A Michelson-type Radio Interferometer for University Education

5.2 MB·

by  Jin Koda

Accepted for publication in American Journal of Physics, January 14, 2016
PreprinttypesetusingLATEXstyleemulateapjv.01/23/15
A MICHELSON-TYPE RADIO INTERFEROMETER FOR UNIVERSITY EDUCATION
Jin Koda1, James Barrett1, Tetsuo Hasegawa2,3, Masahiko Hayashi3, Gene Shafto1, Jeff Slechta1, and
Stanimir Metchev1,4
Accepted for publication in American Journal of Physics, January 14, 2016
ABSTRACT
We report development of a simple and affordable radio interferometer suitable as an educational
laboratory experiment. The design of this interferometer is based on the Michelson & Pease stellar
opticalinterferometer,butoperatesataradiowavelength(∼11GHz;∼2.7cm);thustherequirement
6
for optical accuracy is much less stringent. We utilize a commercial broadcast satellite dish and
1
feedhorn. Two flat side mirrors slide on a ladder, providing baseline coverage. This interferometer
0
resolves and measures the diameter of the Sun, a nice daytime experiment which can be carried
2
out even in marginal weather (i.e., partial cloud cover). Commercial broadcast satellites provide

n convenient point sources for comparison to the Sun’s extended disk. We describe the mathematical
a background of the adding interferometer, the design and development of the telescope and receiver
J system, and measurements of the Sun. We present results from a students’ laboratory report. With

2 theincreasingimportanceofinterferometryinastronomy,thelackofeducationalinterferometersisan
2 obstacle to training the future generation of astronomers. This interferometer provides the hands-on
experience needed to fully understand the basic concepts of interferometry.

]
M
I 1. INTRODUCTION §3, telescope setup and measurements in §4, and results
. from a students’ lab report in §5. What we present here
h The future of radio astronomy relies strongly on inter-
is only one realization of the concept. Creative readers
p ferometers(e.g.,ALMA,EVLA,VLTI,aperturemasking
couldmodifyanyparttomeettheeducationalneedsand
- techniques). Fromourexperienceatinterferometersum-
o constraints at their own institutions. For example, the
mer schools at the Nobeyama Radio Observatory and at
r astronomicalmeasurement,theconstructionandtestsof
t the CARMA Observatory, we are convinced that hands-
s onexperimentsarecriticaltoafullunderstandingofthe of telescope, receiver system, and other components can
a be separate lab projects.
concepts of interferometry. It is difficult, if not impos-
[ The best known Michelson interferometer is the one
sible, to obtain guaranteed access to professional inter-
used for the Michelson-Morley experiment (Michelson &
1 ferometers for university courses. Therefore, we built a
v low-cost radio interferometer for the purpose of educa- Morley 1887). It is one of the most important classi-
1 tion and developed corresponding syllabi for undergrad- cal experiments taught in both lecture and laboratory
6 uate and graduate astronomy lab courses. courses(Wolfson&Pasachoff1999;Melissinos&Napoli-
1 This experiment teaches the basic concept of inter- tano 2003; Serway & Jewett 2013; Bennett et al. 2013).
6 Many studies and applications have appeared in this
ferometry using the technique developed by Michelson
0 journal(Fang et al. 2013; Rudmin et al. 1980; Matthys
& Peace in the early 20th century (Michelson & Pease
. & Pedrotti 1982; da Costa, Kiedansky & Siri 1988; Dia-
1 1921). TheymeasuredthediameterofBetelgeuse,oneof
mondetal.1990;Mellen 1990;Norman 1992;Belansky,
0 thebrighteststarsinthesky,withasimpleopticalinter-
Richard & Wanser 1993; Kiess & Berg 1996; Nachman,
6 ferometer. Such optical interferometry needs high pre-
Pellegrino & Bernstein 1997; Fox et al. 1999), and re-
1 cision telescope optics. The same experiment becomes
: much easier when measuring the diameter of the Sun at cently the Michelson interferometer is being applied to
v thedetectionofgravitationalwaves(Kurodaetal.1999;
radiowavelength;theacceptableerrorsintheopticsscale
Xi with the wavelength. Abbott et al. 2009; Harry & LIGO Scientific Collabora-
tion 2010). The Michelson stellar interferometer is an
r Figure1showsaconceptualsketchoftheMichelsonra-
a diointerferometerforeducation. Thistypeofinterferom- application of the same physical concept of interference,
in this case, to a light source in the sky.
