S udy o he de e mina ion o he opocen ic luna
lib a ions by a bes i es ima e o he pla e cons an s om
digi al images wi h a small elescope
Fab izio Pin o
EKOSPACE and Depa men o Ae ospace Enginee ing, Izmi Uni e si y o Economics
Tele e ik Mahallesi, Saka ya Cd. No: 156, 35330 Balço a/İzmi , Tü kiye.
ab izio.pin[email p o ec ed].
Abs ac
The de e mina ion o he luna op ical lib a ions in longi ude and la i ude om di ec obse a ion o he Moon
ep esen s an exci ing, hough possibly challenging, s uden p ojec wi h a p o ound pedagogical po en ial. In
p inciple, i is possible o compu e he op ical lib a ions om he small numbe o obse a ions needed o ob ain
he pla e cons an s. In p ac ice, howe e , he pla e cons an s a e mo e e ec i ely ob ained by a bes i o he
posi ions o a la ge numbe o luna ea u es on a digi al image by means o he well-known ela ions o he
selenog aphic coo dina es a ailable om a luna a las. Fi s ly, he e we epo on an analysis o he ela ionship
be ween he inal accu acy in he de e mina ion o he lib a ions by his me hodology and he se e al ac o s ha
a ec he esul . Fo his pu pose, he undamen al pho og amme ic equa ions a e s udied by means o simula ions
o in es iga e he e ec o he numbe o ea u es used, hei posi ion on he luna ace, and he esol ing powe
o he op ical sys em. Secondly, ac ual measu emen s a e ca ied ou on digi al images acqui ed by he au ho by
means o a Canon 500D a he p ime ocus o a Meade ETX-90. These measu emen s a e also u he augmen ed
by he es ima es o a class o uni e si y sophomo e s uden s wo king independen ly in an In oduc ion o Remo e
Sensing class. Addi ional e inemen s a e conside ed o u u e wo k, such as pa allac ic e ec s, he diu nal
pa allax and he e olu ion o lib a ions du ing he luna o bi .
1. In oduc ion
The op ical lib a ions o he Moon a e, echnically
speaking, su icien ly la ge o be isible by he
unaided eye. Since mul iple obse a ions sepa a ed in
ime a e needed, his o ical esea ch has ocused on he
possibili y ha keen sky obse e s and gi ed a is s,
including Leona do himsel , may be among hose who
i s eco ded su icien ly de ailed images o he luna
disk, al hough likely wi hou ecognizing his
phenomenon (Tucci 2022). A e he in oduc ion o
he as onomical elescope, epo s abou lib a ions
bo h in la i ude and in longi ude apidly appea ed
(Włoda czyk 2011), commencing wi h an ea ly
handw i en no e by Thomas Ha io (c. 1560-1621)
in Decembe 1611, ollowed by he obse a ions by
Flo en ius an Lang en (1600–75), Galileo Galilei
(1564–1642), and, o cou se, by Johannes He elius
(1611-1687).
The pedagogical po en ial o he obse a ion and
measu emen o luna lib a ions in as onomy classes
is now widely ecognized and ull discussions o
possible p ojec s a e a ailable (Buchheim 2015). This
opic has been a long ime in e es o he p esen
au ho (Hughes 1993), who has de o ed e o s o
include i in he cu iculum since he dawn o he
comme cial digi al came a ma ke (Pin o 1995).
I is immedia e o de elop an awa eness o he
phenomenon o lib a ions by iewing images o he
Moon aken a di e en imes, as d ama ically shown
in a e y popula ideo de eloped a NASA Godda d
(2011). Howe e , he goal o ca ying ou eliable
measu emen s o lib a ions om indi idual images o
he Moon aken wi h small elescopes o e s some
challenges o a echnical and ma hema ical na u e well
wo h he ime o as onomy and space enginee ing
unde g adua e s uden s. Geome ical conside a ions
show ha he posi ions on a digi al image o 4 luna
ea u es o known selenog aphic coo dina es a e
su icien , in p inciple, o de i e he in o ma ion
needed o compu e he lib a ions. Howe e , as we
shall see, a emp s o nai ely ollow his minimalis
s a egy lead o unsa is ac o y esul s, due o he
limi ed accu acy a ailable when using o dina y, small
elescope sys ems. Tha his should be o no su p ise
2
Figu e 1. The geome y o he measu emen o he
posi ion o a poin P upon a came a senso . In addi ion o
an ob ious scale ac o , he senso coo dina es (X, Y)
mus be connec ed by ans o ma ion laws o he
s anda d selenog aphic Ca esian coo dina es, (, , ).
