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P e r g a m o n

i s t a s i n

A~ronomy

Vol. 36, pp. 311-361, 1 9 9 3

Printed in Great Brilaim.

0083 .665~ 3 $24 .00

0 0 9 8 - 2 9 9 7 9 3 ) E 0 0 0 3 - 7

S T R O N O M Y N D T H E L IM I T S O F V I S IO N

B r a d l e y E . S c h a e f e r

U n i v e r s i t i e s S p a c e R e s e a r c h A s s o c i a t i o n

N A S A / G o d d a r d S p a c e F l ig h t C e n t er , C o d e 6 6 1 , G r e e n b e l t, M D 2 0 7 7 1 , U . S . A .

A B S T R A C T

Celestial visibility is the study of the limits of o bservability of objects in the sky, with app lication to deducing the truth

about historical events or to the derivation of astronomical information of m odem utility. This study is based on w hat is

seen by ordinary humans, either in their everyday lives or at times of historical events. The reeoits o f ~ studies have

more relevance to non.scientists than does any other area o f astronom y. Celestial visibility is young discipline in the

sense that the num ber o f interesting applications with sim ple solutions outnum ber the solved problems; it is a b road

mterdisoiplinary field that involves work with astronom y, meteorology, optics, physic, p hysiology, history, and

archeology. Eac h of theso discipl ines contribute specialized mathematical formulations which quantify the many p ro ce a~

that affect light as it leaves source, traverses the atmosphere, and is detected by the hunmn eye. These form ulas can then

be com bined as appropriate to create m athematical mod els for the visibility of the source under the conditions of interest.

These model results can thin be applied wide variety of problems arising in history, astronomy, archeology,

meteorological optics, and archeoastronomy . This review also preumts dozen suggestions for observing projects, many

of w hich can be d irectly taken fo r individual study, for classroom projects, or for professional research.

1 . I n t r o d u c t i o n

C e l e s t i a l v i s i b i l i t y i s a n i n t e r d i s c i p l i n a r y r e s e a r c h f i e l d c o n c e r n e d w i t h w h a t c a n a n d c a n n o t b e v i e w e d

i n t h e s k y b y v i s u a l o b s e r v a t i o n s , a n d i s r e l e v a n t to s o l v i n g m a n y m y s t e r i e s a n d p h e n o m e n a f r o m h i s -

t o r y a n d a s t r o n o m y . I n t h e d a y s b e f o r e t h e i n v e n t i o n o f t h e t e l e s c o p e , a l l a s t r o n o m y w a s d o n e w i t h

t h e u n a i d e d e y e . I n b o t h a n c i e n t an d m o d e r n t i m e s , h i s t o r i c a l e v e n t s h a v e b e e n i n f l u e n c e d b y v i su a l

o b s e r v a t i o n s o f s k y e v e n t s. T h e r e a r e m a n y s i g h t s i n t h e sk y f o r w h i c h v a l i d e x p l a n a t i o n s h a v e

b e c o m e a v a i l a b l e o n l y r e c e n t ly . F o r s t u d i es o f a n c ie n t c h r o n o l o g y a s w e l l a s f o r c a l e n d a r - m a k i n g

e v e n t o t h e p r e s e n t d a y , s t u d i es o f l u n a r a n d p l a n e t a r y v i s i b i l i ty a r e v i t a l . T h e r e f o r e , i f w e s e e k t o

u n d e r s t a n d o l d a s t r o n o m i c a l d a t a , t h e e f f e c ts o f c el e s t ia l h a p p e n i n g s o n h u m a n a f f a i r s , w h a t u n t r a i n e d

o b s e r v e r s h a v e s e e n i n t h e s k y , o r t h e d e t a i ls o f a n y c a l e n d a r , t h e n w e m u s t l e a r n a b o u t ~ c e l e s t ia l

v i s i b i l i t y . "

B y t h e n a t u r e o f t h e t o p i c s c o v e r e d , c e l e s ti a l v i s i b i l i t y i s c o n c e r n e d w i t h t h e a s p e c t s o f

a s t r o n o m y t h at h a v e r e le v a n c e to o r d i n a r y p e o p l e . F o r e x a m p l e , a q u a r te r o f t h e w o r l d ' s c u r r e n t

3 1 1

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3 1 2 B. E. Schaefer

popula t ion se t s i t s f es t iva l s and f as t per iods b y the v i s ib i l it y of t he th in c rescen t m oon and t imes i ts

da i ly pray er s on as t ronomica l event s . Th e anc ien t Egyp t i ans , Mayans , and Aztecs based the i r

ca l endar s and f es t iva ls on the he l iaca l r is ings of S ir ius , Venus , and the P le i ades . Neo l i th i c man

appa r en tl y cons t r uc ted num er ous t em p l e s w h i ch w e r e des i gned w i t h cognoscence o f t he r i s ing and

se t ti ng of t he Sun, M oon , and s ta r s. Th e t iming of most of the grea t ba t t les and invas ions for a t l eas t

t he la s t c en t u r y have been de te r m i ned i n pa r t by l una r phase . Su r e l y m os t hum ans have s t ood ou t

unde r a c l ear sky and w onde red about t he twinkl ing of s ta r s , t he beauty of a sunse t, and the g lory of

a r a inbow.

Th e s tudy o f ce l es ti a l v i s ib i l it y d i r ec t ly r e l a t es t o man y h i s tor i ca l ques t ions tha t f i r e t he publ i c

imagina t ion . Fo r example , t he r esu l ts presented be low are impor t an t t o the de t erminat ion of whether

A dm i r a l Pea r y r eached t he N or t h Po l e , w h e r e w as C o l um bus ' l and fa l l, w he t he r S t onehenge w as an

~ancient obs erv a tor y , " how Paul Reve re s li pped by the Bri ti sh war sh ip in Boston Ha rbor , what was

t he S ta r o f B e t h l ehem , and w hen w as J e sus c r uc i fi ed? Phenom ena in t he sky have a f f ec ted m a j o r

event s , r anging f rom the appearance of a b ig sunspot provoking a genera l amnesty in anc ien t China ,

to an aurora sav ing Byzant ium f rom Phi l ip of Macedon, t o a l unar ec l ipse breaking the sp i r i t o f t he

defender s of Constan t inople in 1453, t o a so l ar ec l ipse s topping the war be tween the Median and

Lyd ian empi res . Exam ples whe re ce l est ia l v is ib i li t y sheds l igh t on h i s tori ca l event s a r e l eg ion and

spread throughout t he en t i r e wor ld and through a l l t imes .

A d e ta i led know ledge of ce l es ti a l v is ib i li ty i s r equi r ed to ex t r ac t t he ma ximum informa t ion out

of v i sua l as t ronomica l observa t ions . Thu s , ex tens ive ef for t has gon e in to the ana lys i s of anc ien t

ec l i p se r epo r t s a s t he p r i m ar y r eco r d on t he acce l e r a t i ons o f t he E a r t h ' s r o t a t i on and t he M oon ' s

r evolu t ion . Exh aus t ive s tudies have a l so been per form ed on the two mi l lennia of or ien ta l sunspot

r epo r t s a s the p r i m ar y m easu r e o f t he l ong t e rm so l ar ac ti v it y . A no t he r m a j o r r e sea r ch p r og r am has

been the ana lys i s of h i s tori ca l supernova s to ex t r ac t ages , pos i t ions , l i gh t curve s , c l ass i f i ca tions , peak

br ightnesses , and r a t es .

Celes t ia l v i s ib i l it y is bo th a very o ld and a very yo ung d i sc ip l ine . Th e anc ien t Greeks w ere

cons t ruc t ing model s for he l i aca l r i se da t es , a tmospher i c r e f r ac t ion , and ec l ipse predic t ions , whi l e

medieval I s l amic as t ronomers were much concerned wi th predic t ing the f i r s t c r escent of each month

and the t imes o f twi ligh t . W hi l e these e f for t s r esu l ted in empi r i ca l ru l es-of - thumb tha t usua l ly

produce d r easo nable predic t ions , t he qual i t y of t he anc ien t methods i s wel l be low m odern s tandards.

Unt i l mode rn t imes , t he r equi r ed m athemat ica l t oo l s and phys ica l m odel s we re not ava i l ab le , so tha t

ao s igni f i cant advanc es on the o ld empi r i ca l ru les were poss ib le . Ho we ver , t he in t e res ts of modern

as t r onom er s ha s been sha r p l y f ocused on h i gh t echno l ogy m e t hods e . g . , spec t roscopy, r ad io

t e l escopes , and gamma- ray observa tor i es) and exot i c new di scover i es

e . g . ,

supernovas , quasar s , and

gam ma - ray bur s t s ). Thu s , for the las t century , mode rn as trophysics has passed by ques t ions r e l a ting

to v i sua l observa t ions o f t he sky . Non ethe less , r esearch in to ce les t ia l v i s ib i l it y has been en joying a

renai ssance in the l as t decad e , wi th an ever i ncreas ing nu mb er of worke r s in the f i e ld . M any i ssues

can be app roached for t he f i rs t t ime wi th modern methodo logies , so tha t the num ber o f s imple , usefu l ,

and s t a r tl ing r esu l ts i s l a rge , j us t as for a y oung d i sc ip line .

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Astronomy and the Limits of Vision

313

Th is review art icle wil l cov er the whole f ield of celestial visibi l i ty. Th e nex t section w il l give

deta iled equations for most of the phenom ena tha t a f fec t what can be seen in the sky . Th e m iddle

section presents m odels of specif ic visibi l i ty questions, of relevance fo r man y applicat ions. Th e fourth

section applies the se mod els to specific questions o f historical or astron om ical interest. Th e last

section giv es twelv e suggestions f or research projects o f curr ent interest.

2 . Too l s

Th e theoret ical calculat ion of celestial visibi l i ty requires accurate m odels o f man y effects . Thes e

inc lude models of the source

p o s i t i o n

refract ion, air mass, ext inct ion, source brightness, sky

brightness, glare, shadows, resolut ion, opt ics, thresholds o f hum an vision, color vision, and we ther

stat is t ics . In this sect ion, I wil l present the basic equations an d references for each topic. Thes e

equations can then be used as tools and com bined appropriately for each quest ion regarding celestial

visibility.

2.1 Source position

Calcu lat ions o f celestial v isibi l i ty almo st always begin with an evalua t ion of the sou rce 's posit ion in

the sky . In som e cases ( for example , w i th the v is ib i l ity of s ta rs in ch imneys or w i th the green f lash) ,

the posi t ion can be assum ed or low accuracy is adequate. But in man y cases, especially tho se

involvin g historical incidences, the exac t posit ion is impo rtant . Th en, the task can be brok en into two

parts; f i rs t , determ ining the sou rce 's celestial posit ion in r ight ascension and decl ina t ion, and second,

determ ining the apparent posi tion with respect to the horizon f or the observer .

In gen eral , the m ost defini t ive reference fo r celestial posi tions o f al l sources is

the Astronomical

Almanac

(or its predecessor the

American Ephemeris and Nautical Almanac , the MICA

compu te r

program, and the Explanatory Supplement to the Astronomical Almanac (Seidelmann 1992).