eter,addingsignalsinsteadofmultiplyingthem,iscalled
The theoretical basis of the Michelson stellar inter-
an adding interferometer. We discuss the mathematical
ferometer was already established in the Michelson and
background of the adding interferometer in §2, design
Peace’s original work (Michelson & Pease 1921) and has
and development of the telescope and receiver system in
been used in radio interferometry, especially in its early
[email protected] history(Pawsey&Bracewell1955;Steinberg&Lequeux
1DepartmentofPhysicsandAstronomy,StonyBrookUniver- 1963; Christiansen & H¨ogbom 1985; Wilson, Rohlfs &
sity,StonyBrook,NY11794-3800 Hu¨ttemeister 2013). This adding interferometer is the
2National Astronomical Observatory of Japan, NAOJ Chile
type used in modern astronomy at optical and near-
Observatory,Joaqu´ınMontero3000Oficina702,Vitacura,San-
tiago763-0409,Chile infrared wavelengths (Shao & Colavita 1992; Quirren-
3NationalAstronomicalObservatoryofJapan,2-21-1Osawa, bach2001)thoughmodernradiointerferometersareofa
Mitaka,Tokyo181-0015,Japan
differenttype,multiplyingsignalsinsteadofaddingthem
4Department of Physics and Astronomy, The University of
Western Ontario, 1151 Richmond St, London, ON N6A 3K7, (TaylorCarilli;Thompson,Moran&Swenson2007). For
Canada educationalpurposes,somestudiesinthisjournalshowed
2
that the concept of the stellar interferometer could be The radio frequency ν is typically large compared to a
demonstrated inan indoor laboratorysetup usingan ar- data sampling rate. Hence, the total power P(θ), de-
tificiallightsource(Pryor 1959;Illarramendietal.2014). tected by a receiver, is a time average (or integration).
In professional optical astronomy, the technique is now Using the notation < ... > for the time average, we ob-
being applied for advanced research (Shao & Colavita tain
1992; Quirrenbach 2001; Monnier 2003).
P(θ)=(cid:10)E2 (θ)(cid:11) (5)
tot
2. MATHEMATICALBACKGROUND =(cid:68)E2(θ )(cos[2πνt]+cos[2πν(t−τ)])2(cid:69) (6)
0
The mathematical basis of the stellar interferome-
ter was presented in Michelson and Peace’s original =(cid:10)E2(θ )(cid:0)cos2[2πνt]+cos2[2πν(t−τ)]
0
work(Michelson&Pease1921)andcanbefoundintext-
+2cos[2πνt]cos[2πν(t−τ)])(cid:105) (7)
books (Pawsey & Bracewell 1955; Steinberg & Lequeux
1963; Christiansen & H¨ogbom 1985; Wilson, Rohlfs & =E2(θ0)[1+cos(2πντ)] (8)
Hu¨ttemeister 2013). Here we describe the basic equa-
In going from eq (7) to (8) we used the transformations:
tions at a mathematical level that college students can
cos2A=(cos2A+1)/2 for the first and third terms and
follow.
2cosAcosB = cos(A + B) + cos(A − B) for the sec-
We start from the geometric delay calculation (§2.1)
ond term. In addition, because of the high frequency,
andexplainthetotalpower,theparameterthatwemea-
ν, all terms with (cid:104)cos(∗νt)(cid:105), (cid:104)sin(∗νt)(cid:105), etc, vanish when
sure, in §2.2. We will show an example of how a point
time averaged, and only the terms with no t dependence
source (i.e., a commercial broadcast satellite) appears in
remain. Usingequation(1)withthesmallangleapprox-
§2.3. Wewillthendiscussthecaseofanextendedsource.
imation, this becomes
We prove that an interferometer measures Fourier com-
ponentsanddefinevisibilityin§2.4. Wewillexplainhow
P(θ)=E2(θ )[1+cos(2πB (θ−θ ))] (9)
visibility is measured with our interferometer, and how 0 λ 0
the Sun’s diameter is derived in §2.5. where B ≡ B/λ is a normalized baseline length and λ
λ
is the wavelength (λ=c/ν).