This in ol es all angles ha cause depa u es om he
idealized geome y o his Figu e, ha is, lib a ions in
la i ude and longi ude, and o a ion o he ield o iew
(see ex ). No ice he de ini ion o longi ude P, posi i e
eas wa d, owa ds Ma e C isium, shown by he gene ic
angle , and o he la i ude P, posi i e owa ds he
hemisphe e con aining Ma e Imb ium (no indica ed
abo e o a oid clu e ) (see Kopal (1966), p. 169).
can be bes app ecia ed by conside ing ha such a
amous obse e as Saunde (1900, 1901, and 1905)
epo ed con inued challenges measu ing he
selenog aphic coo dina es o luna ea u es, as
di icul as de ec ing he pa allaxes o s a s. Saunde
was using he mos ema kable o he F ench Coudé
equa o ials, he G and Coudé e lec o a Pa is, an
imp essi e ins umen wi h a 60 cm objec i e and a
ocal leng h o 18 m. The ascina ing a icle by
Lequeux (2011) includes his o ic pho og aphs o hose
ins umen s, including o he sophis ica ed con ols a
he ocal poin . The images o he Moon ob ained by
such Coudé e lec o s we e so exquisi e as o be
p oduced a 80 cm o diame e “on special pla es
supplied o ee by he Lumiè e b o he s,” and hey
we e “s ill ound o be use ul when choosing he
landing si es o he Apollo p ojec ” (Lequeux 2011).
In his pape , we analyze such challenges and
ocus on a speci ic issue wi hin he b oade goal o
ca ying ou s uden luna lib a ion measu emen
p ojec s. This is he e o analysis ha mus
accompany any inal es ima e. In o de o ex ac mo e
eliable in o ma ion om a ailable digi al images, a
s a is ical app oach based on he measu emen o he
coo dina es o a much la ge numbe o su ace
ea u es has been de eloped. By a bes i , he Pla e
Cons an s can be es ima ed, leading o alues o he
lib a ions, accompanied by an unde s anding o he
sou ces o e o .
One u he elemen ha has enabled b inging
such ac i i ies wi hin each o s uden s in
unde g adua e classes has been he de elopmen o
compu e algeb a sys ems (CAS), such as ha
a ailable in he Ma hema ica language. Indeed,
pe sonal disco e y o Ma hema ica by his au ho
occu ed speci ically du ing he sea ch o a simple
app oach o sha ing da a analysis o po en ially
hund eds o coo dina es om a single image o
lib a ion calcula ions. The inhe en powe o he
Ma hema ica language (Wol am 2013), along wi h
he a ailabili y o he no ebook pla o m, also
in oduced by S e e Wol am, ha e ep esen ed
pe sonal ‘game change s’ in he pedagogical ea men
o his p oblem. In ecen yea s, he F ee Wol am
Engine has become a ailable o use on he highly
lexible Jupy e No ebooks, a de elopmen o his o ic
impo ance ha has u he immensely inc eased he
pedagogical po en ial o his app oach (Rome 2018,
Some 2018, Rule 2019)..
In he spi i o sha ing he p esen app oach ia
such powe ul, mode n ools, a Jupy e No ebook was
p epa ed and is a ailable as Supplemen a y Ma e ials,
wi h LaTeX- o ma ed equa ions, wo ked examples,
and plo s (Pin o 2025). This Supplemen a y Ma e ials
ile, s o ed in he Open Science F amewo k, is w i en
in Jupy e No ebook o ma (ipynb), ully execu able
and edi able upon ins alla ion o he F ee Wol am
Engine; i can be also downloaded in h ml language
(no execu able) o simple iewing.
The wo k low leading o he image p o ided and
o he measu emen s, including o he so wa e used, is
b ie ly desc ibed below. Howe e , ou emphasis is on
he analysis o he sensi i i y o lib a ion calcula ions
o a ious impo an ac o s, explo ed by using
syn he ic da a; la e , one sample image o he Moon is
used along wi h da a p oduced by almos 500 s uden
measu emen s, which a e analyzed by means o he
algo i hms p esen ed in he Supplemen a y Ma e ials.
