A FORMULA FOR THE BRIGHTNESS OF LUNA

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P u b l i c a t i o n s

o f

th e

A s t r o n o m i c a l

S o c i e t y

o f th e P a c i f i c

1 0 3 : 1 0 3 3 - 1 0 3 9 ,

S e p t e m b e r 1 9 9 1

© 1 9 9 1 .

T h e A s t r o n o m i c a l

S o c i e t y o f th e

P a c i f i c .

A l l

r i g h t s

r e s e r v e d .

P r i n t e d

i n

U . S . A .

A

MODEL

OF

THE

BRIGHTNESS

OF MOONLIGHT

K EV I N K R I S C I U N A S

Joint Astronomy Centre,

66 5

Komohana

Street,

Hilo, Hawaii 96720

A N D

B R A D L E Y E . S C H A E F E R

NASA/Goddard

Space Flight Center,

Code

661,

Greenbelt,

Maryland

20771

Rece ived 1 9 9 1

Apri l 1 ,

revised 1 9 9 1

June

3

ABSTRACT

A

knowledge

of

the brightness

of

moonlight

is

needed fo r detailed

calculations of

the

limiting

magnitude of astronomical

detectors, whether

they

be

visual, photographic, or

electronic. The

previous iterature

contains

no

method or

making

even

crude

estimates

and has

ew

actual

measurements of moonlight brightness. n his paper we report new measurements of the ky

brightness

from

the

2800-m

level

of

Mauna

K e a .

In

addition,

we

present

a

model

fo r

predicting

the

moonlight as a

function

of

the Moon's phase, the zenith distance

of

the Moon, the zenith distance of

the

ky

position, he angular

separation of

the Moon

and

ky

position, and

the

local

extinction

coefficient.

The model equations can be

quickly

calculated

on a pocket

calculator.

A

comparison

of

our model with our

unar

data

and with

some

Russian solar data hows he

accuracy

of our

predictions

to

range from

8% to

23%.

K e y

words: ight-sky brightness-moonlight-atmospheric extinction

1. ntroduction

It

is

of

fundamental interest to

know

the limiting mag-

nitude

of visibility for finding

objects

with

a telescope.

T h i s

subject

has

recently

been

addressed

by

Schaefer

1990a based

on

a

database of 31 4

visual

observations

of

stars viewed in small- to-medium-sized telescopes with

a

variety of

magnifications.

For large

telescopes

using

video

acquisition

and guiding systems, he question is still im-

portant. t

s

also

relevant

o he

prediction

of proper

exposure times fo r optical

CCD

cameras. A related ques-

tion i s : what

is

the

magnitude

limit

when

the

Moon

is out?

With the

eye,

video system, or

CCD

camera the detec-

tion

ofan

object

depends

on

the

contrast

ofthe

brightness

of

the object in question versus the

brightness

of the

sky.

We

can think of

this

as

achieving

a

certain

signal-to-noise

ratio.

Whatever

brightens

the

sky

degrades the limiting

magnitude

of

the

eye

or

video

system

by approximately

the same amount.

In

a recent

paper

Krisciunas 990)

we

presented a

sample

of

photometric

measurements

of the sky bright-

ness,

with and without

moonlight.

Most

of the measure-

ments and

all of the moonlight

observations)

were

ob-

tained at the 2800-m level ofMauna K e a . The telescope is

a

15-cm

Newtonian

reflector,

and

he

photometer em-

ploys standard U B V

filters

and

an

uncooled R C A 931A

photomultiplier. The beam size is 6.5 square

arc

minutes.

A t a

site

not hampered by light pollution it is found that

the inherent dark sky

level

at

the zenith

(but

away

from

the galact i c plane and more than

two

hours after the end

of

twilight) is about V

=

21.8

to

22.0

mag sec

, ß =

22.8

to

23.0

mag e c

. t

solar

maximum

hese nherent

levels

are 0.4

to

0.6

mag

sec

brighter.

The

lunar sky

brightness

s he

arithmetic

difference

between

he bserved ky brightness

with

he

Moon

above

the horizon and the inherent dark sky

value,

given

the

phase

of

the solar yc l e .

n

Figure

8

of Krisciunas

1990

these

differences are plotted fo r

21

measurements,

where

he nominal dark

ky

levels

are aken

o be

he

yearly

averages

fo r the zenith. A brief summary of some

measurements

s also

given

by Walker 1987 based

on

measurements

at

Cerro

Tololo. Walker

s

measurements

made within 4 days

of

full

Moon were taken 90 degrees

or more away from the Moon.

The effect of the

Moon on the

sky brightness

at

some

position

is

a

function of

the

phase of

the

Moon,

the

zenith

distance

of the Moon, he zenith distance

of the ky

position,

he angle between the Moon and the sky posi-

tion,

and

he

atmospheric

extinction.

n

his

paper we

report additional moonlight

brightness observations from

the

2800-m

level of

Mauna

K e a .