Un fortuna tely, these materials are n ot the m ost accessible references to m any observers and only cover

the chronology from the mid-1800 ' s to the near fu ture . A m ore convenient resource might be any of

the many commercial computer programs avai lable for personal computers , of which the

Voyager

progra m h as a good reputat ion for accuracy. Th e best reference depends on the objects involved and

the needed accurac y. Fo r the Sun, M oon, and planets , the defini t ive resource is to access the

computer tapes of the JPL DE-200 ephemeris or Bretagnon and Simon (1986), while Van Flandern

and Pulkkinen (1979), Meeus (1991), Tuckerman (1962), Tuckerman (1964), and Gingerieh and

W elther (1983) are conven ient and sui tably accurate for most purposes. Fo r the M oon , the tables of

Chapront -Touze and Chapront (1991) a re of h igh accuracy , whi le Golds t ine (1973) and Morr i son

(1966 and 1968) present times of the lunar phases . For lunar ec lipses, the canons of M eeus and

M ucke (1979), Espen ak (1989), and Liu and Fiala (1992) are conven ient and accurate. Fo r solar

eclipses, M uck e and M ecus (198 3) or Espenak (1987) should be used.

Once the celest ial posi t ion of the source is known, then the apparent posi t ion needs to be

deduce d. This requires a series of t r igonom etric calculat ions which are described in ma ny astrono my

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3 4 B. E. Schaefer

and navigational textbo oks e .g . , Sm ar t 1977, Mee us 1991). No te tha t var ious cor rec t ions might be

needed , such a s f o r r e f r ac ti on , pa r a ll ax , and t he d i f f e r ence be t w een U n i ve r sa l T i m e and D ynam i ca l

Tim e. Thi s l as t cor r ec t ion i s uncer ta in in anc ien t e r as and m ay change the loca l t imes and v i s ib il it ies

of ec l ipses and conjunct ions by s igni f i cant amount s ( see S tephenson and Mor r i son 1984 or Meeus

1991).

2.2 Refraction

W hen l igh t tr ave l s t hrough the a tmo sphere , i t s pa th i s bent by r ef r ac t ion . Exc el l en t h i s tor ies of

r ef r ac t ion s tudies a r e presented by M ahan (1962) and Bruin (1981) . Fo r a l t it udes wel l abov e the

hor i zon , t he c l ass ica l t r ea tment s of Sm ar t (1977) and Green (1985) a re va lid . Th e apparent pos it i on

o f t he sou r ce i s r a i sed by an am oun t R , t he r e f r ac t ed ang l e , w h e r e

R 5 8 2 [ 0 3 7 2P 1

2 " ~ "+"~ ;j t a nZ

(1 a)

and wh ere Z i s the apparen t zeni th d i s t ance (in uni ts of degrees-of - arc) of t he source , P i s the

a tmospher i c pressure ( in uni t s of mi l l imeter s of mercury) and T i s

Ta ble 1 . Ref rac t ion the a i r t empe ra ture ( in uni ts of degre es Cel s ius). Fo r a l ti t udes

down to the hor i zon , Saemundsson (1986) g ives two a l t e rna t ive

formulas for R which are r easonably accura t e :

1 '

R

ta n h + h + 4 . 4 }

(1 b)

(1 c)

h h' R

(*) (*) (3

90 90 0.00

60 59.99 0.57

30 29.97 1.71

20 19.95 2.70

15 14.94 3.64

R ~ _

1.02 '

10.3 )

t an h ' + h ' +5 .11

10 9.91 5.39

5 4 . 8 4 9 . 8 8

4 3 .8 0 11.7

3 2.76 14.3

2 1 .70 18 .2

1 0.59 24.3

0 -0.57 34.5

H er e h ( = 9 0 - Z ) i s t he sym bo l f o r t he appa r en t i .e . , as seen)

a l t it ude above an idea l hor i zon , w hi l e h ' (= 90 - Z - R) s tands for

the true a lt it ude ; t he uni ts o f (h , h ' ) a r e degrees-o f - arc . Th e

am oun t o f r e f r ac ti on on t he ho r i zon i s t yp i ca ll y 34 ' o r 0 . 57 . Fo r

a l t i t udes near and even be low the hor i zon , t he complex a lgor i thm

of Gar f inkel (1967) i s h ighly accura t e i f t he a tmospher i c t hermal

s t ruc ture is s imi l ar t o tha t of t he U. S . S t andard A tmosph ere . In

Table 1 , R-values a re presented as de t ermined f rom independent

va lues h , and as ca l cu la t ed f rom Equat ion lb .

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stronomy and the Limits of Vision

315

: Th e values of the terrestrial refract ion, t h a t i s the he ight of a ray o f l igh t emi t ted hor izonta l ly

a t sea leve l as taken f rom the program in Schaefer (1989a) , i s presented in Table 2 . How ever ,

Scha efer and Lil ler (1990) h ave shown that s ignif icant thermal inversions are ubiquitous and w il l

drast ical ly chan ge the refract ion close to the horizon. Scha efer (1989a) presents a prog ram fo r

calculat ing refract ion (and air mass) for an arbi trary therm al s tructure in an atmosph ere of any planet .

2.3 Air Mass

The d imm ing of a beam o f l ight en ter ing the a tmosphere

depends on how mu ch a i r i s t raversed by the beam. The

quant i fica t ion of d imm ing can be broken in to two com po-

nents: the f i rs t , a geometrical term which is primari ly a

funct ion of the sourc e 's ap parent zeni th distance (Z), and

the second, a meteorological term that varies with time

and place (see the next sect ion). The geometrical term

(X) is t radi t ional ly measured in units of ~air masses,"

which is the optical path length to the source in units of

the optical path length tow ards the zenith.

For a l t i tudes wel l above the hor izon , a good

approximat ion i s g iven by .

X see Z)

2 a )

Near (but not be low) the hor izon , the formula of

Rozenberg (1966) is conven ient and fa ir ly good fo r

reasonab le elevat ions abov e sea level and aerosol d ensi ty;

X =

[ c o s ( Z )

+ 0.025e-~1~ ~z)]-1

( 2 b )

Table 2. Terrestrial Refraction

Height N O R M A L ~ With Inve=~ions

( m e t e r s ) D i s t a n c e u n ) D i s t a n c e k ln )

0 0 0

16 14.6 21.7

33 21.1 27.3

50 26.1 32.1

67 30.4 36.3

84 34.2 40.1

100 37.6 43.6

150 46.4 53.0

200 53.9 61.0

250 60.4 68.0

300 66.4 74.3

350 71.8 80.0

400 76.9 85.3

Equations (2 a,b) pro vide descriptions of the air m ass for

the a tmosphere taken as a whole .

Ho we ver, th e optical path length through the atmosphere has contr ibut ions from three sources

of ext inct ion: Ra yleigh scat ter ing by gas m olecules, M ie scattering b y aerosols , an d absorption by

ozon e in the stratosphere. Each o f these components ha s a dist inct and characteris t ic depen dence on

height , so each wil l contr ibute a differen t opt ical path length for a specif ic valu e of Z. Th e air mass

for the Rayleigh and aerosol component can be calculated for the appropriate scale heights with the

prog ram o f Scha efer (1989a) for arbi trary atmospheric co ndit ions and viewin g direct ions. Th e ozone

ai r m ass can be ca lcula ted in a c losed form f rom s imple t r igonom etry .

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31 6 B. E. Schaefer

The a i r mass fo r an ext inc t ion componen t tha t i s d i st r ibuted w i th an exponent ia l sca le he ight

of H. in k i lom eters may be ca lcula ted wi th the use of Equation 3a . Fo r Rayle igh sca t te r ing , Ho i s

close to 8.2 ki lom eters (Allen 1976), wh ile for aerosol scat ter ing the scale height varies substant ial ly

wi th a typica l va lue of 1 .5 k i lometers (Hayes and Latham 1975) . Equat ion 3b i s used to ca lcula te the

ext inc t ion layer w hich i s HL above the observer (w i th R . be ing the equator ia l rad ius o f the Ear th , i .e . :

Ro-- 6378 k m ), which fo r the stratospheric ozo ne is rou ghly 20 ki lom eters (Meinel and M einel 1983).

The use o f Equat ions (3 a ,b) i s v i ta l for any accura te ca lcula tion of ex t inc tion near the hor izon .

x ( s ) - - ( z ) } ] - 1

i ]

L I t L ) = 1 = ( ' 1 * ( R z / R . ) ) ]

3 a , b

In Table 3 a re presented the var ious a i r m asses as a func t ion of Z .

Table 3. Air Mass

z h X X.(S.2 km) X.(I.5 kin) Xt(20 kin)

(0) (0) sot (Z) Eqn(2b) (Gases) (Aerosols) (Ozone)

0 90 1.0 1.0 1.0 1.0 1.0

60 30 2.0 2.0 2.0 2.0 2.0

80 10 5.8 5.6 5.6 5.7 5.3

85 5 11.5 10.3 10.1 11.3 8.5

87 3 19.1 15.1 14.5 17.9 10.6

88 2 28.7 19.3 18.3 24.9 11.6

89 1 57.3 26.3 24.2 39.3 12.4

90 0 --- 40.0 34.9 81.6 12.7

a ~

2 .4 Ex t inc t ion

The ext inc t ion coef f ic ien t (k) i s a measure o f the to ta l optica l pa th length tow ards the zeni th , and i s

measured in uni t s of magn i tudes per a i r mass; i t i s related to the opt ica l depth ( r ) as 1 . 0 8 6 r and to

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Astro nomy an d the Limits f Viaion

3 1 7

t h e a t m o s p h e r i c t r a n s m i s s i o n T= e T ) as -2.51Og~o T}. Th e xfu~tion ~ on both the

w a v e l e n g t h o f t h e i n c i d e n t l i g h t a n d t h e c o m p o n e n t o f e x t i n c t i o n b e i n g c o n s i d e r e d .

T h e R a y l e i g h s c a t t e r i n g c o m p o n e n t o f t h e e x t i n c t i o n i s r e l i a b l y c a l c u l a t e d a s a f u n c t i o n o f

w a v e l e n g t h , a l t i t u de , a n d t h e i n d e x o f r e f r a c t i o n b y H a y e s a n d L a t h a m ( 1 9 7 5 ) :

k ~ = I 1 3 7 0 0 ( n - l ) 2 e . a ~ [ - ( H / H , ) ] ~ . - 4

( 4 a )

w h e r e n i s t h e r e f r a c t i v e i n d e x o f a i r a t s e a l e v e l f o r t h e d e s i r e d w a v e l e n g t h , ) . , i n m i c r o n s ( s e e A l l e n

1 9 7 6 ) , a n d t h e o b s e r v e r s h e i g h t is H ( i n u n i t s o f k i l o m e t e r s ) a b o v e s e a l e v e l . F o r v i s u a l w a v e l e n g t h s

(7.= 0.55 microns), Equation (4 a) is reduced to

k l

= O . 1066e - s / s ~ )

4 b )

with units of magnitudes per air mass

The ozone com ponent of the ex tinc tion i s re liab ly ca lcula ted f rom the ozone depth f rom Bow er

and War d (1982) and the cross sect ions in Allen (1976). In the visual bandpass, the ozone depth, Do, ,

( in m m ), and ext inc t ion (in magni tudes per a i r mass) f rom ozon e can be approxim ated wel l by

D ~ = 3 .0 + 0 . 4 ( O c o s [ a s ] - c o s [ 3 4 ~ ] ) ,

koz = 0.031 D ~

3.0

5 a , b )

Here , is the obs erve r 's lat i tude (in radians) a nd a s is the r ight ascension of the Sun. Ozo ne holes,

ozone deple t ion , b iennia l cyc les , and even major s torm sys tems wi ll a l l change the ozone depth f rom

the avera ge values given by Equation 5a. Equation 5b applies fo r the central wa veleng th o f day

vis ion , wh i le the coef f ic ien t i s only 30% as la rge for n ight v i s ion . S ince k ,~ i s smal l compared to

other com pon ents, i ts uncertainties have l i t t le impa ct upon the overal l ext inct ion problem .