2. Theo e ical backg ound
Following he s anda d app oach (Saunde 1900,
1901, and 1905; Kopal 1966, Kopal and Ca de 1974),
he undamen al equa ions de ining he selenog aphic
Ca esian coo dina es o a poin P on he luna su ace)
in e ms o he selenog aphic longi ude and la i ude,
(P, P) a e:
P = cos P sin P , (1)
P = sin P ,
P = cos P cos P,
P 2 + P2 + P2 = 1,
assuming ha he Moon is sphe ical. Wi h e e ence o
Fig. 1, i is clea ha he o igin o he (P, P) e e ence
ame would be ep esen ed by a poin a (X=0, Y=0)
on he senso in he absence o lib a ions and ield
o a ion (we de ine he ield o a ion angle as he angle
made by he Y-axis wi h he pola axis o he Moon,
ha is, he depa u e om an ideal No h-up
o ien a ion). I he disk o he Moon is no cen e ed on
he o igin o he senso e e ence ame, u he e ms
mus be added. In gene al, apa om a scale ac o
equal o he a io o he physical adius o he Moon o
i s adius on he senso , he abo e equa ions simply
imply ha , i he Moon is cen e ed on he senso , he
coo dina es o he image o poin P on he senso ,
poin , Q, a e connec ed o i s selenog aphic Ca esian
coo dina es as:
XQ = P , (2)
YQ = P .
In he p esence o lib a ions (lib, lib), howe e , hese
equa ions become:
XQ = P cos lib - P sin lib , (3)
YQ = - P sin lib sin lib + P cos lib - P cos lib sin lib.
Finally, i he ield is o a ed by an angle o du ing he
exposu e, he abo e equa ions mus be u he
modi ied by he s anda d o a ion ans o ma ion,
R( o ), a ound he -axis.
X’Q = XQ cos o - YQ sin o , (4)
Y’Q = XQ sin o +YQ cos o .
By subs i u ing Eqs. (3) in o Eqs. (4), we ob ain he
ull esul (neglec ing e ms ha desc ibe non-
cen e ing). Fo succinc ness, he e we do no ep oduce
he ull exp ession (Supplemen a y Ma e ials, Sec.
2.1). The conclusion o be d awn om his ea men
is ha Eqs. (2) mus be comple ely gene alized o
conside he coo dina es o he senso as linea
unc ions o all selenog aphic Ca esian coo dina es.
Including cons an s ha will also desc ibe bo h he
scale ac o and non-cen e ing, we ha e:
X’Q = A P + BP + C P + D, (5)
Y’Q = E P + FP + G P + H.
whe e (A, …. H) a e e e ed o as he Pla e Cons an s.
As appa en om he abo e discussion and p o en
nume ically in he Supplemen a y Ma e ials, a
minimum o 4 measu emen s is needed o de e mine
each o he wo se s o cons an s, which allow o he
compu a ion o he op ical luna lib a ions.
The Pla e Cons an s can hen be used, o ins ance,
o compu e he adius o he Moon on he senso , in
pixels, by using he wo equi alen equa ions:
RMoon, X = (A2 + B2 + C2)1/2, (6)
RMoon, Y = (E2 + F2 + G2)1/2,
RMoon, X = RMoon, Y .
Also, he cen e o he luna disk in senso e e ence
ame in he absence o lib a ions and ield o a ion is
gi en by:
XCen e = D , (7)
YCen e = H .