2.1. Geometric Delay Equation(9)canbegeneralizedforanextendedobject
as
Interferometers mix signals received at two different
positions(position1&2inFigure2). Inourradiointer- (cid:90)
ferometer, the signals that arrive at the two side mirrors P(θ)= E(θ0)dθ0[1+cos(2πBλ(θ−θ0))], (10)
(Figure1)areguidedtotheantennaandmixed. Thesep-
aration between the two mirrors, called baseline length where E(θ ) is an intensity/energy density distribution
0
B, causes a time delay τ in the arrival of the signal at oftheobject. OuraddinginterferometermeasuresP(θ);
position2becauseofthegeometry(Figure2). Usingthe we slew the telescope across the object in the azimuthal
anglesofthetelescopepointingθ andtoanobjectinthe direction and obtain fringes, i.e., variations in the power
skyθ ,asimplegeometriccalculationprovidesthedelay, as a function of θ.
0
Bsin(θ−θ ) B(θ−θ )
τ = 0 ∼ 0 (1) 2.3. Point Source
c c
The energy density of a point source is a δ-function
wherecisthespeedoflight. Weusedthesmallangleap- at the position of the object θ = θ . By adopting the
0 c
proximation,sin(θ−θ0)∼θ−θ0,sincemostastronomical coordinate origin to make θc =0, it is
objects have a small angular size.
E(θ )=E δ(θ ). (11)
0 0 0
2.2. Total Power
Combining with eq. (10), we obtain
Radiosignalsareelectromagneticradiationandcanbe
P(θ)=E [1+cos(2πB (θ−θ ))]. (12)
described in terms of an electric field E and a magnetic 0 λ 0
fieldB. Forsimplicity, weconsideronlytheelectricfield
As we sweep the telescope from one side of the object to
E in the following calculations (but this simplification
the other, we should see a sinusoidal power response as
does not lose the generality of the discussion). If we
a function of θ.
define the radio signal at frequency ν that is detected at
Figure3(top)showsthetheoreticalfringepatternfrom
position 1 (or reflected if a mirror is there) at time t as,
a point source. Our satellite dish (and any other radio
telescope) has a directivity; its response pattern tapers
E (t)=E(θ )cos[2πνt], (2)
1 0
off away from the center. The pattern that we actu-
thesignalthatisdetectedatposition2atthesametime ally obtain is attenuated by the dish response pattern
is, (beam pattern) as shown in Figure 3 (bottom). Com-
E (t)=E(θ )cos[2πν(t−τ)], (3) mercial broadcast satellites are very small in angle and
2 0
approximate point sources.
because of the geometric delay τ. Fringe measurements are useful in determining the
An adding interferometer adds the two signals and baseline length B . The total power is zero when the
λ
measures total power of the two. The total electric field normalized baseline is B (θ−θ )=n+1/2, where n is
λ 0
is an integer. The separation between adjacent null posi-
E (t)=E (t)+E (t). (4) tions is δθ =1/B =λ/B.
tot 1 2 λ
3
Fig. 1.—ConceptualsketchoftheMichelsonradiointerferometer.