Atmo spher ic aerosols com e f rom m any sources , inc luding sea spray , w indbo me deser t dus t ,

and tree pollen. Th ey vary in complicated manners that are diff icul t to predict with accu racy.

Nevertheless, there are a variety of t rends which can be used to provide a reasonable guess for the

aerosol ex t inc t ion coef f ic ien t (k ,) . From an eva lua tion of t rends for t ime o f year , re la t ive humidi ty ,

al t i tude, and wavelength, I f ind that

oo = 0.01 + OA k

[ 1 + 0 . 3 3 s / n ( a s ) ] ,

6 a , b )

Here , S is the relat ive hum idity and a , is the uncertainty in ka. Th ere are addit iona l t rends with

la t itude and da te , w hi le the humidi ty te rm depends on aerosol type . The num er ica l var ia tion around

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3 8

B. E. Schae fer

t hese es t imates i s skewed by the l i ke l ihood of having much be t t e r haz iness be ing smal l e r t han the

l ike l ihood of having much worse haz iness .

There may be addi t iona l sources of ex t inc t ion not i nc luded in the above d i scuss ion , such as

vo l can i c ae r oso l l aye rs ( s ee M e i ne l and M e i ne l 1983 , L ockw ood and T hom pson 1986 , and R u f ene r

1986 for t yp ica l e f f ec t s ) and urban pol lu t ion (see Joseph and M anes 1971, Hu sar and H ol low ay 1984,

and Flow ers , M cCo rm ick , and K ur f i s 1969 for typ ica l va lues) .

T h e u se o f t he equa t ions above w il l

Tab le 4. ExtinctionCoefl iciant al low fo r the ext inc t ion to be reaso nab ly

es t imated for any s i t e i n the wor ld .

Elevation < k> N ever t he l e s s , t he ex t i nc t i on va r i e s

Site (meters) winter summer unpredic t ab ly from site-to-site. T her e f o r e ,

for spec i f i c appl ica t ions , i t wo uld be bes t t o

Mauna Kea, HI 4205 0.11 0.11

have ac t ua l da t a f r om o r nea r t he s i t e o f

Los Angeles, CA 100 0.2 8 0.46 interest . Ap pro pr iate data i s avai lab le in

the l i te r a ture for muc h of t he wor ld . I have

Kitt Peak, AZ 2064 0.15 0.21

been ab le to f ind r e l i ab le ex t inc t ion

Tucson, AZ 770 0.22 0.28 me as ur es spread throughout t he year for

225 loca t ions sca t t e r ed over a l l o f t he

Tallabasee, FL 40 0.24 0.44

cont inent s . Thu s , any l and s i te i s usua l ly

Atlantic City, NJ 10 0.27 0.47 wi th in severa l hundred mi l es of someplace

tha t has provided r ea l da t a to the ex t inc t ion

Boston, MA 150 0.23 0.31

problem . Publ i ca t ions wi th r e l i ab le and

Athens, Greece 107 0.2 5 0.31 useful me asure me nts f ro m mu l t iple locales

inc lude Abbot t ( I 908-1954 ) , C har l son e t a l .

Jerusalem 775 O. 18 0.28

(1974) , Joseph and Manes (1971) , and

Sou th Po le 3000 0 . 14 0 . 1 4 Y am am ot o , T anaka , and A r ao ( 19 68) . I n

Table 4 a re presented average ex t inc t ions

f rom a var i e ty of s it es . Th e to ta l ex t inc t ion

coef f i c i en t wi ll be the sum of the var ious

k

cont r ibut ing component s ,

(7)

w h i ch r ep r e sen ts the d i m m i ng t o w a r ds t hez en i t h . I n o t he r d i r ec t ions , t he t o t a l l o s s i n bngh t nes s w ill

be g i ven by :

A m= k X

A m = k s X , 8 . 2 k m ) + k ~ X L ( 2 0 k m ) + k o X , ( l . 5 k m ) ,

8 a ,b)

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A s t r o n o m y a n d the Limits of V i s i o n 3 1 9

i n u n i t s o f m a g n i t u d e s . E q u a t i o n 8 a i s v a l i d f o r p o s i t i on s a w a y f r o m t h e h o r i z o n , w h i l e E q u a t i o n 8 b

i s v a l i d a n y w h e r e , y e t i s r e q u i r e d n e a r t h e h o r i z o n f o r a c c u r a t e a n s w e r s . T h e a p l a U e m i n t e ns i ty f

a s o u r c e v i e w e d t h r o u g h t h e a t m o s p h e r e i s g i v e n b y t h e e x p r e s s i o n

1 = 1 1 0 a = 1 2 ~ ,

( 9 )

whe re I* i s t he br ightness abo ve the a tmosphere and I i s the apparent br ightness fo r t he observer .

Fo r an objec t low on the hor i zon , t he apparent br ightness var i es roug hly as 104u~s wi th X " 40 ,

so tha t even s l igh t var ia t ions or uncer t a in ti es i n k wi ll chang e the apparen t br ightness by a l a rge f ac tor .

Fo r exam p l e , I w il l change by 45% as k changes f r om 0 . 24 t o 0 . 25 m agn i tudes pe r a i r m ass . T h e

dom inant sou rce of uncer t a in ty in k is f rom the aerosol comp onen t , and o , i s t yp ica l ly l a rger t han 0 .05

magni tudes per a i r mass or h igher , so tha t t he deduced va lue of I wi l l vary by roughly an order of

magn i tude or mo re . Thu s , t he uncer t a in ty in the aerosol ex t inc t ion is usua l ly the dominant e r ro r i n

near -hor i zon v i s ib i li t y .

2 5 S o u r c e b r i g h t n e s s

A fundamenta l f ac t for v i s ib i l i t y of any as t ronomica l ob jec t i s i t s apparent br ightne ,~ above the

a t m osphe r e . T h e b r igh t ness can be exp r e s s~ a s a m agn i t ude, f o r w h i ch t he v i sua l m agn i t ude (m ~ )

i s re l evant for norm al day v i s ion . Thi s v i sua l magn i tude i s re l a t ed to the i ll uminance out s ide the

a tmosp here (1" ) as m easured in foot - candles by

m , = - 1 6 . 5 7 -

2.5 log

( I )

(10)

T h e f o l l o w i n g e q u i v a i e n c i e s f o r f o o t - c a n d l e s m a y b e d e r i v e d f r o m Al len ( 1 9 7 6 ) .

l f o o t - c a n d / e = 1 0 . 7 6 / u s

= 1 0 . 7 6 x I 0 - 4 p h o t

= 1 /umenj~-2

= 4 .240 x 106 s t a r s of m v = 0.00

I f t he source i s ex tend ed, t hen i t s i ll uminance can be ob ta ined by in t egra t ing i t s l uminance , B ( the

sur face br ighmess in uni t s of nanoLamber t s ) over t he source ,

i.e.:

I " = 2 .95 x

l O ~ f B d ~ l

J

wh ere f l i s the sol id angle in steradians (Ho l laday 1926) .

The fo l lowing equiva ienc ies

Garstang 1989) .

1 nano / .am b er t =

f f i

(11)

f o r " nanoL am ber t " m ay be de r i ved f r om A l l en ( 19 76) and

3.18 x 10-10 st i /b

3.18 x 1 0 -1 candelacm 2

3.18 x 10-e n / t

1 0 - s a p o ~ / b

4.61 x 10 -4 stars of m v = 0.00

26.33 magnitudes per square arc second.

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32 0 R E. Schaefer

T h e b r i g h t n e s s a n d c o l o r o f s ta r s a n d p la n e t s h a v e b e e n t a b u la t ed i n m a n y b o o k s , w i t h t h e

Astronomical Almanac o r Astrophysical Q uantities ( A l l e n 1 9 7 6 ) a s re l i a b l e r e f e r e n c e s . T h e b r ig h t n e s s

o f a w i d e r a n g e o f s o u r c e s i s p r e s e n te d i n T a b l e 5 , o v e r l e a f . T h e S u n ' s b r ig h t n e ss , s p e c tr a l e n e r g y ,

a n d s u n s p o t i n f o r m a t i o n a r e w e l l p r e s e n t e d i n A l l e n ( 1 9 7 6 ) . T h e M o o n ' s n~ , i s g i v e n b y

m w ~ a ) = - 1 2 . 7 3 * 0 . 0 2 6 I 1 4 1 o - 9 1 ' ,

1 2 )

T a b l e 5 . S o u r c e B r i g h t n e s s

Source mv /*

(foot-candles)

Sun -26.74 1.17 x 10 l

Setting Sun -16 5.9 x 10-1

Full M oon -12.73 2.91 x 10 2

Qum 'ter Mo on -10.13 2.65 X 10 3

Supernova (1000 pc) -8 3 .7 x 10 4

Crescem M oon -7.89 3.37 x 10 4

a = 135")

Ve nus at brightest) -4 .6 1 .6 x 10-5

60W bulb -3.6 6.4 x 10 6

(860 iunwm, km)

Sirius -1.46 9.04 x 10 7

Veg a 0 2 .36 x 10 7

Polaris 2.02 3.66 x I0"s

D a y vision limit 4.1 5.4 x 10 9

Typical naked eye limit 6 9.4

x 10-10

Typical binocular limit 9 5.9 x 10- II

Brightest quasa r 13.0 1.5 10-12

(3c 273)

Pluto (at opposition) 13.7 7.8 x 10 13

Approximate HST limit 26 9.4 x 10- s

w h e r e a i s t h e p h a s e a n g le o f th e M o o n i n

d e g r e e s , w i t h a f f i 0 * f o r f u l l M o o n ( A l l e n

1 9 7 6 ) . T h e s p e c t r a l r e f l e e ti v i t ie s f o r t h e

M o o n a r e g i v e n i n M c C o r d a n d J o h n s o n

( 1 9 7 0 ) . T h e l u n a r s u r f a c e b r i g h t n e s s i s

g i v e n b y t h e h i g h l y c o m p l e x H a p k e

e q u a t i o n s ( H a p k e 1 9 8 4 ) a n d t h e p a r a m e t e r s

g i v e n b y H e l f e n s t e i n a n d V e v e r k a ( 1 9 8 7 ) .

