos(theta). In fact the solid angle may subtend another cos(theta') due to the orientation of S' surface, but from now onwards we will consider both theta and theta' angles as 0, so their cosines are 1.
The illuminance, that is the received illumination on surface S', related as well to the irradiance, is just E = dF / dS' = L · dS/dS' · pi·D^2/4·r^2, measured in [lux] (or [lumens/m2]).
Now to determine the dS/dS' areas ratio we must consider the focal length and the (linear) magnification equation of a lens.
Take a spheric mirror (simpler than a lens -no refracting indexes involved- and will do as well) of radius of curvature r. Geometrically, the image point P' at distance s' can be deducted from the different angle relations. Notice the beta angle (at the center of the mirror sphere) is the sum of alpha (drawn from the object P) and theta (the reflecting angle). Similarly gamma = alpha + 2·theta. Combining both equations and removing theta we obtain 2·beta = gamma + alpha.
As these angles are considered "small" (sinus ~ angle), they can relate the P object distance alpha ~ sin(alpha) = l/s; the P' image distance gamma ~ sin(gamma) = l/s'; and the C center of curvature distance beta ~ sin(beta) = l/r. So, finally, 2·l/r = l/s' + l/s, that is 2/r = 1/s' + 1/s. When the distance to the object is quite larger (at the "infinite", so incoming rays are parallel -paraxial-) than the mirror radius of curvature, 1/s is negligible, and s' = r/2. This s' distance is then the focal length f of the mirror (or lens), and P' the focal point: 1/f = 1/s' + 1/s.
The (linear) magnification m quantifies the apparent ratio between the image and the object sizes m = h'/h. Notice that according to the lens, h/p = h'/q, or the paraxial rule h/(p-f) = -h'/f, thus m = h'/h = -(p-f)/f.
In our case, the dS/dS' areas ratio is proportional to (h/h')^2 (squared because we are talking about surfaces). Replacing in the illuminance equation E = dF / dS' = L · dS/dS' · pi·D^2/4·r^2 = L · [ -(r-f)/f ]^2 · pi·D^2/4·r^2 = pi/4 · L · [ (r-f)/r ]^2 · [ D/f ]^2. For astronomical observation (r much greater than f), illuminance can be approximated to E = pi/4 · L · [ D/f ]^2.
The D/f amount stands for the relative aperture (adjustable with the camera diaphragm or the pupil iris), and its inverse for the diaphragm number N = f/D. The aperture limits the incoming amount of brightness (pupil or diaphragm size) to the eye or camera. But same apertures will produce equal illuminance even varying focal length f and diameter D.
Apertures are commonly expressed as fractions of the focal length, called f-numbers or f-stops, and each stop represents half the light intensity from the previous one, that is f/1 = f/sqrt(2)^0, f/1.4 = f/sqrt(2)^1, f/2 = f/sqrt(2)^2, f/2.8 = f/sqrt(2)^3 and so on (root-squared because it is lately squared-powered in the illuminance equation to halve the incoming light). Then lower f-numbers denote greater apertures, which means more light to the camera sensor. Maximum aperture (or minimum f-number) defines the (lens) speed: The greater the aperture, the faster the lens, as it lets in more light (higher illuminance). So the shutter speed will be faster as well.
Bearing this in mind, astrophoto may require different focal ratios (relative apertures) according to the astronomical objectives to be shot. Among others, they can be divided into planetary or deep sky.
The apparent (or angular, or visual) magnification M of the (refracting) telescope is determined by the ratio of tangents of the angles under which the object is seen with (beta(i), apparent field of view) and without (beta(s), true field of view) the lens, respectively.
Thus, tangent( beta(i) ) = h / fe = (D/2) / (fo+fe), where fo is the objective focal length and fe the eyepiece's, h the image height, and D the objective lens diameter; and tangent( beta(s) ) = h / fo = (d/2) / (fo+fe), here d means the eyepiece lens diameter. So the telescope magnification power M = tangent( beta(i) ) / tangent( beta(s) ) = fo / fe = D / d.