λ/B
Fig. 3.—Exampleplotsofthetotalpowerasafunctionoftele-
scopepointingθinthecaseofapointsource. Top: Fringepattern
Fig. 2.—Schematicillustrationofsignaldetectionwithtwode-
(eq. 12). Bottom: Fringe pattern attenuated by the telescope
tectors separated by the baseline length B. The direction of the
beampattern. Thedotted-lineisaGaussianbeampatternwitha
telescopepointingisθandthattoanobjectinskyisθ0,bothfrom FWHMof1degree.
anarbitraryorigin. 1 (cid:20) (cid:90)
= cos(2πB θ) E(θ )cos(2πB θ )dθ
S λ 0 λ 0 0
0
2.4. Extended Source and Visibility (cid:90) (cid:21)
+sin(2πB θ) E(θ )sin(2πB θ )dθ (17)
An astronomical object is often extended. In general, λ 0 λ 0 0
an interferometer measures the Fourier transform of the
≡V (B )cos[2πB (θ−∆θ)]. (18)
energy density distribution E(θ ). Here we prove this. 0 λ λ
0
From eq. (10) we define the visibility V (B ) as fol- Here, the visibility V (B ) and the phase shift ∆θ are
0 λ 0 λ
lows: defined as
(cid:90) 1 (cid:90)
P(θ)= E(θ0)dθ0 V0(Bλ)cos(2πBλ∆θ)=S E(θ0)cos(2πBλθ0)dθ0(,19)
0
(cid:90) 1 (cid:90)
+ E(θ0)cos(2πBλ(θ−θ0))dθ0 (13) V0(Bλ)sin(2πBλ∆θ)=S E(θ0)sin(2πBλθ0)dθ0(,20)
0
≡S [1+V(θ,B )], (14) which lead to
0 λ
1 (cid:90)
where (cid:90) V0(Bλ)=ei2πBλ∆θS E(θ0)e−i2πBλθ0dθ0. (21)
S ≡ E(θ )dθ (15) 0
0 0 0
The first term ei2πBλ∆θ is a phase shift ∆θ of a complex
and visibility. The visibility amplitude is therefore
V(θ,Bλ)≡S1 (cid:90) E(θ0)cos[2πBλ(θ−θ0)]dθ0 (16) |V0(Bλ)|=(cid:12)(cid:12)(cid:12)(cid:12)S1 (cid:90) E(θ0)e−i2πBλθ0dθ0(cid:12)(cid:12)(cid:12)(cid:12). (22)
0 0
4
Pmax
Pmin
Fig. 4.—Exampleplotsofthetotalpowerasafunctionoftele- Fig. 5.—Visibilityamplitudeasafunctionofbaselinelengthin
scopepointingθincaseofdisk(liketheSun). Top: Fringepattern thecaseofadisk.
(eq. 12). Bottom: Fringe pattern attenuated by the telescope The Sun’s E(θ ) can be approximated as a top-hat
beampattern. Thedotted-lineisaGaussianbeampatternwitha 0
function. Assuming the Sun’s diameter is α, it is
FWHMof1degree.
(cid:26)
ThisisaFouriercomponentoftheobjectE(θ )atabase- 1, if |θ |<α/2
0 E(θ )= 0 (27)
line length of B . The inverse 1/B is the angular size 0 0, otherwise
λ λ
of the Fourier component in radians. Observations at
The Fourier transform is
long baseline lengths detect structures of small angular
size(i.e.,Fouriercomponentscorrespondingtosmallan- sin(πB α)
gular structures), while those at short baselines capture |V0(Bλ)|= πBλ . (28)
structures of large angular size. λ
Figure 4 (top) shows the theoretical fringe pattern This is a sinc function (Figure 5). By fitting, we de-
for the top-hat function (e.g., the Sun’s disk in 2- termine the parameters of this sinc function, which can
dimensions). Thepatternisalsoattenuatedbythebeam be translated to the diameter of the Sun α. (This is
pattern (Figure 4 bottom). a 1-dimensional approximation of the Sun’s shape. A
moreambitiousexercisewouldbetouseamoreaccurate
2.5. Visibility Measurements and Sun’s Diameter treatment of its 2-dimensional shape.)
WemeasureP(θ)andcalculatethevisibilityamplitude
3. INSTRUMENTS
|V (B )|. From eqs. (14) and (18), we have
0 λ
We describe the construction of the telescope and re-
P(θ)=S0[1+V0(Bλ)cos[2πBλ(θ−∆θ)]] (23) ceiver system. The budget is often the main limitation
in the development of student lab experiments. Hence,
Figure 4 (bottom) is what we see toward the Sun –
weutilizedlow-costpartsandmaterialsandusedacom-
we sweep across the Sun by slewing the telescope in the
mercial broadcast satellite dish and feedhorn operating
azimuthal direction (i.e., changing θ). The fringe pat-
at radio X-band. The system was constructed in our
tern is attenuated by the antenna response pattern, but
machine and electronics shops. Fabrication of the com-
we assume that the antenna response is approximately
ponents could be offered as a student lab projects.