Le us now p oceed o ex ac he desi ed in o ma ion
om he Pla e Cons an s. F om Eqs. (3), we see ha
he coo dina e P appea s only in he exp ession o
YQ. The e o e, by di iding he Pla e Cons an s o he
coo dina e P in Eqs. (5), B and F, by each o he and
wi h a sign change due o he nega i e sign a he igh -
hand-side o he o me o Eqs. (4), we can elimina e
o :
o = - a c an 𝐵
𝐹 (8)
Since we now ha e he o a ion angle, we can apply
he in e se ans o ma ion o bo h Eqs. (5), so as o
e u n o he un o a ed exp essions a Eq. (3):
R-1( o )(𝑋’
𝑌’) = R(- o ) (𝑋’
𝑌’) . (9)
The longi ude lib a ion can hen be ob ained om he
obse a ions in a ious, equi alen ways by
ecognizing ha , o ins ance:
lib, OBS = - a csin C’ , (10)
whe e C’ is he -coo dina e coe icien a e he
o a ional back- ans o ma ion. Since his lib a ion
only depends on he P–coe icien , unnecessa y
complica ions may be a oided by le ing he R(- o )
ope a o ac only upon a column ec o , (C, G). Once
lib, OBS is known, o he lib a ion in la i ude we ha e:
lib, OBS = - a csin (G’/cos lib, OBS), (11)
4
Figu e 2. The comple e op ical sys em (see ex ).
whe e 𝐶′ and 𝐺′ a e he ‘un o a ed’ coe icien s:
(𝐶’
𝐺’) = R(- o ) (𝐶
𝐺). (12)
3. The ha dwa e
In o de o c ea e applicable syn he ic da a, he
ollowing in o ma ion ega ding he ha dwa e
a ailable o ac ual image acquisi ion is p o ided. The
elescope used is a Meade ETC-90 a Maksu o -
Casseg ain e lec o wi h a nominal ape u e lis ed as
D = 90 mm (3.5”) and ocal leng h FL = 1,250 mm,
co esponding o an /13.8 ape u e ( o echnical
specs, ollowing he demise o Meade Ins umen s,
see, o ins ance, Meade Ins umen s, Meade ETX
Telescopes, and Op icsS a , 2025, in he Re e ences).
Imaging is ca ied ou by a Canon 500D came a body
(Pin o 2024, and Re e ences he ein) placed a he
p ime ocus by a TelescopeAdap e s ing moun
(h ps://www. elescopeadap e s.com/ ) and powe ed
by an ex e nal powe supply. The sys em is assembled
on an equa o ial moun bu no acking is used in his
case (Fig. 2). The came a senso is a CMOS Type
APS-C (14 bi ) wi h a maximum esolu ion o (4,752
3,168) pixels and a pixel pi ch o spix = 4.68 m.
Focusing is achie ed by isually moni o ing he image
on he back LCD sc een magni ied elec onically a up
o 10 imes, while p o iding adjus men s by means
o he Meade elec ic ocuse (Fig.3).
Figu e 3. LCD sc een a 10 magni ica ion (see ex ).
By using he abo e da a and using an a e age
wa eleng h equal o ligh = 500 nm, he heo e ical
esol ing powe o he sys em (Rayleigh c i e ion) is:
es 1.22 ligh /D = 1.40”. (13)
4. Syn he ic da a analysis
In his Sec ion, a ew examples a e discussed in
o de o es ima e he e ec o unce ain ies on he inal
esul s by gene a ing syn he ic da a o assumed ield
o a ion and lib a ion angles and by moni o ing he
accu acy o eco e y o hose assump ions.
As men ioned in he In oduc ion, syn he ic da a
analysis was ca ied ou by means o he Wol am
Ma hema ica language. The F ee Wol am Engine
(WFE), . 11.3, was ins alled on Jupy e No ebook,
se e . 6.5.4, unning on a ela i ely old bu ully
ope a ional hp Pa ilion lap op unde Windows 8.1.
Commencing in 2014, he Ma hema ica language
has ea u ed he powe ul GeoG aphics unc ion,
which can po en ially aid in ca ying ou syn he ic
analyses using ac ual luna e ain (see Supplemen a y
Ma e ials o examples). Howe e , he e we shall no
associa e syn he ic da a o exis ing o ma ions bu
shall ins ead gene a e he selenog aphic coo dina es
by using a andom numbe gene a o . Fo his pu pose,
he equa ions o Sec. 2 we e coded in Ma hema ica and
used o p oduce simula ed measu emen s, (X’i, Y’i)
co esponding o hypo he ical su ace ea u es o
selenog aphic coo dina es (i, i), wi h i = 1 … Nsyn h.
The a bi a y inpu s o he simula ion a e he o al
numbe Nsyn h o such en ies, he anges in
selenog aphic longi ude and la i ude
Figu e 4. Randomly ex ac ed posi ions on he luna disk
o Nsyn h = 4. The axes ep esen he senso e e ence
ame in no malized coo dina es.