In

addition, we

derive

a

simple model

that

g i v e s

he

relation

between

the

ca t -

tered moonlight and the f ive

variables

mentioned above.

This model results n an easy-to-use equation hat will

predict he moonlight brightness with an

accuracy

of

8% to

23%.

1033

©

Astronomical Society

of the

Pacific Provided

by

the

NASA Astrophysics

Data

System



1034

KRISCIUNAS AND

SCHAEFER

2. Observations

A dozen observations of

the

brightness

of the moonlit

sky

were

made

from

the

2800-m

level

of

Mauna

Kea

with

the

dentical

nstrumental

setup

as

used by Krisciunas

1990.

These dozen

observations

are

to

be joined with

the

21 observations

in Table

5

of

Krisciunas

1990.

In Table 1

we

gi ve the

new measurements

of

the

observed V-band

sky

brightness

on

moonlit

nights. The

first

s ix

columns

gi ve he U T date and UT time of the observation, he

azimuth and

zenith

distance of the

sky

position,

and the

azimuth and

zenith distance

of the Moon.

Note

that

al l

angles

have

been

rounded o

he nearest

degree

since

greater

accuracy

is

not needed. The seventh column g i v e s

p ,

which is

the

angular

distance

between

the Moon

and

the sky

position.

The eighth column gives the phase angle

(a )

of

the Moon,

which

is

defined as the angular distance

between

the Earth and the

Sun

as seen from the Moon.

(When

the

Moon

is

between full and

new,

<

0. )

The

phase

angles

werecalculated using

the

simpler

algorithm

of

Meeus

1980 e q.

31.4)),

which

gnores

he

noncon-

stancy

of the

e c l i p t i c )

atitude

of the Moon. The

ninth

column

g i v e s

he

observed

V

magnitude

of

the

ky

as

measured in magnitudes per square

arc

second.

The

last

column

in

Table

1

gi ve s

the

sky brightness of

the

moonlight alone as

measured

in

nanoLamberts

(nL) .

The

reason

fo r

converting

from a

logarithmic

unit

(magni-

tudes per square arc second) to a

linear

unit

(nanoLam-

berts) s

because a magnitude difference compares

he

moonlight to

the

background which is variable

from site to

s i t e . In addition, the

use

ofa linear unit allows fo r the easy

subtraction ofthe background sky brightness. The choice

of

the

nanoLambert

as

the

linear

unit

is

to

follow

the

lead

of

Knol l ,

Tousey

&

Hulburt 1946,

Weaver

1947,

Hecht

1947,

Garstang

1989,

and

Schaefer

1990a. The relation

T A B L E

V - b a n d

ky

rightness

on

ights

i t h

moonlight

3

U T k y posn.

o a n

B^on

( O b s . )

between

the

sky

brightness in nanoLamberts

{ B

) and in

magnitudes per square arc second (V )

is

D a t e im e

A Z

o

m a g / s e c

2

1 9 9 0

1 2

e b

:11

2 8

D e c

:05

1 9 9 1

4

an

4

Jan

4

an

4

an

4 an

4

Jan

4 an

4

Jan

2 1 Jan

2 1 Jan

8 : 0 7

8 : 1 5

8 : 2 3

8 : 3 0

8 : 3 4

8 : 4 2

8 : 4 9

8 : 5 4

6 : 2 6

6 : 3 3

2 8 0

8 9

1 2 5

63

55

8 2

1 4 3

9 9

7 9

1 6 5

2 6 3

1 1 3

9

1 8

1 7

3 8

3 0

5 2

3

61

68

1 1

4 9

2 4

9 4

3 0 0

8 5

8 5

8 6

8 7

8 7

8 7

8 8

8 8

2 6 5

2 6 6

7 9

4

8 2

8 0

7 8

7 7

7 6

7 4

7 2

7 1

6

64

8 8

2 2

68

4 5

53

2 4

69

1 6

9

69

1 4

8 5

- 3 0

4 9

- 4 8

- 4 8

- 4 8

- 4 9

- 4 9

- 4 9

- 4 9

- 4 9

1 1 9

1 1 9

1 9 . 8 1 3 0 8

1 8 . 9 0 7

3 3

2 0 . 1 4 1

1 9 . 6 0 5

1 9 . 6 8 4

1 8 . 6 1 5

1 9 . 9 0 2

1 8 . 4 5 6

1 8 . 2 2 8

1 9 . 7 7 1

2 0 . 5 7 2

2 1 . 0 3 1

2 0 6

3 8 0

3 5 4

1 0 8 7

2 8 1

1 2 5 5

1 5 5 7

3 2 9

7 1

3 0

Β =

34.08

exp(20.7233 0.92104

V )

(1)

Observations

m a d e at

8 0 0 - m

elevation of

M a u n a

K e a ,

H a w a i i .

as given in equation

(27)

of Garstang

1989.