constant around the peak of the response pattern. The
maximum and minimum powers of the sinusoidal curve
3.1. Telescope and Optics
(see Figure 4 bottom) are
Figure1showsthedesignoftheMichelsonstellarradio
Pmax=S0[1+V0(Bλ)] (24) interferometer. Radio signals from the Sun hit two flat
P =S [1−V (B )]. (25) mirrors at the sides and are reflected to a satellite dish
min 0 0 λ
antennabythecentralflatmirrors. Thesignalsfromthe
From these, we calculate
two sides are mixed as detected. Figure 6 shows photos
P −P of the telescope. It was built with mostly commercial
|V0(Bλ)|= Pmax+Pmin. (26) products and materials. A broadcast satellite dish and
max min feedhorn(blueinFigure1;Figure6a,b)operatesatafre-
This is the visibility amplitude at a baseline length of quency of ν ∼11 GHz (λ∼2.7 cm in wavelength). The
B . required accuracy of optics at this wavelength is about
λ
ThetwosidemirrorsslideontheladderinFigure1and ∼3-5 mm, which is relatively easy to achieve with flat
change the baseline length. We repeat measurements of mirrors (without curvature).
|V (B )| at different baseline lengths and make a plot The flat mirrors (green in Figure 1) are made of fiber-
0 λ
of |V (B )| as a function of B . |V (B )| is a Fourier board with wooden framing structures (Figure 6e). The
0 λ λ 0 λ
componentof E(θ ); therefore, weshouldseetheFourier mirror surfaces are all angled 45deg from the optical
0
transformation of the emission distribution in the plot. path. We originally covered their surfaces with kitchen
5
aluminum foil, which has an appropriate thickness with The output from the receiver box goes to a commer-
respect to the skin depth (∼ 0.8µm) at the operating cial LabPro A/D convertor. The LabPro is connected
wavelength (reflectivity ∼ 96% from our lab measure- via USB to, and controlled by, a laptop computer with
ments). Later, we replaced it with thin aluminum plates LabPro software installed. It takes care of time integra-
as student-proofing (Figure 6d). The two side mirrors tion and sampling rate for voltage measurements.
slide on a ladder to change the baseline length. Table 2 lists the electronics components that we pur-
The azimuth-elevation mount structure is made with chased. The square-law detector (Schottky diode detec-
plywood (red in Figure 1 and blue and yellow in Figure tor) was purchased through eBay, and similar devices
6). The azimuthal and elevation axes are driven with seem almost always on sale there. We then found and
motors(Figure6c),whicharecontrolledbyapaddle(i.e., purchased the amplifier and attenuators to adjust the
handset in Figure 6b). The protractor (Figure 6f) is signalvoltageamplitudetoadjusttheinputrangeofthe
placed at the center of the bottom mount plate (yellow detector and the output range of the LNBF when the
in Figure 6b) for measurement of the azimuthal angle of telescope is pointing toward the Sun and satellites.
thetelescope. Figure6ashowsthewholestructureofthe
telescope. A metal pole is mounted perpendicular to the 4. SETUPANDMEASUREMENTS
topmountplate(Figure6b)andaluminumframe(Figure 4.1. Setup
6c), and supports the dish. Note that the pole should
The mount structure, ladder, and mirrors of the tele-
be perpendicular, which makes the pointing adjustment
scope (Figure 6) are detached when it is stored in our
easier as discussed later.
physics building. We move them witha cart tothe front
The azimuthal rotation is facilitated by greased hand-
of the building and assemble them there on the morn-
crafted ball bearings in circular grooves around the az-
ing of experiment. We make sure that the flat mirrors
imuth shaft on the base (blue in Figure 6a,b - below the
areangledat45deg withrespecttotheopticalpathand
yellow structure) and on the bottom mount plate (yel-
90deg vertically, using a triangle. We then attach the
low).
