The Astronomical Telescope

Transcription

1 The Astronomical Telescope Any Chmilenko, Instructor: Jeff Gariner Section (Date: 2:30 pm Tuesay October 8, 203) I. PURPOSE The purpose of the experiment is observe an measure the magnification of compoun lenses, in two special cases, the Astronomical Telescope which uses two positive lenses, an the Galilean Telescope which uses a positive an a negative lens. We will be measuring the magnification using several ifferent methos such as the ratio of focal lengths, using irect observation, an by measuring an comparing the ratio of the entrance to exit pupil. We will also be investigating an measuring brightness an fiel of view as the size of the lens changes to try to measure the iameter of the pupil. We will also be measuring the angular resolving power of the astronomical telescope an the unaie eye an compare them with each other an theoretical expectations. II. ANALYSIS A. Measurement of Magnification. Magnification from the Ratio of ocal Lengths ( ) f Measurements (± 0. cm) Average (± 0.06 cm) 9.66, 9.300, f 4.626, 4.70, Inter-lens Distance 23.9, 24.0, TABLE I: Measurements an averages for the objective focal point, the ocular focal point f, an the inter-lens istance between the objective an ocular when both focal points overlap Sample Calculations for using row of Table I. n i = i n = = Sample Calculations for = ± n = ± 0. 3 = ±0.06cm Sample Calculations for f f = f = 4.096

2 2 Sample Calculations for f f = ( ) 2 + ( f) 2 f = (0.06) 2 + (0.06) 2 f = ±0.08 The magnification was calculate to be ±0.08. When viewing the image though the eyepiece you can see a large, bright target compare to looking at the light source irectly without the aie of the lenses. The inter-lens istance which was measure to be 24.0 ±0.06cm is equal to f+ = = ±0.08cm. 2. Magnification by Direct Observation ( L l ) We looke through the telescope at a wall with bricks. Using the unaie eye to look at the wall an looking through the telescope through the other, we observe 4 bricks in the space where one brick appears in the telescope image, meaning the magnification observe is 4 ±0.08. This coincies with the result from earlier with out measure magnification of ±0.08, only eviating by 2.3%. Sample Calculations for % eviation of observe magnification with f % eviation = % eviation = 2.3% 3. Magnification from the Ratio of Entrance to Exit Pupil ( Do ) Measurements (± 0. cm) Average (± 0.06 cm) D o 4.736, 4.776, ,.246, DE 6.480, 6.270, TABLE II: Measurements an averages for the iameter of the objective lens D o, the minimum iameter (the exit pupil), an the the istance from the ocular lens to the exit pupil DE. Sample Calculations for Do D o = D o = Sample Calculations for Do Do = ( D o ) 2 + ( ) 2 Do = (0.06) 2 + (0.06) 2 Do = ±0.08 Sample Calculations for % eviation of Do with f % eviation = % eviation = 4.9%

3 3 Sample Calculations for % eviation of Do with L l % eviation = % eviation = 2.7% The magnification by metho of measuring the ratio to entrance to exit pupil ( Do ) was measure to be ±0.08 times magnification. This is also in line with our previous results only eviating 4.9% a 2.7% for f an L l respectively. Sample Calculations for DE using measure values for an f to be ±0.06cm an ±0.06cm respectively DE = f ( + f ) DE = ( ) DE = 5.839cm Sample Calculations for % eviation of measure an calculate DE % eviation = % eviation = 8.6% Our calculate an measure value for DE are similar, only iffering by 8.6%, this is enough to confirm that the istance DE obeys the relation above. B. Brightness of the Image Using the iaphragm in front of the telescope to vary the light until a point where the image, while looking through the telescope, seems to get arker, we were able to fin a iameter for the iaphragm D o eff. I measure D o eff for my eye to be.324 cm. Sample Calculations for p, the iameter of my pupil using measure values for D oeff of.324 cm an M to be p = Doeff M p = p = 0.323cm = 3.23mm Sample Calculations for p p = p ( Doeff D oeff )2 + ( M M )2 p = 3.23mm ( 0.cm.324cm )2 + ( )2 p = ±0.25mm My pupil iameter, p, was calculate to be 3.23 ±0.25mm which is on the orer the size of what a pupil shoul be. D e (± 0.cm) : 3.774, 3.802, 3.796, D e = 3.79 (± 0.05cm) C. iel of View