(min i max, min i max) in which
measu emen s a e assumed o ha e been aken, he
ield o a ion angle, o , and he lib a ions, (lib , lib ).
The selenog aphic coo dina es (i, i), a e andomly
d awn om uni o m dis ibu ions gi en by he desi ed
coo dina e anges. The co esponding alues
(X’i, Y’i), o he measu ed coo dina es, a e compu ed
as we ha e p e iously discussed. In o de o explo e
he e ec o measu emen unce ain y, noise is
injec ed in o such measu emen se by adding andom
e o s dis ibu ed acco ding o a no mal dis ibu ion
(Taylo 1997) o expec ed alue = 0 and s anda d
de ia ion exp essed in e ms o a dimensionless
mul iplica i e pa ame e o he heo e ical esolu ion
o he op ical sys em ound a Eq. (13), ha is,
= es. A e compu ing he pixel coo dina es
(X’i, Y’i), an a bi a y cons an is added o each,
ep esen ing he o -cen e posi ion o he luna disk.
In o de o connec his simula ion o he op ical
sys em used, he adius o he Moon on he senso was
ob ained om he dis ance o he Moon a he epoch
o he obse a ion o be analyzed in he ollowing
Sec ions (2020-04-30 22:54:04). The epheme ides
used a e hose o he Vi ual Moon A las, which gi es
a dis ance EM = 3.72939506 105 m. Since he physical
adius o he Moon, i assumed sphe ical, is RM =
1.7373 103 km, he adius in pixels is compu ed o be:
Rpix = (RM / EM ) (FL /spix) = 1469.77 pix. (14)
I is impo an o conside ha , in p ac ice, depa u es
o he ac ual ocal leng h and pixel pi ch om hei
nominal alues do exis and a s udy mus be ca ied
ou o de e mine he e ec i e ocal leng h o he
ins umen a he ime o obse a ion, which should be
conside ed a ime a iable e en du ing one session (as
indeed is epea edly men ioned by Saunde ). This ac
can be cap u ed by ano he a bi a y ac o o mul iply
he esul o he p e ious equa ion and o be ob ained
om he measu emen s.
In wha ollows, we b ie ly su ey he esul s. Full
algo i hms and plo s a e in he Supplemen a y
Ma e ials.
4.1 Exac measu emen s
In his i s example, we conside he ideal case
o 4 exac measu emen s ( = 0) and we e i y ha , in
his case, we eco e he assumed a bi a y
pa ame e s. The e o e, Nsyn h = 4, lib = - 7.4 deg, lib
= -3.7333 deg, o = 21.7048 deg. The longi udes and
la i udes a e assumed o be -40 i 40, -40 i
40. The o -cen e cons an s a e Xc = 1250.6 and Yc =
2538.2. By using a andom seed (SeedRandom)
equal o 1234, o epea abili y, we ind, o ins ance
( hese alues a e ounded below o p ac icali y, all
angles a e in deg ees):
X1= 2037.4, Y1= 2981.5, 1 = 30.13, 1 = 1.76
X2= 1489.6, Y2= 2688.9, 2 = 3.500, 2 = 1.65
X3= 1027.0, Y3= 3345.3, 3 =- 3.278, 3 = 30.78
X4= 2141.1, Y4= 3156.6, 4 = 39.98, 4 = 7.035.
The loca ion o he andomly ex ac ed poin s is shown
a Fıg. 4. Upon execu ion o he algo i hm o es ima e
he ield o a ion angle and he lib a ions, all alues a e
eco e ed wi h absolu e nume ical e o s 10-13–10-12.
Also he condi ion ha he wo adii es ima ed om
Eqs. (6) be equal o each o he is sa is ied wi hin an
absolu e alue o 10-11.
The ema kable s abili y o he p ocess can be
app ecia ed by epea ing he compu a ion o 4 poin s
clus e ed in s a e y small a ea o he luna disk. Fo
ins ance, e en by choosing coo dina es bound as 70
deg i 75 deg, 80 i 85) o be conside ed
ex eme om he obse a ional poin o iew he
Ma hema ica algo i hm eco e s he assigned
pa ame e s wi h he same accu acy.