ladder and mirrors to the mount structure using clamps
Sweeping across the Sun in azimuth permits fringe
mounted on the structure.
measurements. This telescope can be converted to a
Theelectroniccomponentsarealsoconnected: thesig-
single-dish telescope by flipping the satellite dish by
nal from the feedhorn goes to the receiver (Figure 7),
180 degrees around the metal pole (see Figure 6b).
then to the A/D converter LabPro, and finally to a
Single-dishandinterferometermeasurementscanbeeas-
computer via USB. We use software which comes with
ily made and compared, which is essential for apprecia-
LabProtocontrolsamplingfrequency(integrationtime)
tion of the high angular resolution possible with the in-
and duration of recording.
terferometer.
Telescope pointing adjustment is the next step be-
Table1liststhecommercialproductpartsthatwepur-
fore the experiment. We prepare a table of the
chased. The other parts, mostly the support structure,
Sun’s azimuthal and elevation angles as a func-
are made in the machine shop.
tion of time (e.g., at 10min interval) using an on-
3.2. Receiver System line tool provided by the U.S. Naval Observatory
(http://aa.usno.navy.mil/data/docs/AltAz.php). The
The signal detection system in radio astronomy is a
antenna is set to the single-dish mode (i.e., dish facing
series of electronic components. Figure 7 shows the de-
towardtheSun). Wealigntheplanesofthemount’stop
sign and photos of the receiver. Again, these are mostly
plateandladderparalleltosunlightusingtheirshadows.
commercial products.
The azimuth is set to that of the Sun, and we adjust the
Signals from the sky are at too high a frequency (∼11
elevation angle of the dish to maximize the signal from
GHz) to be handled electronically. Hence the Low Noise
the Sun on the voltage meter. [Our dish is an off-axis
Block Feedhorn (LNBF) down-converts the frequency to
paraboloid antenna, and the direction of the dish looks
alowerfrequency, calledtheintermediatefrequency(IF;
very offset from the direction of the Sun. We therefore
950-1950MHz),bymixingtheskysignalwithareference
need to use the voltage meter. We later installed a foot-
signal at a slightly-offset frequency and producing a sig-
longrodonthedishandmarkedapoint(onthedish)at
nalatthebeatfrequencyoftheskyandreferencesignals.
which the shadow of the rod tip falls when pointed to-
This is called heterodyne receiving. The LNBF works as
ward the Sun.] We then flip the dish by 180deg around
a heterodyne mixer.
the metal pole for interferometer measurements.
Figure 7 shows the flow of signal. In sequence, an am-
The signal amplitudes from the two side mirrors need
plifier, two attenuators, and bandpass filter adjust the
to be balanced. We check the voltage readout from each
signal amplitude to the input range of a square-law de-
side mirror separately by blocking the optical path of
tector. We combined two commercially available atten-
the other (or by removing the other mirror). We move
uators to achieve the desired attenuation of ∼ 16 db.
the central mirror toward the side of stronger signal to
A filter with a 100 MHz width narrows the frequency
decrease its effective surface area.
range, since the bandwidth of the IF (1GHz at the oper-
ating frequency of ∼11 GHz) is too broad for detection
4.2. Measurements
of null fringes in interferometry. Output from the de-
tector is then amplified to the whole dynamic range of Once the mirrors are set and the telescope is pointed
the analog-to-digital (A/D) converter. We assembled all toward the Sun, we start interferometer measurements.
