# 3.1 Lab VI: Introduction to Telescopes / Optics [i/o]

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1 3.1 Lab VI: Introduction to Telescopes / Optics [i/o] Please answer the following questions on separate paper/notebook. Make sure to list the references you use (particularly for the last questions). For this assignment, working in small groups is permitted. Reminder: For any observation you do (naked eye/binoculars/telescope/ccd) please record the details of your observation. These include: the weather/sky conditions; rough estimate of the stability of the seeing (twinkling); location of object in the sky; location and nature [city lights? trees blocking part of the view? etc.] of the ground site where you observe from; time/date of the observation; integration time/filters/telescope/etc. [if applicable]; and the members of your observing team Simple Astronomical Refracting Telescope The simplest of astronomical telescopes are built of two converging lenses, one typically of long focal length (f ob ;objective)andthesecondofshortfocallength(f ep ;eyepiece),separatedbya distance, f ob + f ep. Figure 3.1 labels the geometric setup of a simple astronomical refracting telescope. From figure 3.1 we see that the lens combination acts to angularly magnify (m α /α) and invert the image of a object. Figure 3.1: A simple astronomical refracting telescope. The is a shorthand notation for a converging lens. The simple astronomical telescope is an inverting instrument. From the leftmost triangle we see that in the small angle approximation α h/s iep.fromthe central triangle we further see that α h/s oep h/(f ob +f ep ). Making use of the basic lensmaker s equation: 1/f ep =1/s iep +1/s oep,itcanbedemonstratedthatm = f ob /f ep. Namely for a given eyepiece focal length, f ep,alongobjectivefocallength,f ob leads to high magnification, while for a given objective focal length, a short eyepiece focal length leads to high magnification. Furthermore f/ratio (written f/#) can be defined as, f/# f ob /D ob,whered ob is the diameter of the objective lens. The f/# is solely of function of the design of the objective lens. For a given D ob,bigger f/# imply high magnification, while for a given f ob,biggerf/#implysmallerlightgrasp(seebelow). 28

2 3.1.2 Schmidt-Cassegrain Telescopes Many of the telescopes you will use are not simple refracting telescopes, but the above concepts can be fairly easily adapted to apply. For simple (Newtonian) reflecting telescopes the focal length of the objective is just the distance from the mirror to the focus point of the converging (or diverging) rays. Etscorn Observatory has a number of Schmidt-Cassegrain telescopes. The focal path of Schmidt-Cassegrains are folded and so a bit more complex. Here we just use its reported focal length as f ob.howeveritcanbedeterminedfromtheopticsofthemirrors,with f ob corresponding to extending the converging rays from the secondary lens back along the line until they reach the diameter of the telescope, D ob. SCT f ob f m f b /(f m d), where f m is the focal length of the primary mirror, f b is the distance between the secondary mirror and the focal plane, and d is the distance from the secondary mirror to the primary mirror. Figure 3.2: Schematic of a Schmidt-Cassegrain, with its focal length drawn. The TA will demonstrate the use of the Schmidt-Cassegrain telescopes at Etscorn, including use of the domes, checking the collimation of the telescopes, focusing the telescope and pointing the telescope using the TheSky6.0 software Field of View The field of view (FOV) of a telescope depends on its optics, both the objective and the eyepiece. To a crude approximation the FOV of the simplest eyepieces are, FOV ep D ep /f ep,whered ep is the diameter of the eyepiece lens (or more properly any limiting aperture stops inside). However, modern eyepieces have become optically quite complex and so nominally we take the FOV ep (often referred to as apparent FOV ) of an eyepiece as a given. They are often written on the eyepiece directly. Low quality eyepieces, like the Hyugens, RamsdenandKellnertypestypically have a FOV ep o. Very high quality, wide field eyepieces, such as Erfles and Naglers have FOV ep 65 > 82 o. However the typical common use eyepieces, such as Plössls and Orthoscopics, have intermediate FOV ep 50 o. Due to magnification the FOV of the telescope system, FOV tel,ismuchsmaller. Thetelescopicmagnification zoomsin onthe FOV ep by a factor equal to the total magnification. So telescopic FOV (often referred to as true FOV ) is given by 29