4.2 Random e o s
In his Sec ion, we explo e he e ec o injec ing
andom e o s in o he measu emen s. Fo his
pu pose, we shall epea he compu a ions o p e ious
6
Figu e 5. Showing he log-log plo o he absolu e alue
o he e o in he lib a ion in longi ude as a unc ion o
he pa ame e . Measu emen s spanning 70 deg i
75 deg, 80 i 85 (black) lead o la ge e o s han
hose in a much wide a ea, -40 i 40, -40 i 40
(blue).
cases bu assume ha 0. The cases conside ed a e
shown in Fig. 5 in a ew illus a i e examples. As can
be seen, he choice o measu emen poin s wi hin a
smalle a ea leads o nume ical e o s o de s o
magni ude la ge han in he case o a na ow a ea.
In e es ingly, howe e , i he numbe o poin s
employed is much la ge , he e o again dec eases in
any case, as in ui i ely expec ed. Fo ins ance, by
employing Nsyn h = 100 poin s, he lib a ion in
longi ude e o in he case o a wide su ey a ea (blue
cu e) o = 10-3 dec eases om 2.7 10-3 deg o
5.6 10-4 deg. This app oxima ely co esponds o a
ac o o 25 imes, equal o he a ios o poin s
employed, 100/4. These nume ical es ima es do no
ise o he le el o a s a is ics heo em bu o e
guidance as o whe he he measu emen we seek o
ca y ou is possible wi h he smalle ins umen a ion
a ailable. F om he p ac ical poin o iew, choosing
such small alues o is clea ly unphysical bu , in ou
case, i helps o p esen he d ama ic beha io o he
e o o ex emely small su ey a eas.
As one las example, we conside he case o a la ge
sample (Nsyn h = 497 poin s) he au ho secu ed o e
ime by p esen ing he same image o s uden s en olled
in an In oduc ion o Remo e Sensing class. By
choosing pa ame e s app op ia e o he i s qua e
Moon (0 deg i 90 deg, -90 i +90), we ind,
o a physically pe missible, hough qui e op imis ic,
pa ame e = 1.0, e o s |lib| 0.012 deg, |lib|
0.011 deg, espec i ely. E en o much highe alues
o = 102, he e o s o e such a wide su ey a ea as
hal o he luna ace emain as low as 0.05 deg in
bo h longi ude and la i ude. As we shall see, his is
incompa ible wi h he esul s om s uden
measu emen s, which yielded e o s 0.6 deg in bo h
longi ude and la i ude. This is no physically
consis en wi h he op ical sys em a ailable, as i
Figu e 6. Analogously o Fig. 4, showing andomly
ex ac ed posi ions on he luna disk o Nsyn h = 497 wi h
(0 deg i 35 deg, -60 i +60). The axes ep esen
he senso e e ence ame in no malized coo dina es.
would equi e un ealis ically high alues o he
pa ame e . On he o he hand, o = 20, and
conside ing a es ic ed a ea de ined by (0 deg i
35 deg, -60 i +60), leads o syn he ic samples
ha p oduce he obse ed e o s.
In his Sec ion, o b e i y, we ha e ocused on he
lib a ion in longi ude. Howe e , ou conclusions also
apply o he lib a ion in la i ude, o ield o a ion, and
o he simila i y o he adii along he wo senso axes.
Such da a can be ob ained by accessing he
Supplemen a y Ma e ials.
5. Image Acquisi ion and p ocessing
Fo simplici y o execu ion, unlike deepe sky
objec s (Pin o 2024), no calib a ion ames we e aken.
All images we e sa ed o he p op ie a y Canon RAW
CR2 o ma . This p ojec was ca ied ou bo h wi h
single images and wi h much highe quali y images
de i ed by s acking a la ge numbe o ames. No
a emp has ye been made o de e mine whe he
p ocessing o luna images leads o imp o ed
es ima es o he lib a ions. Al hough ha appea s
easonable, pho og amme y also equi es expe ience
on he side o he obse e . In his case, s uden s
wi hou p e ious as onomical expe ience open a FITS
ile, o c ea e one by image s acking p og ams, and
ca y ou he measu emen s desc ibed by using
ASTAP and he Visual Moon A las as a guide.