these components inside a metal box for protection. A We should see fringes from the Sun (e.g., Figure 4) as
power supply is also in the box, providing the power to we slew the telescope and sweep across the Sun in the
the LNBF and amplifiers. azimuthaldirection. Wetypicallyspend10-30secondon
6
(a) (b) (c)
(d) (e) (f)
Fig. 6.— Photographsofthetelescope. (a)Overallview. (b)Mountstructure. Theblueboxatthebottom(withhandles)andyellow
plates are made of wood. The entire yellow part rotates in the azimuthal direction on the blue box. The two yellow plates are attached
withhinges,andthetopplatemovesuptochangetheelevationangle. Thetelescopeisshownina”single-dish”mode,andthedishwould
be rotated by 180deg for an ”interferometer” experiment. (c) Support structure. The aluminum frame supports the telescope. A screw
rod and elevation drive motor are also visibile. (d) Side mirror from the front side. Kitchen aluminum foil is thicker than the required
skindepth,butwegluedathinaluminumplateinstead,asstudent-proofing. (e)Sidemirrorfromthebackside. It’ssupportedbyawood
frame. (f)Protoractortomeasuretheazimuthalangleofthetelescope.
each”sweep”observation,andthencorrectthetelescope and cannot pass through exit doors of our building. We
pointingbeforethenextsweep. Thepatternmaybeseen have to carry it out via a loading deck. This could have
as variations of the voltage readout, or as a fringe pat- beentakenintoaccountwhenthetelescopewasdesigned.
terninaplot(Figure4),iftheLabProandcomputerare The telescope can be used as a single-dish radio tele-
already started. The LabPro and the computer do not scope by pointing the dish directly toward the sky. The
knowabouttelescopepointingandrecordonlytheread- beam size of our dish is roughly ∼ 1deg in X band,
out voltage as a function of time. We therefore need to with which we can barely resolve the Sun (∼ 1/2deg
convert the time to azimuthal angle after the measure- diameter). We can compare the profiles of the Sun and
ments. We record the start and end azimuthal angles a commercial satellite (a point source) to find this ex-
in sweeping the Sun – we start from a far-off position, perimentally. The Sun’s diameter can be resolved and
say 10-20deg away in azimuth, and sweep the Sun in az- determined with the interferometer. The comparison of
imuth. We assume that the telescope slew speed is con- thesingle-dishandinterferometermeasurementspermits
stant (approximately correct when we record for a long students to appreciate the superiority of interferometry
time, e.g. 20-30 seconds). The projection effect, i.e., the in terms of spatial resolution.
cos(elevation)term,mustbeaccountedforincalculation
of arc length in the sky. 5. RESULTSFROMALABREPORT
We change baseline length by sliding the side mirrors Figure 8 shows results from a student group’s lab re-
ontheladderandrepeatfringemeasurements. Thebase- port, Panel (a) is an example of a fringe pattern of the
linelengthshouldbedeterminedfromthefringepattern, Sun. They determined the baseline length by measuring
but for reference, we record the side mirror separation the interval between peaks and troughs (and from their
using a tape measure fixed to the ladder. readings of the side mirror separation). This group re-
peatedfringemeasurementsthreetimesateachof10dif-
4.3. Miscellaneous
ferent baseline lengths. Panel (b) shows a fit of the sinc
Radio interference was initially a problem. We con-
function, i.e., theFouriertransformoftheSun. Thenull
ducted a site search across the campus. We brought the
point at B = 96 in the fit suggests that their measure-
λ
dishandacommercialreceiver(calledasatellitefinder∼ ment of the sun’s diameter is ∼ 36(cid:48) at ∼ 11 GHz. Note
$10-20,whichisusedtofindcommercialtelevisionsatel- thatitsreporteddiameterat∼10GHzisabout34(cid:48) with
liteswhenadishisinstalled)andcomparedthestrengths
littledependenceonsolaractivity(i.e.,sunspotnumber);
of the Sun and ambient radio signals. We conveniently
this diameter is calculated from the observed radio-to-
found that one spot in front of our building was radio
optical diameter ratios (Das, Sarkar, & Sen 2000) and
quiet. theopticaldiameterof∼30(cid:48). Theseresultsdemonstrate
Geosynchronoussatellitesarelocatedalongathinbelt
a proof of concept demonstrated by our students, and a
in the sky. The Sun’s sidereal path gets aligned along
variety of exercises can be developed for a student lab
this belt in some seasons, which hinders the experiment.
beyond what is described here.
This should be checked at the planning stage of the ex-
periment.