The surface-

brightness equivalent o one enth magnitude star per

square

degree

(that

i s ,

the

S 10

unit)

equals

0.263

nL.

A n

alternative

expression,

equivalent

to

equation

(1), which

includes

the

definition ofthe

S 10

unit

and

the

s ca l e factor

to convert to nanoLamberts, is

B = 0.263 0

where

100)°

2

«

.51189 nd = 0. 0 .5

log(3600

2

)

«

27.78151.

The

dark nighttime sky brightness

(ß

0

)

as a

function

of

zenith

distance

(Z )

is

B

0

(Z)

=

610 ° 2)

where

X

=

(1

-0.96sin

2

Zr

0

·

5

3)

The extinction coefficient

is

k in units

of

magnitudes per

air mass . X is the optical

pathlength

along

a

line ofsight in

units

ofair

masses.

Note

that

the

formula

fo r X

in equation

(3)

is

appropriate

fo r

the

night glow. Equations (2)

and

(3)

are implifications

f quations 29) nd 30)

rom

Garstang 1989

that

are

adequate fo r

the

present

purposes.

The dark zenith sky

brightness (B

ze n

)

is taken as

the

yearly

average

reported

in

Krisciunas

1990 or

as

discussed be-

low.

The extinction coefficients were taken to equal the

value

measured fo r that night

or

the median

value

at

the

2800-m

level

of

Mauna K e a , namely 0.172

mag/air

mass ,

obtained

by

Krisciunas

1990.

The

ast

column

of

Table has he background ky

brightnesses

subtracted

out.

That

s ,

he value or

he

background

alone,

B

0

{

Z),

is

subtracted

from the

observed

brightness for the moonlight plus background, Β

,

leaving

the

brightness

of the

moonlight

alone,

B

moon

. When

the

moonlight is dim compared to the background,

the natu-

ral

variations in

the

background will

cause

large

relative

errors n

derived

values

of B

m

oon

. Therefore, we have

excluded nine

observations

for

which

the

moonlight

is

fainter

than

the

dark

sky

background) from

our

statistical

analys i s of the model compared to

actual

observations.

The

observations

of

1991

January

4

(UT)

are

particu-

larly

useful,

since a

concerted

effort was undertaken to

narrow

down

the

functionality of the

components

of the

lunar sky-brightness ffec t . This nvolved obtaining

a

dark-sky

value on

that

night

(V

=

21.444 mag

sec

)

an

hour before he Moon ose. This value compares

well

with the

1990

average

(the

year

of sunspot maximum) of

V

=

21.377

±

0.095 mag sec

,

based on measurements

of

5

nights.

For

the two observations in Table 1 from 1990

we

shall adopt he 1990

yearly average

as he

nominal

dark-sky value. We s h a l l adopt the

actual

dark-sky

value

©


of the

Pacific Provided

by

the

NASA Astrophysics

Data

System



MOONLIGHT-BRIGHTNESS

MODEL

for

1991

January

4

fo r

reference

in

calculating the

lunar

e f fe c t on that night. On this night we also measured the

V-band extinction to

be

A :

=

0.116 ± 0.019 mag/air mass ,

based on 8 measurements

of

5

stars

over a range

of

1. 5

air

masse s .

or

the

1991

January

21

data we shall adopt a

dark-sky

value of

V

= 1.388 mag

sec

2

,

qual

o

he

average

of

the

s ix

measurements

of

the

previous

year.

3. Atmospheric

Scattering

The atmospheric

scattering

of moonlight is

a

complex

problem. The

sky brightness will depend on

the

extinc-

tion coefficient,

he

magnitude

of the Moon, he

zenith

distance

of

the Moon, the angular separation between the

Moon

and the

sky

position, and the

zenith

distance

of the

sky position. n

hese

next

wo sections we describe

a

simple

scattering model (elaborated by one

of

us,

B .

E.

S .)

so

as

to

establish functional forms for the variation ofeach

parameter that are fairly

accurate.

When looking out

along

a

line of

sight,

an observer

will

detect

scattered

light

from

the

Moon. Let

the field

ofview

be

a

circle

with an angular radius of φ , o

that

the

s ca t -

tered

light

must

come

from

nside

a

narrow

cone.

The

volume of

the

cone

can be broken

up

nto differential

volumes

dV

=

π (R φ )

2

dR 4)

where R

is

the

distance from

the

observer

to

the

volume.

Thi s volume will have some density of scatterers, D,

which will

be

proportional to

the volume

extinction

coeffi-

cient b

D

=

cb 5)

where

c

i s

some

constant.