Na u ally, pa icula a en ion is d awn o he
e mina o a ea whe eas a eas whe e he Sun is high
on he ho izon on he Moon a e neglec ed.
Figu e 7. The image o he Moon analyzed in his wo k.
The image conside ed he ein is pa o a se acqui ed
on 2020-04-30:19:54:04 (UTC) om he au ho ’s
obse a o y in he Izmi egion. The exposu e ime
was 1/20 s a 200 ISO, while he Moon was
app oxima ely a an al i ude o 43o.
6. Pho og amme ic image analysis
The image shown a Fig. 7 was p o ided o
s uden s in FITS o ma and analyzed by hem by
means o he ASTAP p og am and he Vi ual Moon
A las. This allowed hem o build a ile equi alen o
ha c ea ed syn he ically by Ma hema ica and
discussed ea lie . E e y s uden wo ked on
app oxima ely 20-100 ea u es bu i was no iced ha
some s uden s sha ed choices o he same ea u es,
being p edic ably a ac ed o clea ly iden i iable
o ma ions nea he e mina o , as shown a Fig. 8.
All iles we e hen me ged wi hou pa icula
cu a ion and analyzed by he same algo i hm used o
p oduce he simula ions p esen ed in Sec. 4. The esul
ound is:
o = 21.7048 deg, lib = - 7.99 deg, lib = -4.40 deg.
The e o s on he lib a ions, es ima ed om he model,
a e 0.6 deg. Impo an ly, he s a is ical analysis
ca ied ou by Ma hema ica also shows ha he
con idence le el in e als o he pla e cons an s
ob ained om he model a e o he same o de o
magni ude as hose om he obse ed da a (1-10).
Figu e 8. The luna o ma ions chosen by he s uden
analys s o lib a ion de e mina ion. No ice he much
highe densi y o en ies along he e mina o ( his is an
un o a ed image, No h up, Eas o he igh ).
7. Conclusions
Possibly he mos impo an quan i a i e
conclusion o his pape is ha he c ucial ac o o
de e mine he accu acy o he lib a ion es ima es is he
dis ibu ion o he ea u es o be measu ed. The
syn he ic model shows ha a la ge numbe o
measu emen s can indeed lead o e y accu a e esul s.
Howe e , such po en ial disappea s i pa s o he
luna disk a e no p ope ly sampled. In he u u e, his
au ho in ends o u he explo e hese conclusions in
o de o achie e de ec ion o he diu nal opocen ic
lib a ions and o he physical lib a ions o he Moon.
An addi ional imp o emen o he model, al eady
being es ed, is he inclusion o he ini e dis ance o
he Moon in he da a analysis. The pedagogical
po en ial o his ac i i y is being exploi ed by
ansi ioning in e es ed s uden s o mapping o small
bodies o he sola sys em, such as as e oids and
moonle s, o he pu pose o modeling hei
g a i a ional ield.
3. Acknowledgemen s
I hank my as onomy s uden s a Boise S a e
Uni e si y who, o e h ee decades ago, a emp ed he
i s s eps o his p ojec , pa icula ly Jenni e Sype,
Debbie S ee , and Ca olyn Hughes (Hughes, 1993). I
8
am g a e ul o all s uden s in my In oduc ion o
Remo e Sensing class (AE-202) a he Izmi
Uni e si y o Economics in he yea s 2017-2025,
mos ly sophomo es in ou space enginee ing p og am,
and o some in my As ophysical Sys ems class (AE-
415), mos ly senio s, o hei wo k on digi al luna
pho og amme y. My hea el g a i ude also goes o
all p og amme s who de eloped and made a ailable
he ema kable so wa e used in his wo k unde
a ious licenses.
4. Re e ences
Buchheim, R. K., As onomical Disco e ies You Can
Make, Too! (Sp inge , Cham, Swi ze land, 2015).
Ch.2, P ojec 14.
Hughes C. L. “Pho og amme ic de e mina ion and
in e p e a ion o geocen ic and opocen ic luna
lib a ions,” (1993). Announce : Join APS/AAPT Ap il
Mee ing Highligh s, Boise, ID, 23, 45 (F. Pin o,
Supe iso ).
Kopal, Z., An In oduc ion o he S udy o he Moon
(Sp inge -Science+Business Media, B. V.,
Swi ze land AG, 1966). Ch. 3.