For planetary observation a telescope with greater magnification power is worth (longer focal, usually a refractor -dioptric- telescope, which uses lenses), and for deep sky higher illumination is needed (wider diameter, mostly a reflector -catoptric- telescope, with curved mirrors), as nebulae or galaxies are extended objects and their apparent magnitude is distributed over a wider angle than planets or stars.
The magnitude of an object is a logarithmic measure of its relative brightness. Relative to the star Vega, which has a (almost) 0 magnitude. Sun has a -26.74 magnitude (brighter), Moon -12.74 (less brighter), or Mars ranges from -2.91 (brighter than Vega) to 1.84 (fainter than Vega). So, even the M42 nebula (Orion) has a 4.0 magnitude, it is less visible than a star of the same apparent magnitude because its dimensions are 65x60 arcminutes.
Or even a catadioptric (lens and mirror) telescope for a combination of both planetary and deep sky observation. But telescopes are not perfect, as paraxial optics laws applies strictly to light rays that are infinitesimally displaced from the optical axis of a system, and a series of optical imperfections (aberrations) must be considered.
Refracting telescopes suffer from chromatic aberration, a distortion by which the lens cannot focus all colors at the same (converging) point. This dispersion is caused by different refractive indexes depending on light wavelengths.
It can be partially fixed adding more lenses (achromatic Fraunhofer doublet, the sum of a convergent -crown- lens plus a divergent -flint-; or apochromatic, adding more lenses, better focus correction of wavelengths), or minimized with greater quality lenses (lower dispersion, made of fluorite).
But also reflecting telescopes do have aberrations. They gather light with a mirror, and it is primarily parabolic, not spheric, to avoid spherical aberration, where light at the edges of the mirror focus closer than that reflecting from the center. This is corrected with a parabolic mirror instead, as in Newtonian telescopes.
|Spheric mirror||Parabolic mirror|
But parabolic mirrors trouble with coma aberration, that's a change of magnification for incoming light closer to the edges (off-axis) of the curved mirror. It can be partially fixed closing the aperture 1 or 2 stops, along with an increase of exposure time to photograph.
This lack is better solved in Schmidt-Cassegrain (and Maksutov-Cassegrain) catadioptric telescopes, which combine a correcting lens with a primary spherical mirror and a secondary parabolic convex, that multiplies the focal length, thus getting a compact telescope with high magnification power and wide angle, optimal for both planetary and deep sky.
Because of all these side effects, and also due to diffraction, the image of a point becomes a spot (an Airy disc). The angular resolution (or power resolution) of a telescope is a measure of the minimum angular separation between distinguishable objects in an image, according to the Rayleigh criterion sin(theta) = 1.22 · lambda/D, where 1.22 is nearly the first zero of Bessel function, angle theta is measured in [arcseconds], and lambda (light wavelength) and D (aperture diameter) in same units (i.e. [mm]).
Since theta will be a "small" angle, the expression can be approximated by sin(theta) ~ theta = s / f, being s the separation of both objects in the image (focal) plane and f the focal length. Thus s = 1.22 · lambda · f/D = 1.22 · lambda · N, where N is the diaphragm number.
Once mounted the telescope on an equatorial platform (i.e. GEM -German Equatorial Mount-, much better than alt-azimuthal for shooting, easier following position movement of celestial objects), one just needs a camera (attaching it to the telescope as primary focus) to begin with astrophoto. Even a webcam will do, though a digital CCD is highly recommended.
|Orion and Running Man nebulae (M42 and NGC1977)|
(c) Sergi Verdugo Martínez (astrophoto-sv.com)
Unlike it is commonly believed about the dutch origin of the telescope around 1608 credited to Hans Lippershey, the oldest reference about its existence is a noble's inheritance written legal document dated as of April/10/1593, and his inventor was the catalan optician (from Girona) Joan Roget, as published in a book authored by Girolamo Sirtori in 1609.
Acknowledgement to Sergi Verdugo Martínez (astrophoto-sv.com) for his awesome images.