T h e l u m i n a n c e o f t he M o o n ' s d a r k s i de ,

B~s i.e., o f t h e E a r t h s h i n e ) i s g i v e n i n

n a n o L a m b e r t s b y :

B z s = 2 . 5 9 x a le x [ - 0 . 4 m , ~ ( a ' ) ]

(13 )

w h e r e a ' = 1 8 0 - a ( S c h a e f e r , B u l d e r , a n d

B o u r g e o i s 1 9 9 2 ) . E q u a t i o n 1 3 w a s d e r i v e d

w i t h a n a v e r a g e l u n a r a l b e d o o f 0 .0 6 7 ,

w h e r e a s a l b e d o s f o r p a r t ic u l a r f e a t u r e s c a n

b e f o u n d i n M i n n a e r t ( 1 9 6 1 ) .

2.6 Sky brightness

T h e b r i g h t n e s s o f t h e s k y v a r i e s o v e r a

r a n g e o f s e v e n o r d e r s o f m a g n i tu d e . T h e s e

v a r i a t i o n s a r e c o m p l e x f u n c t i o n s o f t h e

z e n i t h d is t a n c e s o f th e S u n , M o o n , a n d s k y

d i r e c t i o n ( Z ~ , Z m o,~ , a n d Z ) , t h e s e p a r a t io n

b e t w e e n t h e s k y d i r e c t i o n a n d t h e S u n a n d

M o o n ( p , ~ . a n d P m o o . ), t h e a l t it u d e o f t h e

S u n ( h ~ ) , t h e a p p a r e n t m a g n i t u d e o f t h e

M o o n ( m m ~ . ) , a n d t h e e x t in c t i o n c o e f f i c i e n t

( k ). I d e a l l y , t h e s k y b r i g h t n e s s s h o u l d b e

c a l c u l a t e d w i t h m u l t i p l e s c a t t e r i n g , a r o u n d

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stronomy and the Liraits of Vision

321

Earth, and realistic scattering as a function of height in the atmo sphere. Un fortun ately, such studies

(see Rozenberg 1966, Py~skovskaya-Fesenkova 1957) only provide resul ts of such complexi ty as to

have small practical util ity. Fortun ately, with som e simplifications it is pos sible to estimate the

brightness of the c loudless sky a t any t ime to perhaps 20% accuracy. The scat ter ing of l ight by the

atmosphere is the pr imary source of skyl ight ; an approximate phase funct ion descr ibing this

phenomenon

i s :

f ( p ) = 105~s [1.06 + c o s 2 ( p ) ] 1 0 6 I s- (p /4 0.) + 6 . 2

X 10 7

p -2

(14)

with p m easured in degr ees CKxisciuuas and Schaefe r 1991). The sk y brightness in nanoL am berts (nL)

is given by the fol lowing expressions:

B,, , . . , = B o [ O . 4 + / l _ O - ~ - ~ s in 2 z ] l O A t x ( z)

B . o o

- f ( p . o . ) a le x { - 0 . 4 [ m . 0 0 . + X 6 .Y l k X Z . o o . ) l } l - 1 0 "x < z ~ )

B ~, - muax{ 1, _alex ( p s , J g 0 ) - 1.1 ]} 10 4s .o.4 ~. (1 - 10 "~.tx(z))

8 , ~ = i t . v o o / f p ~ , , ) , Z e x [ - o . 4 k x ( z ~ . , ) ) ] ( I - t o ~ ~ )

B ~ = B . ,s h , + B . oo ~ + m i n [Bn~ , B4,., ] + B e ,,

Bo is a parameter which var ies with the

t ime of day, the solar cycle , volcanic

erupt ions , and other pheno mena (Krisciunas

1990); a typical value for a dark site Bo is

180 nL (Schaefer 1990b). The evaluat ion

of Bcuy from first principles is a com plex

problem, with Garstang (1989) giving a

definitive solution.

For some problems, an empir ical

approach might be preferable to the above

theoretical calculation, so it is useful to

have a ser ies of measures under many

condi t ions for comparison. The br ightness

of the day t ime sky i s g iven by Weaver

(1947), Koomen

et a l .

(1952), and Tousey

and Hulbu rt (1948). The night sky

brighmess is g iven observat ional ly by

Table 6. Sky Brightness Values

Source

Sun's surface

Full M oon's surfsoo

Typical daytime sky

Quarter Mo on's sur~ce

Zenith at s~asec

Typical sky in big city

Zmith at Civil Twilight

Day vision/night vision

Typical sky at Full Moon

Zenith at Nautical Twilight

Near horizon for dark sky

Zenith for dark sky

( 1 5 a , b , e , d , e )

n., (~L)

6 x 1014

lx lOP

5xl O =

3x 10=

3xlO ~

l x l ( f

lx l O s

1500

1400

300

24O

20

Darkest ever observed 54

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3 2 2

B. E. Schaefer

Krise iunas

e t a l .

(1987), Pi lachowski

et a l .

(1989), an d Garstang (1989), w hile Schaefer (1990b) gives

genera l formu las and methods for es t imat ion . The sky br ightness dur ing tw i l ight has been observed

b y K o o m e n

e t a l .

(1952). The ef fec ts of c i ty l ight pol lu tion have been observa t iona l ly addressed by

W alker (1973) , G ars tang (1989) and Lockw ood, F loy d, and Thom pson (1990) . Sky br ightness va lues

under typica l condi t ions a re summ ar ized in Table 6 .

2 . 7 G l a r e

Glare occurs i f a br ight source of l igh t masks a nearby fa in ter source . The paper b y Ho l laday (1926)

remains the bes t and most comprehens ive repor t on the phys io logica l measurem ent of the e f fec ts of

g lare . His bas ic conclus ion was tha t g la re can be t reated mere ly as an increase in background l ight .

This ar ises w hen l ight from the glare source is scattered (in the atmosph ere, the telescope, or the

eyeball) so that the back groun d l ight aroun d the source of interest r ises s ignif icant ly. Th e scattering

in the telescope can arise from both diffract ion and scattering o n the mirror . Using the fol lowing

expressions, Sch aefer (1991c) calculated the added back groun d from the four types of scat ter ing:

atmo spheric (atm), diffract ive (d/t), m irror

( m i r ) ,

and e yeball (eye).

B , ~ , = 6 . 2 5 x I 0 7 1 0 - 2 [ I 0 - 0 4 i x - I 0 0 l lk x ,

B a =

1 . 1 3 x l 0 ~ l . D - i 0 - s 1 0- 0 . ,t kx ,

B ~

2 . 6 0 x l O S l * e [ 0l'412 ,

B ~ = 4 .63x 1071" 0 " z 10 " '4kx and

B s ~ e = B a t s B ~ B m u B ~

( 1 6 a , b , c , d , e )

H e r e , 0 i s t h e a n g l e t o t h e g l a r e s o u r c e ( i n u n i t s o f

d e g r e e s n d D

i s t h e a p e r t u r e ( i n u n i t s o f

c m .

I f 0 i s l e s s t h a n 0 . 0 0 2 ( ~ 6 ) , t h e n t h e s e e q u a t i o n s m u s t b e m o d i f i e d a s d e s c r i b e d b y S c h a e f e r

( 1 9 9 1 c ) . I f t h e g l a r e s o u r c e i s e x t e n d e d , t h e n a d d i t i o n a l m o d i f i c a t i o n s a s p r e s c r i b e d b y S c h a e f e r ,

B u l d e r , a n d B o u r g e o i s ( 1 9 9 2 ) a r e n e e d e d . F o r v i e w i n g w i t h o u t a t e l e s c o p e , t h e v a l u e s o f b o t h B d i

a n d B m ~ r s h o u l d b e s e t t o z e r o .

T h e v a l u e o f b a c k g r o u n d b r i g h t n e s s , B b , k , f r o m t h e s k y a n d g l a r e i s, t h e n , g i v e n b y :

B b a c k B s k y B s t a r

( 1 7 )

Other addi t ive te rms

( e . g . ,

from nebulosity) m ight have to be included in Equation 17.

2 . 8 S h a d o w s

Shadows co ns is t of an umbra l reg ion w here the l ight source i s comp le te ly h idden and a penumbra l

region where the l ight source i s only par t ia l ly h idden. The l ight source i s b locked by an o ccul te r and

the shadow is cast upon a screen. Fo r the Sun, the width , W, of the penum bra l reg ion on the screen

can be expressed as:

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Astronomy and the Limits of Vision

W = 2 L c o , . .

c o s n )

323

1 8 )

where L i s the d is tance be tween the occul te r and the screen , a m i s the angular rad ius of the Sun in

radians (0.00465 radians), and r l is the ang le between the screen 's norm al and the Sun. Fo r a s traight

edged occu lt e r, t he i l luminance o f t he sc reen i n the pe nu m br a , /~ , i s g iven by :

I . . . = ( ~ ) [X - c o s ( X ) s / n ( X ) ]

1 9 a , b )

wh ere d is the perpend icular distance to the um bra/penu mb ra l ine, and X is in radians. Fo r d < 0

(in the um bra) the i l lumina nce is zero, wh ile for d > W (in direct sunlighO the il lumina nce equals the

direct solar i l luminan ce, ~. , the value for which is:

1~.n = 11700 x 10" '4tx

2O)

In the special case of a pinholein units o f foo t -cand les as found f rom Equat ion 9 and Table 5 .

camera , the il luminance va lue of the image , l p i . ~ , i s g iven by:

2 1 )

wh ere D is the sym bol for the aperture d iame ter o f the pinho le and L stands for the distance between

the pinhole and the screen.

A screen w il l also be i lluminated by the skylight or scattered l ight o ff the surrounding terrain.

Ideal ly, this extra l ight can be calculated with Equation 11 integrat ing the sky brightness from

Equation 15, but this is diff icul t to do accurately. Nevertheless, an approxim ate answer is adequate

for mo st applicat ions. Koo m en et al. 095 2) present the i lluminance on a f la t sur face poin ted in

Carious direc tions and times. Fo r a horizontal surface, the illum inan ce valu e (in foot-candles) w ill be

3000 in daytim e shadows, 40 at sunset , a nd 0.125 at civi l twil ight .

Th e screen wil l scatter so me fract ion of the incident l ight in al l direct ions. Fo r a perfect ly

diffusing (Lambert ian) surface, the surface brightness o f a screen, B= ,~, , i l lum inated with a total

in tens ity of I w i l l be g iven by:

B j = 1 . 0 7 6 x I O ~ A I

2 2 )

where A is the screen's albedo, I is i n f o o t -c a n d l e s , a nd B .~ , i s in nanoLam ber ts .

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3 2 4

2 9 R e s o l u ti o n

B F~ Scha e fer

A di s t an t po in t sourc e of l igh t can be broaden ed by a tmo spher i c t urbulence , d i f f r ac t ion by op t i cs , and

sm ea r ing ov e r t he ape r t u r e o f a p i nho l e cam er a . I dea ll y , the se e f f ec ts shou l d be supe rposed b y t he

ca lcu la t ion of a two-dimensional convolu t ion of t he or ig ina l image wi th a l l t he smear ing funct ions .

In prac t i ce , i t i s d i f f icu l t t o ca l cu la t e t he conv olu t ion and imposs ib l e to presen t t he r esu l t i n a s imple

and genera l m anner . As an a l t e rna t ive , i t i s poss ib l e to ca l cu la t e t he charac t er is t i c a r ea o f t he

conv olu t ion , which can be expressed as the second m om ent (~) of the apparent br ighm ess d i s t r ibu t ion .