3 FOV tel = FOV ep /m. The larger the FOV the larger fraction of diffuse astronomical objectsthatcanbeviewedsimultaneously. The angular drift rate of an object on the sky is 15 o /hr cos δ, (δ = declination). So for simple telescopes without tracking motors, larger FOVs also mean longer times for the object to be viewed without readjusting the pointing of the telescope. However,largerFOVnaturallyimply low magnifications Resolving Power In the (better) wave theory of light, the point source response function (or point spread function; PSF) is the Fourier transform of the aperture function (the shape of the aperture). For circular apertures of size, D ob,thepsfisan Airydisk.Fromthecentralpeaktothefirstnull of an Airy disk is θ 1/2 =1.22λ/D ob,soastarviewedinthevisible(λ =5500Å) will exhibit a full width zero intensity (FWZI) size of 2 θ 1/2 ( ) 280/D ob (mm). No objects spaced by less then this can theoretically be separated completely. Small telescopes, with perfect optical systems, well focused, on sturdy mounts, and in very stable atmosphere can reach close to the theoretical limit. But since these are difficult conditions to obtain, practical resolving power of an aperture rarely is this good. For larger apertures, the atmosphere limits resolving powertoabout1-2.atypicalapproximation to estimate the ability to resolve two point sources (up to the atmosphericlimit)istoassumeone FWZI PSF separation between the two sources. With this assumption then point sources (stars) that are separated by s 4θ 1/2 =560/D ob (mm), oughttoberesolvedbyanobjectiveofsize D ob.thisroughlycorrespondsto 20/20 visionindaylight Maximum & Minimum Useful Magnification / Exit Pupil The eye s aperture at night ranges from 5-7 mm, so from s 560/D ob (mm), we obtain the resolving power of the unaided eye to be roughly s 100,orabout 1 18th of the size of the Full Moon. This practical limit of the eye implies a maximum useful telescope magnification. Any telescopic magnification that magnifies the maximum theoretical limit of theaperture( 140/D ob (mm)) greater than 100 is of no practical use. Doing so would just result in zooming in ontheunresolved blob of light limited by the telescope optics and not lead to seeing any finer detail (and just make it appear fainter see the Surface Brightness subsection). Inserting numbers, one obtains m max D ob (mm), or phrased in terms of the eyepiece focal length: f ep (min) f/#(objective). Therefore for small hobby telescopes, the maximum useful magnification is Thelarger,stably-mountedtelescopesatEtscorncansupport roughly twice this magnification. Note: these numbers are approximate and depending on the observer, site and quality of the telescope, these numbers vary somewhat. The exit pupil, D ex,isthephysicalsizeoftheimageoftheobjectiveasseenthrough the eyepiece. From Figure 3.3 it can be seen that α =(D ex /2)/f ep =(D ob /2)/f ob,hencetheexitpupil size is given by D ex =(f ep /f ob ) D ob or D ex = D ob /m. Thehigherthemagnificationforagiven objective focal length, the smaller the exit pupil. This is why it often takes some effort to get your eye aligned properly to see the image when working at high magnification. The exit pupil also controls the minimum useful magnification of a system. If the exit pupil gets bigger than 7 mm, then the entire light collected by the telescope is not focused down tight enough to completely enter the 30