4 4 Sample Calculations for D e. n i D e = De i n D e = D e = 3.79 Sample Calculations for D e Distance to bricks from Objective: 67.5cm ±cm Brick = 8.2cm 6 = 3.5 cm ±0.25cm D e = ± De n D e = ± 0. 4 D e = ±0.05cm D s Number of Bricks in View l b l b Ds % D s % eviation % % % % TABLE III: Measurements for the iameter of the stop D s, the number of bricks observe, an the calculations for fiel of view, l b an Ds which are the length of observe bricks ivie by the istance to the bricks from the telescope an the iameter of the stop ivie by the objective focal length respectively. Sample Calculations for l b using Row of Table III, an a conversion rate of brick = 3.5 cm ±0.25cm an the istance to the bricks as 67.5cm ±cm l b = l b = 0.2 Sample Calculations for l b l b = l b l b = 0.2 ( l b l b using Row of Table III ) 2 + ( ) 2 ( )2 + ( 67.5 )2 l b = Sample Calculations for Ds using Row of Table 3 D s = D s = 0.4

5 5 Sample Calculations for Ds Ds Ds = Ds ( Ds D s using Row of Table III ) 2 + ( )2 = 0.4 ( )2 + ( )2 Ds = Sample Calculations for % eviation of measure an calculate iel of View using Row of Table III % eviation = % eviation = 6% The measure iel of View with the calculate theoretical fiel of view line up rather nicely, are are close to the margin of the estimate uncertainty. The measure an calculate eviate by only 6-23% in our 5 trials. D. Resolving Power Hole Separation (mm) D telescope (± cm) α measure (ra) % eviation from α theoretical % 2 3 TABLE IV: Measurements for the istance at which two holes separate by some istance were no longer istinguishable as two separate holes when viewe through the telescope using an objective Diameter D o = ±0.06cm. The lab was not large enough to test the resolving power for the hole separation greater than mm. Hole Separation (mm) D unaie (± cm) α measure (ra) % eviation from α theoretical % % 3 TABLE V: Measurements for the istance at which two holes separate by some istance were no longer istinguishable as two separate holes when viewe through the unaie eye using an objective Diameter p = 3.23 ±0.25mm. The lab was not large enough to test the resolving power for the hole separation greater than 2mm. Sample Calculations for α measure using Row of Table IV α measure = 2 arctan separation 2 D telescope α measure = 2 arctan α measure = ra Sample Calculations for α theoretical for the telescope using a wavelength of 550nm an D o = 4.755cm λ α theoretical =.22 D o α theoretical = α theoretical = ra

6 6 Sample Calculations for α theoretical for the unaie eye using a wavelength of 550nm an p = 3.23mm α theoretical =.22 λ p α theoretical = α theoretical = ra Sample Calculations for % eviation of measure an calculate α % eviation = % eviation = 200% There were large iscrepancies between the measure an theoretical angular resolving power for both the telescope an the unaie eye. With the telescope angular resolving power eviating by 200% an the unaie eye eviating by 30% for the mm case, an the unaie eye eviating by 200% for the 2mm case. This may be ue to the subjective nature of the test, an as the pupil of a human s eye will contract an ilate naturally, the parameter calculating the theoretical angular resolving power at the time of the test was most likely ifferent. E. Galilean Telescope. Determining the ocal Length of the negative lens The negative lens was place 5cm away from the objective an the screen was place at a point on the other sie of the lens such that the image S 2 was focusse on the screen. The istance to the screen from the objective was measure to be 22.9 ±0.cm. IG. : Diagram of the Galilean telescope (not to scale) with relevant points an istances labelle.