The current mount structure is slightly wider than a We thank Peter Koch, the previous Chair of the De-
standarddoorway. Itdoesnotfitonmostofourelevators partmentofPhysicsandAstronomyatStonyBrookUni-
7
(a) (b)
(c)
(d)
+5 VDC
PPOOSS
1A
SQUARE LAW OUTPUT TO
DETECTOR DC AMPLIFIER
(9) (10) LABPRO (14)
3) $30 $20 1A $220
ER(1 NNEEGG
T
EC
MD
TV
L
O5
V- +5 VDC +5 VDC PWR SUP
ANALOG 0 $0 G+N5D VDC (11$)8 1A I2N4P UVTDC ODNIES HM EDTSE-R3100
(1)
$90
+5 VDC
B.P. FILTER ATTENUATOR ATTENUATOR ZX60-2534M+ 75-50 OHM
1.35 - 1.45 GHZ 6 DB (7) 10 DB (6) AMPLIFIER (5) ADAPTOR PWR INSERTER 1A LNBF (2)
VBFZ-1400-S+ $19 $19 $65 PE7075 (4) (3) $55
(8) $40 $83 $2
Fig. 7.— Photographs and schematic of the receiver. (a) Interior of receiver box. Most components are commercial. (b) Front side of
the receiver box. Two critical plugs are for an input from the feedhorn and output to the LabPro (commercial analog/digital converter
oftenusedinphysicslabcourses,whichoutputstoacomputerthroughaUSBconnection). (c)Backside. Weinstalledananalogvoltage
meter,sothatsignaldetectioncanbeeasilycheckedduringobservations. (d)Schematicdiagramofreceivercomponents.
versity, for providing funds to develop this experiment. us use plots from their lab report. This work is sup-
We also thank Munetake Momose for useful discussions. portedbytheNSFthroughgrantAST-1211680. JKalso
We also thank students in the lab course, Kendra Kel- acknowledges the supports from NASA through grants
logg, Melissa Louie, and Stephanie Zajac, for letting NNX09AF40G, NNX14AF74G, a Herschel Space Obser-
vatory grant, and a Hubble Space Telescope grant.
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TABLE 1
Commercial Products Purchased for Telescope Mount
No Description Quantity Manufacturer PartNo. Vendor Price
1 ManholeLadder16ft 1 Werner M7116-1 Lowe’s $226
2 Motor 2 Dayton 1LPZ7 Walmart $248
3 Lev-O-Gage 1 SunCompany,Inc. NWH-0152-1003 opentip.com $18
TABLE 2
Purchased Receiver Parts
No Description Quantity Manufacturer PartNo. Vendor Price
1 1-MeterSatelliteDish 1 WINEGARD DS-3100 SolidSignal $90
2 QuadPolarLNBF 1 INVACOM QPH-031 SatPro.tv $55
3 PowerInserter 1 PDI PDI-PI-1 SolidSignal $2
4 75-50OhmAdaptor 1 PASTERNACK PE7075 Pasternack $83
5 Amplifier501/20.5to2.5GHz 1 Mini-Circuits ZX60-2534M+ Mini-Circuits $65
6 AttenuatorSMA3GHz50Ohm10db 1 Crystek CATTEN-0100 Digi-Key $19
7 AttenuatorSMA3GHz50Ohm6db 1 Crystek CATTEN-06R0 Digi-Key $19
8 BandpassFilter1350to1450MHz 1 Mini-Circuits ZX60-2534M+ Mini-Circuits $40
9 Square-LawDetector1.0-15.0GHz 1 OmniSpectra Model20760 eBay $30
10 5XOP-Amp 1 CustomBuilta $20
11 ICBuckConverterMod5.0VSIP3 1 ROHM BP5277-50 Digi-Key $8
12 BoxAluminum4’×6’×10’(HWD) 1 LMBHeeger UNC4-6-10 DigiKey $45
13 0-5VAnalogMeter4¨ 1 Salvaged $0
14 DataConverter&Collection 1 Vernier LabPro Vernier $220
aThiscomponentcouldbesimplysomebatteriesthatprovidethevoltageof∼5V.
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