4 Figure 3.3: The geometry for determining exit pupil. eye. For a completely dark-adapted eye aperture of 7 mm, this implies a m min = D ob (mm)/7. It is for this reason that most astronomical binoculars tend to be manufactured such that the ratio of the magnification to the objective is 7 (suchas7 50, 12 70, 15 80), and terrestrial binoculars have the above ratio of 4 (D eye in daylight; such as 8 25, 10 42) Light Grasp / Surface Brightness Light grasp, G L,representshowmuchmorelightanobjectivecollectscompared to the eye. It is simply given by the ratio of the area of the objective to the area of the eye, and hence is roughly G L =(D ob (mm)/7) 2. Light grasp is the main benefit of large telescopes, not so much magnification, as seen in the previous section. When magnifying by m, ascopespreads(roughly)thesameamountoflightoverasurface area m 2 larger. Hence the surface brightness, SB, ofanobjectism 2 fainter. However a scope also collects more light, in proportion to G L. So the surface brightness of an object when viewed through a telescope is: SB tel = SB eye G L m 2 = ( Dob D eye ) 2 ( fob (as long as D ex <D eye ), where SB eye is the surface brightness the object would have with the unaided eye. The maximum SB tel corresponds to the case when the magnification is minimum, m = m min,sosb max = SB eye.thebestsurfacebrightnessatelescopecanprovideisthatof the unaided eye! A telescope just makes that same surface brightness be observed over a larger area. (Note: this does not account for integration time. Surface brightness sensitivity can be improved by integrating longer than the eye does or by using more efficient photon detectors like CCDs more later.) We define SB eye as 100%. Thus: SB tel (%) = 100% G L m 2 = ( fep D eye ) 2 (f/#) 2 f ep ) 2 = 2% f ep(mm) (f/#) 2. High magnification makes objects large but dim, while low magnification keeps objects bright but compact. 31

5 3.1.7 Limiting Magnitude (Telescopic) The limiting magnitude of a telescope, tel m Vlim, (in this case m V is the V-band apparent magnitude, not the magnification apologies for the collision in notation) is the faintest magnitude seen by the eye, through a telescope. Since: ( ) tel m Vlim eye Ilim m Vlim = 2.5 log eye = 2.5 log(g L ), I lim then: tel m Vlim = 2.5 log(g L )+ eye m Vlim 5 log(d ob (mm)) 4.2+ eye m Vlim. If the site you are observing from has a limited V-band magnitude for the unaided eye of 6, then tel m lim,6 5 log(d ob (mm)) Exercises 1) Using a pair of binoculars, observe β Cygnus (Alberio). Calculate whether you should be able to resolve this binary given the D ob of the binoculars? Do you? If yes, please sketch. If not, and your calculation indicates you should, suggests reasons why you do not. 2) Repeat your limiting magnitude experiment from the beginning of the semester with the binoculars. Observe M 45 (Pleiades) and use the following reference chart (Figure 3.4) to determine stellar magnitudes. Sketch the (six) backbone bright stars then add a number of faint stars to the sketch based on your binocular view. Identify the stars and their magnitudes, and determine your limiting magnitude. How does the determined m lim compare to eye m lim? Is this consistent with your theoretical expectations? 3) Select two eyepieces, one with as long a focal length, f ep,asispracticable,andone with a short f ep,(preferablynearm max ). Attach each eyepiece to a telescope. Measure and record the exit pupil of each eyepiece (your will need arulerforthis). This can be most easily done during daylight or with the dome lights on. Calculatethe magnification of each eyepiece for the telescope setup you use. Calculate the expected exit pupil, D ex,andcomparetoyourmeasurement. Table 1 displays a collection of famous astronomical objects. Category A contains selected double/multiple stars. Category B contains selected bright objects with interesting structure amenable to high magnification. Category C contains a collection of faint, extended, diffuse nebulae/galaxies amenable to large collecting area telescopes and wide FOVs. 4) Select one object from each category (bold/italics gives you hints as to the time of year each object is visible). Sketch the view through each of the above two eyepieces. Include comments on brightness, color and orientation. For each category describe which eyepiece gives you the preferred view and why? 5) For the category A object (double star), turn off the telescope tracking and let the star drift across the center of the FOV of the eyepiece. Time the interval required for 32

6 Figure 3.4: A close up view of the Pleiades (M45) with associated stellar magnitudes (NASA). it to drift across. Calculate expected drift time given m, FOV ep,andδ, andcompare to your findings. Table 3.1: Astronomical Objects Category A Category B Category C β Cygnus Moon M8 γ Andromeda Jupiter M31 β Scorpius Saturn M57 ɛ Lyra Venus M42 α Hercules M 45 M81 β Monoceros NGC 869/884 M49 ι Cancer M13 α Gemini M 3 γ Leo M 44 θ 1,2 Orion variable summer/fall winter/spring 33

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