7 7 Sample Calculations for using the thin lens law = = s 2 + s 2 = s 2 + s 7.9cm + (9.222cm 5cm) = 7.9cm + (4.222) = 9.07cm Sample Calculations for = (0.06) 2 + (0.) 2 + (0.) 2 + (0.) 2 = ±0.062cm 2. Magnification by the Ratio of ocal Lengths Measure Inter-lens istance when focusse image is seen: 8.56 ±0.cm The Inter-lens istance still seems to follow the relationship of f +, but in this case is negative so the inter-lens istance is equal to 9.222cm-9.07cm = 0.5cm Sample Calculations for M using f = 9.222cm an = -9.07cm M = f M = M = 2. Sample Calculations for M M = 2. ( )2 + ( )2 M = ±0.06 The image of a istant object prouce by the Galilean telescope is not inverte in the case of the Astronomical telescope. 3. Measurement of Magnification Using irect observation on a brick wall, we were able to observe the magnification to be about 2 times which is in line with the calculate magnification o.±0.06. We coul not employ the metho of magnification calculation using the ratio of the entrance to exit pupil because the Galilean telescope prouces a virtual image which cannot be measure. 4. Comparisons between the Astronomical an Galilean Telescopes The length of the Galilean telescope is shorter than the length of the Astronomical telescope ue to the negative focal length of the bi-concave lens. The position of the exit pupil for the astronomical telescope is always centre because the image it prouces is real. However the Galilean telescope prouces a virtual image in the on the opposite sie of the lens. The exit pupil can be off centre by some egree when not viewing an object irectly centre in the fiel of view of the telescope. The fiel of view of the Galilean telescope however, is smaller than the Astronomical telescope which gives the astronomical telescope an avantage in this regar. The magnification for both cases is the ratio of the focal lengths of the lenses use.

8 8 III. CONCLUSION We were able to observe the magnification properties of the Astronomical an Galilean telescopes using several ifferent methos. Using the ratio of the focal lengths, we were able to calculate the Magnification of the Astronomical telescope to be about 4.096±0.08, an the Galilean telescope to be about 2.±0.06. We also employe irect observation an foun the magnification factor to be about 4 an 2 for the Astronomical an Galilean telescopes respectively, which were in line with our calculate values. We use a thir metho, by comparing the ratio of the entrance to exit pupil an were able to measure the magnification for the Astronomical telescope to be 3.894±0.08, which only eviate from the previously calculate value by only 4.9% an was close to the margins on uncertainty. The metho of comparing the ratio of the entrance to exit pupil coul not be employe for the Galilean telescope because of the image it prouces is virtual, an therefore coul not be measure. Using a stop, we were able to vary the effective iameter of the objective lens an were able to measure an calculate the size of my pupil. We calculate a value of 3.23±0.25mm for the size of my pupil which is on the orer of a regular pupil size. We also investigate the fiel of view of the astronomical telescope by using bricks. The fiel of view measurements were in line with our expecte theoretical values for the varie effective objective iameter. Our measure values for the fiel of view only eviate by 6-23% from the theoretical values. We took the time to test out the resolving power of the Astronomical telescope an the unaie eye an compare those values with theoretical. We foun that our values varie quite a bit with the theoretical limits. Our measure values eviate by 200% for the Astronomical telescope, an % for the unaie eye. However ue to the subjective nature of this test it is ifficult to come up with conclusive measurements, especially with the unaie eye as the pupil constantly ilates an contracts between measurements. The size of the lab was also not optimal in fining a istance within the lab at which 2 objects coul no longer be resolve. Chromatic aberration an the assumption that the light we were viewing was that of a fixe wavelength also incurre significant error in our comparisons as it is unrealistic to our scenario. We were able to measure the focal length of the negative lens in the Galilean telescope an foun it to be ±0.062cm We were able to iscern some avantages an isavantages between the Astronomical an Galilean telescopes. The Galilean telescope has a shorter length than the Astronomical telescope ue to its negative focal length. The position of the exit pupil on the Astronomical telescope is fixe which the Galilean can be off centre when not viewing an object hea on with the telescope. The fiel of view is also smaller as a result for the Galilean telescope. The magnification for both telescopes followe the same property, of being proportional f. Uncertainty in our measurements were largely limite to the mounting equipment. Such as the mount which hel the lenses an trying to measure istances between lenses, there were no conclusive start an en points which we coul measure between, so an uncertainty of 0.cm was assume. When measuring longer istances using several meter sticks, between an object an the telescope, the start of the telescope coul not be accurately compare to the values on the meter stick so a larger uncertainty of cm was use for these measurements.