I t c an be p r oved t ha t t he s econd m om en t o f a convo l u t i on o f c i r cu l a r l y sym m et r i c no r m a l i zed

di s t ribu t ions i s j us t t he sum o f the second m om ent s of t he funct ions be ing conv olved . Thu s (Schae fer

1993b) , t he second moment of t he smear ing funct ion , ~ ,m~, , i s de t ermined by the fo l lowing

express ions :

~ s , , l s t =

0 . 3 6 1 ( M c c s * * l n # ) 2 ,

1~6 , / - 2 16 .0"

c m M z

D

~ P " 8 / L J L '

~ .me a. = P seetat + ~ dl// + ~tap#r

2 3 a , b , c , d )

H e r e , M i s t h e m a g n i fi c a t i o n o f a n y o p t ic s b e i n g u s e d , a , ~ i s t h e F W H M ~ o f th e s e e i n g d i sk , ~.

~ s t h e w a v e l e n g t h o f li g ht w i t h ~. 0 . 5 5 m i c r o n s f o r t h e r i g h t h a n d s i d e o f E q u a t i o n 2 3 b ) , D i s t h e

a p e r t u r e o f t h e o pt i c s, a n d L i s t h e l e n g t h o f p r o je c t io n , b o t h i n t h e s a m e u n i t s. F o r v i e w i n g w i t h n o

opt i ca l a id , t he va lues o f both /~a lf t and ~ ,~ , a r e zero ; a l t e rna t ive ly , for v i ew ing w i th focu s ing opt i cs ,

t he # ~ , va l ue is z e r o . T h e d i s tr i bu ti ons in E qua t ion 23 a r c no r m a l i zed , w h i ch im p l i e s t hat no pho t ons

a r e l o s t i n the sm ea r i ng p r oces se s . T h e second m om en t o f t he sm ea r ed i m age w i l l be g iven by :

] 1 =

I t s = e a r ~ s o u r c e

2 4 )

W h e n c a lc u la t in g t h e s e c o n d m o m e n t o f t h e o r i g in a l s o u r c e ~ ) , t h e d is tr ib ut io n m u s t b e

n o r m a l i z e d . F o r p o o r l y r e s o l v e d i m a g e s o r i m a g e s t h a t a r e f ai r ly n i f o r m , t h e s o u r c e vi si bi li ty a n

b e e q u a t e d t o t h at o f a u n i f o r m c i rc u la r i s k o f i de nt ic al s e c o n d m o m e n t w i t h a d i a m e t e r

25 )

' FWHM f f i Ful l Width a t Hal f -Maximum of the d i s t r ibu t ion funct ion .

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Astronomy and the Limits of Vision 3 2 5

Thi s d i sk o f equal v i s ib i l it y must be no rmal i zed in he ight by the sam e f ac tor as was used to norm al i ze

the or ig ina l sourc e d i s t r ibu t ion .

Fo r p inhole cameras , t here wi l l be some p inhole d i ameter which wi l l op t imize the r eso lu t ion .

T h i s op t im a l dA am e te r can be f ound by m i n im i z ing t he quan t it y ( p ~ + P ~ r ) w i th r e spec t t o D .

Thu s , t he r eso lu t ion is bes t w he n/ )2 - - 4~ff~, which cor resp onds to the p inhole occupy ing exac t ly one

Fresnel zone (Young 1972) .

Th e see ing d i sk towards the zenith has a t yp ica l FW HM of 1 .5" dur ing the n ight or 3" dur ing

the day . Ne ar the hor i zon , t he see ing d i sk is l a rger because the l i gh t t r aver ses a grea t er pa th l ength

of turbulent a ir . Fo r a s ight li ne 60 f rom the zenith , which passes through twice as much a i r as t he

zeni tha l pa th , t he second mom ent of t he seeing d i sk wi ll double . To g enera l i ze :

% , , , , , f f i a ; V t T " ,

26)

wh ere as* i s t he FWI- IM o f the see ing d i sk a t t he zeni th and X i s t he a i r m ass a long the l i ne of s ight

( f r om E qua t ion 3 a f o r H o = 8 . 2 k i n ) .

Th e hum an e ye has an in t rins ic r eso lu t ion or p ixe l s i ze (0cv ,0 tha t depend s on the l i gh t leve l .

The "cr i t i ca l v i sua l angle" or p ixe l d i ameter i s g iven by:

_ 4 0 d e x ( 8 . 2 8 B _ 0 . 2 9 ) i f l o g B ) > 3 . 1 7 ,

O c v 4 - S

[ 380 ex(0.3 ag)] i f / O g , ) < 3 .1 7

cva = min 9 0 0 ,

( 27 a , b )

(c.f . Schae fer , Bulder , and Bou rgeoi s 1992). He re , B i s t he br ightness (in unit s of nanoL amb er t s ) of

the source o f i n te res t , and S i s t he Snel len ra t io ( the Snel len ra t io i s un i ty for 20120 v i s ion , whi l e S =

2 . 0 f o r good 20 / 10 v i s ion ) o f t he obse r ve r .

Ho w far apar t must two identi ca l sources be fo r the eye to r eso lve them? Th e t r ad it iona l

answer i s Ray le igh ' s c r i te r ion , w her e sou r ces a r e p r e sum ed r e so l ved i f

1 . 2 2 ~.

0 > - -

D

28 a)

wh ere 0 i s t he angular separa t ion o f t he center s i n r ad ians , ~. i s t he e f f ec t ive wavelength o f t he l igh t

beam , and D i s the aper ture of t he t e l escope . In the amateur as t ronom y comm uni ty , the popular

D a w e s ' L i m i t gives a simi lar empir ical resul t :

4 . 5

0 > - -

D

(28

b )

~ 1 5 4 4

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B. E, Schaefer

wh ere D i s measu red in inches and 8 in a r c- seconds . Even thoug h these c r i t e r i a a r e prec i se ly s t a ted ,

the prob lem i s poor ly def ined . In par t i cu lar , a wider separa t ion is needed i f backg round must be seen

between the tw o sourc es to prov e dupl i c ity (as opposed to e longat ion) , whi l e devia t ions f rom a poin t

source shape can be de t ec t ed wi th much sm al l e r separa t ions fo r br ight sources . An other problem w i th

these limi t s i s tha t they only apply when the imag e is br ight and d i f f r ac t ion dom inates . Tw o exam ples

where Dawes ' l imi t does not y i e ld cor rec t r esu l t s a r e tha t most naked eye observer s (D ~ 0 .2 inch)

cann ot r eso lve the ma jor separa t ion of t he br ight b inary s t a rs e I and 2

Lyrce

despi t e t he i r 208"

angular d i s t ance f rom each o ther and tha t an observer wi th a 10- inch t e l escope aper ture and poor

see ing ( say , a , . . i~ - 5" ) canno t r eso lve e i ther e I o r 2 L y r ~ with thei r 2.6" angular separat ions.

Perhap s the bes t ru l e is tha t two s imi l ar images can b e " r eso lved" i f t he i r separa t ions a re com parable

to the charac t er i s t i c s i ze o f t he pe r ce i ved i m age .

2 .1 0 O p t i c s

F o r q u e s t i o ns o f vi si bi li ty n m o d e m t i m e s , a m o d e l o f te l e sc o p e o p t i cs m a y b e n e e d e d . T h e o p ti c s

o f t e l e sc o p e s h a v e w e l l k n o w n e ff ec ts , h o w e v e r , t h e r e s e a r c h e r m u s t b e w a r e t h e m a n y s u b tl e

p h e n o m e n a t h a t c a n c h a n g e t h e r es ul t. T h e b e a t d i s c us s i o ns f r o m a v is ib il it y o i n t o f v i e w a r e in

T o u s e y a n d H u l b u r l 1 9 4 8 ) a n d S c h a e f e r 1 9 9 0 b ) . T h e o p t i c s o f a m l e s c o p e w i l l a f fe c t b o t h t h e

p e r c e i v e d s o u r c e i nt en s it y a s w e l l a s t h e p e r c e i v e d b a c k g r o u n d b r i g ht n e s s. T h e s e e f fe c ts c a n b c

m o d e l l e d a s S c h a e f e r 1 9 9 0 b , b u t s e e m i n o r c o rr e ct i o ns i n S c h a e fe r , B u l d e r , a n d B o u r g e o i s 1 9 9 2 a n d

a s b e l o w )

1 = I F ,

B = B c k F

2 9 a , b )

T h e f o r m s f o r t h e o p ti c s c o r r e c ti o n f ac t or s , ~ a n d F B, a r e t h e re s ul t f m a n y c o r r e c ti o n s f o r i n d iv i du a l

e f f ec t s w i t h

F F , F , P s c F .

V , - - F , P , F , V o F s c r c F .

3 0 a , b )

T he i nd i v i dua l com ponen t s o f t he co r r ec t i on f ac t o r s a r e eva l ua t ed w i t h u se o f t he f o l l ow i ng

express ions :

F b = 1 i f b i n o c u l a r v i s i o n

F b = f 2 ~ 1 . 4 1 i f m o n o c u l a r v i s i o n ;

F t -

T, , t

, . _ 1 f

(31 a ,b)

(31 c)

3 1 d )

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327

2

, : I ' ' o .o ' [ 1 2 1 s,

2

(31 e)

(31

F = = M

;

( 3 1 g )

Lhe expressions for the factors Fc, Fc, and F sc d epend upon the f lux-level ; the fol low ing constraints

apply:

Case #1 -

l o g B ) > 3 . 1 7

F :

1 0 * x ,

F c f f i l

3 1 h , l J )

Case 2 - l o g B ) < 3 . 1 7

F, = 10 '~*x ,

1 _ [ P . / 1 2 , 4 4 1 4 ) [

F s c = m / n 1 , 1 - [ D I ( 1 2 . 4 4 M ) ] ) J

F = 1 0 - t - [ a - v ] 1 2 )

.

3 1 k , l ,m )

The correct ion factors ' subscripts are: b for binocular vision, e for atmospheric ext inct ion, t for

transmission throug h the telescope, p for l ight lost outside the pupil , a for l ight gathering power o f

the telescope aperture, r for the resolving power of the eye,

S C

for S t i les -Crawford ef fec ts , c fo r

sourc e col or, an d m fo r the magnification. He re, Ttc, represents the transmission o f the telescope (a

typical value for w hich is 0.8, according to S chaefer 1990b), S stands fo r the Snellen rat io, M refers

to the m agni f ica tion of the opt ics , D to the aper ture o f the opt ics , and [B-V] denotes the v isua l co lor -

index of the source ; the numer ica l fac tor "12 .44" has uni t s of c e n t i m e t e r s . T h e diameter of the

observer ' s pupi l i s g iven by:

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B. F~ Scha efer

32)

H e r e , Y r e p r e s en t s t he obs e r ve r ' s a ge i n ye a r s , B t he ba c kg r ound b r i gh t ne ss i n n a n o L a m b e n s , a n d 1

the in tens i ty of a g la re source p laced a r t angle 0 in degrees a w a y w i t h a m a gn i fi c a ti on o f M . F o r a

s t anda r d obs e r v e r

Y = 25 years ) w i th

no g l a r e s ou r c e ,

P ~ t -~ m a x { O . 2 c m , O . 6 8 c m ) e x p [ - O . 1 6 B l O - e ) ] } ,

(33)

w h i c h i s a n a pp r o p r i a t e e xp r e s s ion f o r c om pa r i s on w i t h t he obs e r ve r s c on t r ibu t i ng t o t he phys i o log i c a l

da ta on v i s ib i l i ty th re sholds .