Kopal Z., and Ca de , R. W. Mapping o he Moon
(Sp inge -Science+Business Media, B. V., Do d ech ,
1974); Ch. 3.
C. Lag ande and P. Che alley, Vi ual Moon A las,
Ve sion 8 (2023). Accessed: 3 May 2025. URL:
h p://ap-i.ne /a l/en/documen a ion
Lequeux, J. “The Coudé Equa o ials,” (2011). JAHH,
14, 191-202.
Meade ETX elescope. Wikipedia. Accessed 4 May
2025. URL:
h ps://en.wikipedia.o g/wiki/Meade_ETX_ elescope.
Meade Ins umen s. In Wikipedia. Accessed 4 May
2025. URL:
h ps://en.wikipedia.o g/wiki/Meade_Ins umen s.
Meeus, J., Ma hema ical As onomy Mo sels (William
Bell, Inc., Richmond, VA, 1997), Ch. 6.
NASA Godda d, “Moon Phase and Lib a ion,”
uploaded by NASA, 15 June 2011.
Accessed: 2 May 2025. URL:
h ps://www.you ube.com/wa ch? =3 _21N3wcX8
Op icsS a , “Meade ETX90 Obse e (ETX-90),”
(2025). Accessed: 4 May 2025. URL:
h ps://www.meadeuk.com/Meade-ETX90-
Obse e .h ml
Pin o, F. “CCDs in he mechanics lab - A compe i i e
al e na i e? (Pa I),” (1995). The Physics Teache , 33,
436-441.
Pin o, F.. “As ome y wi h a DSLR came a.
Obse a ion planning, image acquisi ion, and da a
analysis,” 2024. In P oceedings o he 43 d Annual
Con e ence o he Socie y o As onomical Sciences,
On a io, CA, pp. 29-38. URL: h ps://socas osci.o g/
h ps://socas osci.o g/wp-
con en /uploads/2024/06/2024-
P oceedings_Ve 1.3c.pd
Pin o, F., “S udy o he de e mina ion o he
opocen ic luna lib a ions by a bes i es ima e o he
pla e cons an s om digi al images wi h a small
elescope. Supplemen a y ma e ials,” (2025).
Accessed: 27 Ap il 2025. URL:
h ps://os .io/j myb/? iew_only=053 aaad e84190b4
78824503767624.
Rome , P. “Jupy e , Ma hema ica, and he Fu u e o
he Resea ch Pape ,” 13 Ap il 2018. Accessed: 20
May 2023. URL: h ps://paul ome .ne /jupy e -
ma hema ica-and- he- u u e-o - he- esea ch-pape /
Rule, A. “Ten simple ules o w i ing and sha ing
compu a ional analyses in Jupy e No ebooks,”
(2019). PLoS Compu . Biol., 15, e1007007.
Saunde , A. “The de e mina ion o selenog aphic
posi ions and he measu emen o luna pho og aphs,”
(1900). MNRAS, 60, 174-201.
Saunde , A. “The de e mina ion o selenog aphic
posi ions and he measu emen o luna pho og aphs.
[Second Pape .],” (1901). MNRAS, 62, 41-61.
Saunde , A. “The de e mina ion o selenog aphic
posi ions and he measu emen o luna pho og aphs.
[Fou h Pape .],” (1905). MNRAS, 65, 458-473.
Some s, J. “The Scien i ic Pape is Obsole e,” The
A lan ic, 5 Ap il 2018.
Taylo , J. R. An In oduc ion o E o Analysis
(Uni e si y Science Books, Sausali o, Cali o nia,
1997).
Tucci, P. “The Moon’s ashen ligh and lib a ion in
Leona do and Galileo,” (2022). Quade ni di S o ia
della Fisica, 26, 21-60.
S. Wol am, “The e Was a Time be o e Ma hema ica
…,” 6 June 2013. Accessed: 2 May 2025. URL:
h ps://blog.s ephenwol am.com/2013/06/ he e-was-
a- ime-be o e-ma hema ica/
Włoda czyk, J. “Lib a ion o he Moon, He elius'
heo y, and i s ea ly ecep ion in England,” (2011). J.
His . As on., 42, 495-519.