2 11 Thresholds of Human Vision

All of the preceding too l s can be used to ca lcu la te the appa rent br igh tness of the a s t ronomica l source

as we l l a s the sur face br igh tness of the backgroun d. Th e ques t ion o f v i s ib i l i ty then depends on the

phys i o l og i c a l que s ti on o f c an s uc h a s ou r c e be de te c te d by t he hum a n e ye . E x t e ns i ve e xpe ri m e n t a t ion

unde r a w i de va r i e t y o f c ond i t ions w a s m a de i n t he 19 4 0 ' s a nd 19 50 ' s , a nd i t is th i s w or k w h i c h c a n

be used to quant i fy v i s ib i l i ty th re sholds .

Op e ra t iona l ly , th e r e su l ts c an be d iv ided in to the de tec tion o f po in t sources and un i form

extended sources . Fo r the de tec t ion o f po in t sources of ligh t , Hech t (1947) g ives a convenien t and

accura te f orm ula for the thre shold i l lumina nce va lue , l th , nam e ly :

I t s= c 1 + ~ ) 2 , w h e re

l og ( c ) = - 8 . 35

a n d l o g k ) = - 5 . 9 0 , i f l o g B )

> 3 . 1 7 ,

o r

l o g c )

= - 9 . 8 0 a n d

l og k )= - - 1 . 9 0 , i f l o g B )

< 3 . 1 7 .

34 a ,b , c )

H e r e , B r e p r e s e n t s t he e ff e c t i ve ba c kg r ound i n na noL a m be r t s , a s t a ken f r om

zq afion

29 b . F o r

uni form extended sources , the source v i s ib i l i ty i s cha rac te r i zed wi th a cont ra s t r a t io , C , de f ined a s

c - I B o - n I

B

(3~3

wh ere B ,o ~ , r epre sent s the sur face br igh tness of the source to be de tec ted . No te tha t C does not

de pe nd on w he t he r t he sou r c e i s b r i gh t e r o r da r ke r t ha n t he ba c kg r ound . T he c om pr e he ns i ve a nd va s t

s tudy by Blaekwe l l (1946) rem a ins the de f in i t ive wo rk . H e repor t s on the thre sholds for

h i gh l y - e xpe r i e nc e d young obs e r ve r s o f a ve r a ge v i s i on a l l ow e d t o u s e b i noc u l a r v i s i on w i t h na t u r a l

pupi l s and a l lowed to choose ave r ted v i s ion or d i r ec t f ixa t ion and a l lowed a l e i sure ly examina t ion .

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3 2 9

Fo r uni form sharp-edged c i r cu lar r eg ions o f angular d i ameter ~ , t he threshold cont r as t r a t io , Cth , i s

g iven by:

C tk = m a x [ 2 . 4 B - ' l , 2 0 B -'4

i f e c v A > ~ ~ , o r

= - - i f l o g ( B ) > 6 , o r

l o g ( C , h ) = - [0 . 1 2 + 0 . 4 0 / o g ( B ) ] + [ 0 . 9 0 - 0 . 1 5 l o g ( B ) ] l o g ( l O 0 l O

' t o g ( B ) < 6 a n d ~ ; > -2 0 '

36 a ,b ,c )

Th ese three formu las a r e em pi r i ca l f it s t o the da t a in Blackwel l , and they r ep rodu ce h i s va lues to 5 %.

Equat ion 36a i s an express ion of "Ricco ' s Law," in tha t t he threshold i s propor t iona l t o ( -2 .

Cor rec t ions to these va lues a re neede d for an observ er ' s exper i ence , as di scussed in Schae fer (1993b,

1990b).

He ar the sens i ti v i ty threshold of t he hum an eye , de t ec t ion i s a probabi l i st i c ques t ion; see Lam ar

e t a l .

(1947, 1948) for a def in i t ive ana lys is . Th e quoted phys io logica l th resholds a re a lways fo r a

50% probabi l i t y of de t ec t ion .

2 . 1 2 C o l o r V i s i o n

Th e human ey e can de tec t co lor s f rom sources br ighter than 1500 nL . Th e r eason is tha t p h o t o p i c

v i s i o n ( i . e . ,

day v i s ion which uses the r e t ina l cones ; as opposed to

s c o t o p i c v i s i o n ,

or n ight v i s ion ,

which u ses rods) has three types of photopigment s , each w i th a d i f f e ren t spect ra l sens i ti v i ty . So the

ey e s imul t aneously measures the br ightnesses of an image ove r three d i f f e ren t wavebands , muc h l ike

t ak ing CCD images through three d i f f e ren t f i l t e r s or t ak ing a co lor photograph wi th three d i f f e ren t

chemica l dyes . Th e s i tua tion i s a l so ana logous to photom et ry in the Johnson U B V sys t em , w he r e t h r ee

int ens if ies a r e measured (or a l te rna t ive ly , one br ightness and two color s) . Fo r each photopigment ,

the e f f ec t ive br ightness wi l l be

X = A f x E x d X ,

Y = A f y E x d X ,

Z = A f z E x d ~

( 37 a , b , e )

Th e num er i ca l va lue of t he cons t an t A i s 6 . 8 0 l u m e n s e r g . Th e va lues o f x* , y ' , and z" a r e g iven

as a funct ion of wavelength , k , i n Table 7 ( for example , see Cornsweet 1970 or Weast 1974, for

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B. E. Schaefer

t ab le s w i th h ighe r spec t ra l r e so lu t ion) . Th e br igh tness Ex ha s un i t s

er g /cm2/s / ,4 .

Y w i ll ha ve e gs un i ts o f

p h o t , s o

tha t

m r = - 23 .9 9 - 2 . 5 l og ( Y )

Table 7. Colorimetric Coefficients

h (/~) x* y* z*

3750 0.0006 0.0000 0.0020

4000 0.0143 0.0004 0.0679

4250 0.2148 0.0073 1.0391

4500 0.3362 0.0380 1.7721

4750 0.1421 0.1126 1.0419

5000 0.0049 0.3230 0.2720

5250 0.1096 0.7932 0.0573

5500 0.4334 0.9950

0 0087

5750 0.8425 0.9154 0.0018

6000 1.0622 0.6310 0.0008

6250 0.7514 0.3210 O.O001

6500 0.2835 0.1070 0.0000

6 7 5 0

0.0636

0 0232 0 0000

7000 0.0114 0 0 0 4 1 0.00(30

7250 0.0020 0.0007 0.0000

7500 0.0003 0.0001 0.0000

With the se uni t s ,

38)

The in tens i t i e s in Equa t ion 37 a re usua l ly norma l ized a s

X

X Y Z

Y

y

X Y Z

Z

Z

X Y Z

3 9 a , b , e )

T he t w o va l ue s x a nd y a r e t he n a c om p l e t e de sc r ip t i on o f

any co lor ( the va lue of z i s ignored a s be ing redundant ) .

Th e co lo r whi te (and the co lor of the Sun) i s cha rac te r i zed

as x =

0 . 3 3 a n d y = 0 . 3 3 .

T he c o l o r c a n be

pa r a m e t e r i z e d w i th

~F = A T A N 2 ( y - 0 . 3 3 , x - 0 . 3 3 ) ,

4O)

w he r e t he

A T A N 2

func t ion i s the a rc tangent compute r -

subrout ine which p icks the cor rec t quadrant .

T he de f i n i t i on o f t he p r i m a r y c o l o r s i s a l w a ys quo t e d

di f fe ren t ly in " s tan da rd" r e fe rences , b u t a s e t o f r ea sonable

choice s i s g iven in the fo l lowing t abula t ion (Ta ble 8) . Jus t

the same co lor v i s ion in humans i s qu i t e complex wi th

m a ny non l i ne a r e f fe c t s , and t he a bove m us t s e r ve on l y a s

a p r i m e r . F o r good de t a il e d de s c r ip t i ons , I r e c om m e nd

Cornswee t (1970) .

2 . 1 3 W e a t h e r st a t is t ic s

F or a ny phe no m e non i nvo l v i ng t i m e o r r a t es o f t i m e , t he

f r e que nc y o f c l e a r s k ie s m a y be i m por t a n t . I n ge ne ra l , r e l e va n t da t a c an ge ne r a l l y be ob t a ine d f r om

loca l a i rpor t s , n ewsp ape rs , o r a na tiona l me teoro logica l c en te r . Schae fe r (1987a ) r epor t s on c loudiness

s ta t i s t i c s low on the tw i l igh t sky nea r sunr i se and sunse t ; he found tha t good and ave rage s i t e s have

c loudiness probab i l i t ie s of a th ird and of a ha l f , r e spec t ive ly . Th e c la r i ty odds f ro m one n ight to the

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Astronomy and the Limits of Vision 331

next a re cor re la ted , s ince i t usua l ly t akes a f ew days for a bad wea the r sys tem to pass by . Thus , a

c l oudy n i gh t w i ll be f o l l ow e d by a c l oudy n i gh t 75 % o f t he t i m e ( f o r bo t h good a nd ba d s i te s ), w h i l e

a good s i t e w i l l have a c loudy n ight fo l low a c lea r n ight 25% of the t ime (50% for an ave rage s i t e ) .

I n a ny s t udy o f o ld h i s to r i c e ve n t s , i t s hou l d be r e m e m be r e d t ha t l oc a l w e a t he r pa t te r n s m i gh t ha ve

changed s ign i f i can t ly in the ensu ing yea r s .

Table 8. The P rimary Colors: their ~,'s and q"s.

Color

~. Ran ge (A)

Range

red orange yellow green blue violet

> 6 1 0 0 5 9 0 0 - 6 1 0 0 5 7 0 0 - 5 9 0 0 5 0 0 0 - 5 7 0 0 4 40 0-5 00 0 < 4 40 0

< 0* 0* - 21" 21 - 63* 63 - 148" 148 - 242* > 242

3 R e s u l t s

Th e prev ious sec t ion g ives expl ic i t fo rmu la s which desc r ibe the phenom ena a f fec t ing v i s ib i li ty . To

be use fu l , how eve r , the se equa t ions mus t be put toge the r in a ma nne r tha t makes phys ica l s ense . Fo r

example , to ca lcu la te the he l i aca l r i s e da te of S i r ius f rom anc ien t Egypt , the pa rame te r s needed a re

the pos i t ion of S i r ius , the br igh tness of the tw i ligh t sky , the l ike ly ex t inc t ion f rom Egy pt , the a i r mass ,

and the thre sholds o f v i s ion . In th i s ca se , the r epea ted v i s ib i l ity t e s ting for S i r ius th roughou t the dawn

t w i li gh t a nd f o r m a n y da ys i n tr oduc es a c om p l e x i t y be s t ha nd le d w i t h a c om pu t e r p r og r a m . O t he r

re su l ts m ay be pre sented bes t a s s imple formu la s or a s a s e r i es of equa t ions . In th is s ec t ion a re

pre sented re su l ts on a va r ie ty o f cele s ti a l v i s ib i l i ty problem s , w i th each re su l t having app l ica t ions to

man y h i s tor i ca l and a s t ronomica l ques t ions . In two ca se s ( for sunspot v i s ib i l ity and the ex t inc t ion

angle ) , I w i l l g ive de ta i l ed expos i t ions on how the " too l s " can be used to c rea te working mode l s .

3 I Sunspots

Larg e sunspot s a re ea s i ly v i s ib le to the hum an ey e (wi th prope r pro tec tion) and have been ex tens ive ly

recorded by the anc ien t Chinese (Yau and S tephenson 1988) . Unt i l recent ly , the bes t ru le of thumb

for sunspot v i s ib i l ity ha s been tha t a sunspot m us t be l a rge r than one a rc -m inute in s i ze to be v i s ib le

(Eddy 1980, Ed dy, S tephen son , and Yau 1989). Thi s ru le ha s been s t rongly r e fu ted by obse rva t ion

(Mossman 1989, Kel le r 1980, Kel le r 1986, and Schaefer 1991d).

Schae fe r (1991d and 1993b) has deve loped a de ta i l ed m ode l tha t p red ic t s the v i s ib i l i ty of

sunspot s w i th d i rec t v i s ion , w i th t e le scopic a s s i s t ance , w i th a p inhole camera , and wi th t e le scopic

pro jec t ion . He r e , I w i ll g ive the de r iva t ion o f the d i rec t v i s ion l im i t in de ta i l.

F i r s t , e s t ima te the sur face br igh tness of the Sun a f t e r i t ha s been d immed enough for s a fe

v iewing . An appro xim a te answ er i s adequa te , s ince the depend ence on B in Equa t ion 36b i s so weak .

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3 3 2 B. F_.. Sc ha efe r

Freq uent ly , t he se t ti ng Sun appear s d im eno ugh for good v i s ib i li t y , and th i s d imming can b e es timated

roug hly wi th X = 40 ( for near t he hor i zon f rom Equat ion 21)) and k = 0 .3 ( a t yp ica l va lue , see

T ab l e 4 ) so t he Sun ' s su r f ace b r igh t ness w i l l d im by a f ac t o r o f

1 0 4 s

( f r om E qua ti ons 8a and 9) . Fo r

a more accura t e ca l cu la t ion , Equat ion 8b should

b e u s e d w i t h , s a y , k R = O . 1 0 , k oz f fi 0 . 0 3 , a n d

k o = 0 . 1 7 ,

and a ir m asse s f o r Z = 9 0* f r om T ab l e 3 , so t ha t t he d i m m i ng f ac t o r i s

1 0 7 1.

Al terna t ive ly , t he r ecommended f i l t e r for safe so l ar observ ing i s welder s ' g l ass #14 (Chou 198lab)

which has a t ransmiss ion of e q4 or 10 6 ~. Thu s , t he sur f ace br ightness of t he Sun wi l l be r educed

f r o m

6 x 1014

nL ( see T ab l e 6 ) to r ough l y

I O s n L .

Secon d, t he second mo me nt of t he smear ing m ust be ca l cu la ted . Fo r d i r ec t v i s ion , ~a lf r = 0 ,

/z ,p~r = 0, and M = 1 ; so, f rom Eq uat ion 23:

2

Fsm ar = 0.361 s t e t l q r

A t y p i c a l d a y t i m e v a lu e f o r a , ~ i s 3 , s o t ha t ~ , m ~ is ( 1 . 8 0 ) 2 . (41)

Th i rd , the seco nd mom en t o f the sunspot s t ruc tu re m us t be ca lcu la ted . Fo l l ow in g A l l en

(1976) , I wi ll take the f r ac t ion o f the sunspot a r ea occupied by the umbra to be 17% on average . I

wi l l a l so idea l i ze the umbra and penumbra as concent r i c c i r c l es wi th cont r as t va lues of

0 . 9 1 a n d

0 . 2 6 , r espec t ive ly . (Schae fer [1993b] dem onst r a t es t ha t t he v is ib i li t y i s on ly we akly depend ent on

these assum pt ions . ) Thu s the cont r as t va lue i s 0 .91 fo r 0 < 0 .41Root ,

0 . 2 6

f o r 0 . 4 1 1 ~

< 0 < R,f, ot,

a n d z e r o

for 0 > R,po~, wh ere 0 i s t he angular d i s t ance f rom the sp ot ' s center . But these cont r as ts

must be norm al i zed by a f ac tor N so tha t t he second mo men t s can be added as in Equat ion 24 . Thus

1 ffi 2 n f t N C , p , , ( O ) ] OdO ffi 1.16NRffpo,

42)

T he va l ue o f t he s econd m om en t o f t he con tr a s t, w h i ch cha r ac te r i ze s t he sol id ang l e cove r ed by t he

sunspot , i s t hen g iven by

~,0 , ,, ,= 2nfO 2tNC ,po, O)]OdO = 0.44 /V R,~o, = 0.38R~,0 ,

43)

Four th , we need to f ind the proper t i es of a uni form c i r cu lar d i sk wi th the same v i s ib i l i t y as

T h e pe r ce i ved s i ze o f t he spo t m ay be ca

he sunspot .

l cu la ted b y:

Is = 0.361 c~s,~t + 0.38 R ot ,

44a)

f rom Equat ions 24 , 41 , and 43 . Given tha t a~ i . s < < Rq,~ fo r spot s v i s ib l e wi th unaided v i s ion and

for any r easonable see ing , we have , t hen:

2

p = 0.38

R~pot

44b)

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Astronomy and the Limits of Vision 3 3 3

The d iame te r of the uni form d i sk wi th equa l v i s ib i l i ty w i l l have the same cha rac te r i s t i c a rea a s the

s uns po t , o r f r om E qua t i ons 25 a nd 44b ,

=

1.74 R,p,c

45 )

Th e to ta l da rkne ss in th is d i sk mus t equa l the to ta l da rkness in the (unno rma l ized) sunspot . Thu s ,

2 o d O : 2 0 d O

o r

d j s k

0 4 9

W e now ha ve e nough i n f o r m a t i on t o c he c k w he t he r t he un i f o r m d i s k w ou l d be v i s i b l e .

F i f th , the c r it i c a l v i sua l angle mus t be quant i f ied . Fro m E qua t ion 270 , th i s is

4 4 #

0cv, - S '

4 ~ )

4 6 1 ) )

4 7 )

wh ere S i s va lue for the ob se rv e r ' s Sne ll en r a t io . For f i l te r s which change the Su n ' s sur face

b r i gh t nes s f r om 107 nL to 109 nL , t he 0c v c ha nge s on l y f r om 4 2 / S to 48 /S . But the typ ica l e r ror

in m easur ing S i s 25 %, so the unce r ta in t i e s in no t sp ec i fy ing the f i l te r t e chnique a re r e la t ive ly sm a l l .

F ina l ly , the v i s ib i l i ty of the uni form d i sk i s eva lua ted wi th the da ta f rom Blackwe l l (1946) .

F o r a n a ppa r e n t s u r f ac e b r igh t ne s s o f 10 s nL and the r e su lt s o f Equa t ions 36b , 45 , 46b , and 47 , I f ind

a thre shold sunspot s i ze of

2 2 "

R~Pt - S

4 8 )

T hu s w e s e e t ha t s uns po ts s m a l l e r tha n one a r c - m i nu t e in s i ze m a y be de t e ct e d w i th n o r m a l h um a n

vis ion . Schae fe r (1993b) has co l l ec ted we l l -de te rmined obse rva t iona l lim i t s f rom fou r obse rv e r s and

f rom one ne twork of obse rve r s which va l ida te the l im i t in Equa t ion 48 .

3 .2 S tars and p lane ts dur ing the day time

Th e s ta r s and p lane t s do not tu rn of f when the Sun r i s e s , i t is jus t tha t the g la re f ro m the Sun i s so

ove rwh e lm ing a s to r ende r a l l bu t the br igh te s t o f them invi s ib le. Ho w fa in t a source can be seen in

t he da y t i m e ? F o r a t yp i c al da y t i m e s ky b ri gh t ne ss o f 5 x 108 nL (Table 6) and ex t inc t ion wi th Am =

0 .3 magn i tudes , then Equa t ions 9 , 10 , and 34 show the lim i t for norm a l v i s ion to be -3 .1 magni tude .

Thu s unde r no rma l dayt im e condi t ions , on ly Venus can be seen wi thout t e le scopic a id . I f the Sun i s

c lose to the hor izon ( such tha t B,ky= 3 x 107 nL, see Table 6) , then the una ided norma l eye can see

d o w n t o -0 .3 magn i tudes , which cove rs Venus , Jupi t e r , Mars , S i r ius , and Cano pus . To usey and

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3 3 4 B E Schaefer

Hu lbur t (1948) d i scuss the day t ime v i s ib il i ty of s t a r s wi th t e l escopes and f rom high a l t i t ude a ir c raf t .

Schaefer (1991c) ana lyzes the v i s ib i l i t y of sungraz ing comet s and Venus near conjunct ion wi th and

wi thout t e l escopic ass i s tance . Fo r norm al unaided v i s ion ,

r a n, = - 9 . 2 8 + 5 l o g ( 0 ) ,

4 9 )

whe re O i s t he angular d i s t ance f rom the Sun. S imi l ar equat ions can be foun d for t e l escopic v i ewing .

Th ere have been per s i s t en t myths tha t s ta rs can be seen in dayt im e by looking up a t al l ch imney

or by looking in the r e f lec t ion of water in a deep wel l . Th e l i t e r a ture does conta in such r epo r t s , bu t

they are a l l l i t e r ary devices , t h i rd-hand anecd otes , o r ch i ldhood m em or ies of o ld m en (Hug hes 1983,

Perr ach i 1993) . H ow ev er , al l f i r st -hand repo r ts (Hu m bold t 1851, H yn ek 1951, Smith 1955, Slabinski

1990, and Sander son 1992) a re emp hat i c on the imposs ib i li t y of see ing s t a r s und er such condi t ions .

In addi t ion , s t rong theore t i ca l a rgument s show tha t t he g l are r educt ion i s i ns igni f i cant , t ha t t he

perpen dicular r e f l ec t iv i ty of water i s 2 %, and tha t the small backgrou nd f i e ld-of -v i ew ac tua l ly makes

the threshold up to a magni tude worse (Sch aefer 1991c) . Thu s , bo th observa t ion and theory s t rongly

prove the ch imney/wel l myths to be f a l se .

3 3 S h a d o w b a n d s

Shadow bands are ghos t ly and i r r egular dark l i nes tha t a r e v i s ib l e i n the moment s before and af t e r a

to t a l ec l ipse of the Sun (M arschal l 1984) . Th e bands ar i se f rom di f f r ac t ion and sc in ti ll a ti on of a long

th in li gh t source , wi th Cod ona (1986) provid ing the def in i ti ve theore t i ca l work .

Th e pheno me na of sunse t shadow bands (Ro zet 1906 and Ives 1945) and volcanic shadow bands

(O'Meara 1986) are st i l l myster ies.

3 4 Card ina l or ien ta t ion

Many ancien t monument s and modern c i t i es a r e a l igned wi th axes poin t ing in a nor th / south or

eas t /wes t d i r ec t ion . Such a card ina l or i en ta tion must u l t imate ly com e f rom as t ronom ica l observa t ions .

In mo dern t imes , s t andard survey ing techniques can es tab li sh nor th wi th a rc- second accu racy . In o lder

t imes , be fore good eph eme r ides and ins t rument s were ava i l ab le , t he methods (and m ot iva t ions ) o f

card ina l a l ignmen t a r e genera l ly not know n. M any procedures have been proposed (Zaba 1953,

Edw ards 1961, Neu geb aue r 1980, Ha ack 1984, and Isler 1989) , and these fal l into three catego r ies.

Th e f i r s t ( and s imples t) ca t ego ry i s t o s ight a long so me l ine which poin t s t o the az imuth o f a

s t a r tha t does n ot move. In m odern t imes , Polar i s is about a degre e f rom the pole and i t may be

poss ib l e (wi th very- l a rge- s ca l e so l id ly-mou nted s ights and c arefu l r epea ted ob serva t ions) t o in t e rpola t e

be tween i ts ex t r em e az imuths to perhaps a quar t e r of t he polar d i s tance . In anc ien t times , no s t ar s

were c lose to the pole so tha t t he r esu l t an t accuracy would have been lower .

T h e second ca t ego r y i nvo lves m e t hods f o r f i nd ing t he az i m u t h o f t he sho r t es t shadow . A r ound

noon, t he shadow l ength var i es l i t t l e so i t i s h ighly inaccura t e to d i r ec t ly t ake the az imuth of t he

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Astronomy and the Limits of Vision

3 3 5

shor t es t mea sured shadow. Ins t~O , the az imuths of equal length shadows severa l hou r s befo re and

af t e r noon can be b i sec t ed to f ind nor th . Fo r an accuracy be t t e r t han rough ly 0 .2 , t he observa t ions

m us t be m ade by d i r ec t obse r va ti on o f t he Su n ' s edge ( a s opposed t o t r y i ng t o de f i ne som e pos it ion

wi th in the shado w' s penu mb ra) nea r t he t imes of a so ls ti ce .

Th e th i rd ca t ego ry involves the s ighting of az imuths o f r i s ing and se t ti ng of som e objec t and

then b i sec t ing the an gle to f ind nor th . Thi s l as t ca t egory has the severe prob lem tha t the a l ti t ude on

both hor i zons must be ident i ca l ( c f Schae fer 1986) . An other severe problem f or h igh accurac y

al ignment s by th i s method i s t ha t t he na tura l var i a t ion of r e f r ac t ion wi l l i n t roduce an er ror i n the

measured az imuths (Schaefer and L i l l e r 1990).

3.5 Sky colors

"W hy i s the sky b lue?" i s t he c lass i c young chi ld ' s ques t ion tha t s tumps the i r parent s . A physic i s t

would expla in about Rayle igh sca t t e r ing , where the molecules in our a i r p referent i a l ly sca t t e r b lue

l ight . Thi s i s i ndeed the cor re c t ( a lthough simpl i fi ed) answer . S imple appl ica t ions of Rayle igh

sca t te r ing can expla in m ost of t he co lor s seen in the sky . Fo r examp le , t he r ed co lor of t he set ti ng

sun i s du e to the sca t te r ing of a ll b lue li gh t ou t of t he d i rec t r ays . Th e r ed co lor s of twi l igh t a r e

caused by t he on l y d i r ec t il lum i na ti on o f t he a i r be ing t he l ow r ed Sun . T h e w h i te co l o r o f t he

dayt im e sky c lose to the ho r i zon i s a r esu l t o f t he exac t compensa t ion o f t he skyl ight produ ct ion by

sca t t e ring wi th i ts ex t inc t ion over a l a rge opt i ca l pa th l ength . Thi s can be looked a t quant i t a tive ly

where the luminos i ty of a volume a t a d i s t ance x wi l l be propor t iona l t o the opt i ca l depth f rom

Rayle igh sca t t e r ing ( rR) , w hi l e t he ex t inc t ion wil l be g iven by

exp(-xrrt). The

apparent br ightness

wi l l be an in t egra l a long the l i ne of s ight , which for a near hor i zon pa th i s e f f ec t ive ly

Of. exp(-xrQ x dx and i s a cons t an t i ndependent of wavelength .

Th e l it e r a ture on sky co lor s is ex tens ive . Chandrasekh ar (1960) and R ozenb erg (1966) present

exhaus t ive mathem at i ca l ana lyses and b ib liographies of sca t t e ring in the a tmosphere . Hu lbur t (1953)

gives a mo re access ib l e mathemat i ca l explanat ion for sky co lor s . W alker (1989) present s an exce l l en t

and thorou gh non-m athemat i ca l explanat ion for co lor s and br ightnesses for t he who le sky under many

condi t ions . Minn aer t (1954)an d M einel and Meinel (1983) g ive good pheno men ologiea l descr ip t ions .

3.6 Rainbows and halos

As sunl ight passes through the a i r, i t is r e f r ac t ed , d i sper sed , and r ef l ec t ed by w ater drops and i ce

crys t a ls . Th i s sca t t e r ed l igh t appear s f rom spec ia l d i r ec t ions so as t o produce br ight c i r c l es and spot s

on the sky . Th e m ost f ami li a r phen om ena are the " r a inb ow " seen af t e r a passing s torm (42 f rom

the an ti so l ar d i rec t ion) , t he "ha lo" seen around the Sun or M oon w hen c i r rus c louds are ov erhead (22

f rom the l i gh t source) , t he co lored pa tches of l igh t known as " su ndog s" or ~parhe l i a" ( somewh at more

than 22 f ro m the Sun and a t t he same a l t i tude) , and the co lor fu l c i rc l es cas t on to c loud s be low an

ai rp l ane ca ll ed "g lor i es" ( ex tending typica lly severa l deg rees a round the an t iso l ar d i r ec t ion) . Man y

less-wel l -known e f f ec t s ex i s t , i nc luding " i r idescent c louds , ~ "co ron as , " "Bi sh op ' s r i ngs , " and

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3 3 6 B. E. Schaefer

"c i r cumze ni tha i a r cs" which a l l g ive good d i sp lays of co lor . Sca t t e r ing by i ce c rys t a l s can prod uce

a weal th of a r es , p i l la r s , and ha los , which fo r r a r e d i sp l ays can l igh t up the sky in an in t r ica t e sp ider ' s

w eb o f l igh t . T h e f r equenc y o f r a i nbow s and ha l o s is su r p r is i ng ly l a r ge , w i th obse r ve r s f r om m any

loca t ions logging some such phenomenon typica l ly a t h i rd of t he days throughout t he year .

T he l i te r a t u r e on r a inbow s and ha l o s is ex tens i ve . M y f av o r i t e books a r e M i nnae r t ( 19 54) f o r

a gene r a l de sc r i p t i on o f t he m any phenom ena , K onnen ( 19 85) f o r an i m pr es s i ve d i scuss i on o f

polar i za t ion ef f ec t s , and B oye r (1959) for a de t a il ed h i s tory and explanat ion of t he r a inbow. Despi t e

t he exce l l ence o f t he se sou r ces, Rainbow s Halos and Glories by G r een l e r ( 19 80) i s by f a r t he bes t

genera l r e f eren ce . Thi s boo k g ives def in i t ive and com prehen s ive explanat ions as t o ld in a fun mann er

wi th many beaut i fu l and unique p i c tures .

3 . 7 Mounta in shadows

A sunr i se v i ewed f ro m the top of a t al l moun ta in has an eer i e beauty . So m e mounta ins a t t r ac t

hund r eds o f t ou r is t s da i ly and have been t u r ned i n to com m er c i a l ven t u r e s . H o w ev e r , f ew a r e t he

watcher s w ho turn around af t e r sunr i se to see the shadow o f the moun ta in , which appear s as a conica l

dark r eg ion loom ing above the western hor i zon . As the Sun ge ts h igher , t he shadow con e f la t tens out

and soon d i sappea r s . Fo r an obse r ve r aw ay f r om t he top o f t he m oun t a in , t he s t r uc tu r e can ge t qu i t e

com pl i ca t ed . Th e ef f ec t s a r e mo st prom inent f rom near the top of sharp , i sg la t ed , and t a ll peaks . Th e

obs erve r need not be a mounta in c l imber to see the cone of darkness , s ince man y su i tab le mounta ins

have r oads bu i l t t o t he t op , i ncl ud ing M oun t Wash i ng ton , N H ; P i ke ' s Peak , C O ; M o un t E vans , C O ;

K i t t Peak , A Z ; M auna K ea , H I ; and H a i eaka l a , H I .

T h e m oun t a i n ' s shadow i s ca st on t o the haze in t he a ir . Fo r an obse r v e r a t the peak , any l i ne

of s ight be low the an t i so l ar d i r ec t ion w i ll l i e en t i r e ly in shadow so tha t the on ly l igh t seen w i ll be by

sca t t e ring of t he r e l a t ive ly f a in t skyl ight. An y l ine of s ight above the an t i so l ar d ir ec t ion wi l l be

ent i r e ly in sunl ight , so tha t t he sca t te r ed l i gh t f rom the br ight Sun wi l l resu l t in a br ight sky . Fro m

the top o f a conica l peak , t he shadow w i ll be per f ec t ly sharp . L iv ings ton and Lync h (1979) pro ve tha t

the shadow f rom the top wi l l a lways appear conica l ( r egard less of t he mounta in ' s shape) due to

per spec t ive e f f ec t s , a l t hough the shadow edge may ge t fuzzy; t he i r paper ( and Lynch 1980) g ives a

good m athemat i ca l mod el for t he shadow br ightness f rom anyw here on the mounta in . A com puter

program for ca l cu la t ing shadow images f rom var ious loca t ions near t he top of a conica l mounta in i s

presented by Schaefer (1988a) .

M oun t a i n shadow s a r e s i m i la r t o t he phenom ena o f " c r epuscu l a r r ays , " " sunbea m s , " and t he

" E a r t h shad ow . " T hese o t he r shadow t ypes a r e m ode l l ed in L ynch ( 19 87) and R ozenb e r g ( 19 66).

3 .8 Green f lash

Th e green f l ash appear s w hen the l as t por t ion of t he Su n ' s d i sk s inks be low the hor i zon and changes

to a beaut i fu l eme ra ld co lor . Gree n f l ashes can a l so be seen jus t befo re sunr i se and can ap pear as an

e lec t r i c b lue co lor i f the a i r i s espec ia l ly c l ear . Th e pheno men on f i r s t cam e to the a t t en t ion of t he

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s t r o n o m y a n d t h e L i m i t s o f V is i o n

3 3 7

sc ienti fic co mm uni ty in 1882 wi th the publica tion of a b ook by Ju les Verne

t i t led Le Rayon Vert

Go od descript ions appear in M innaert (1954) and M einel and M einel (1983), bu t the defini t ive w ork

is O'Connell (1958).

Th e exp lanat ion for the green f lash involves the norm al dispersion and ex t inct ion of sunlight

by the a i r . As sunl ight en ters the a tmosphere , re f rac tion bends the l ight pa th by aroun d 0 .5 a round

the t ime of sunset. Each c olor wil l suffer a different dispersion, wi