The

number

of

scatterers

in the

differential

volume, dN, will be

dN

=

DdV

=

7ïcbR

2

^

2

dR 6)

The

brightness

of

scattered light

from

each

volume will be

proportional to d N .

The differential

volume

will be

illuminated

by

moon-

light dimmed by the atmosphere.

If

7 * is the illuminance

of

the Moon outside the atmosphere and I

is

the

illumi-

nance ofthe Moon inside the

atmosphere,

then

these

two

quantities will be

related

as

I

=

l0-

0

Akx

™

,

7)

where k

is

the

extinction

coefficient

and X

m

is

the

optical

pathlength

to

the Moon

as measured

in

air

masses s e e

be low) .

The illuminance of

the Moon (in footcandles) can

be

related

to

the

V

magnitude

of the

Moon

(m)

as

j*

_

2Q -

0

-

4

(

m

+

16

·

57

g)

( s e e

eq.

16)

of Schaefer

1990a).

The

magnitude

of

the

Moon will

be

a function of the phase

angle,

α in degrees,

such

that

1035

m

=

-12.73

+ 0.026

α

+ 4

Χ

Π

9

4

9)

( s e e

page

14 4

of

Allen 1976).

The brightness of

scattered

moonlight

will

be proportional to

I .

Equation (9)

g i v e s

m

= —12.73 fo r full Moon,

in agreement with the results

of

Lane

&

Irvine

1973

extrapolated

to α

=

0.

However, this

ignores

the

opposition

e f fe c t . For al < 7° the bright-

ness of the Moon

deviates from

this relation s e e ,

e.g.,

Whitaker 1969).

When the Moon

is

exactly full

it

is about

35 %

brighter

than

he extrapolation

would

predict,

as -

suming,

of

course,

that

it

is not undergoing a penumbral

or

umbral eclipse.

The

luminous

intensity

of

the

differential volume,

dL,

will be

proportional

o

he

number

of

scatterers n he

volume, the

illuminance

of

the Moon,

and

the

scattering

function.

The

cattering function,

(p), escribes he

intensity

of

scattered

light

as a

function

of

the scattering

angle

p . Therefore,

dL = If{p)dN

10)

In

this

equation

and

below,

the

various

constants

of

pro-

portionality

will be absorbed

into

the scattering

function.

The ight

raveling

from

he

differential volume o he

observer

will

be

attenuated not only

by

the

atmosphere

but

also

by he nverse

square aw

fo r

he

ntensity

of

light.

The

perceived surface

brightness from

the

differen-

tial volume,

dB,

will

be

the

perceived

brightness divided

by

the

subtended sol id angle,

so that

d B

= dLe~

7

/ R

2

=

If{p)be-

J

dR 11)

Here,

corresponds o he

optical

depth between he

observer

and

the

volume.

The total

apparent

surface

brightness

can

be

found

by

integrating along the entire line of sight.

For

the simple

model

that

the atmosphere is

uniform

up to some height

H, then

the

optical

depth will be

simply

b R .

The integral

must

run

rom zero

o

he distance of the op of the

atmosphere that is Hsec(Z)). Thus,

he

surface bright-

ness

will be

B

= If{p)[l-

e

bHsec{z)

] 12)

For

such an atmospheric model, the extinction coefficient

can be related to the

volume

extinction

coefficient as

10 -

O

Ak

=

e-

b

H

13)

The

air

mass is often evaluated as

the

secant

of

the

zenith

distance, and

this

is

accurate

fo r

any direction

not

near the

horizon. S o, the secant

of

the

zenith

distance in equation

(12) will be better approximated as the

air

mass along the

line

of

sight,

X(Z).

Many

expressions

fo r

air

mass

that

are

valid

lo w

on

the

horizon

have

been proposed Hardie 1962; Rozenberg

1966;

Garstang

1989;

Schaefer

1989).

Ideally,

each

extinc-

tion

component must

be

handled

separately

since

they al l

have

different

height

distributions

cf .

Schaefer

1990b).

©


of the

Pacific Provided

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NASA Astrophysics

Data

System



1036

KRISCIUNAS

AND

SCHAEFER

From

these correct

calculations,

we

have found

that equa-

tion

III.

3, 17

from Rozenberg 1966 is the most reasonable

of

the

simple

formulae

fo r zenith

distances

al l

the

way

to

the

horizon.

Therefore,

X(Z) =

[cos(Z) +

0.025

éT

11

1

14)

In

this

formula

the

air

mass

goes

as

s e c ( Z )

when

fa r

from

the

horizon but i s limited to 40 air masses

on

the horizon.

T h i s

is the

best

formula

to use

fo r correcting

the

bright-

ness ofan object directly observed (e.g., the Moon i t s e l f ) .

However,

fo r

scattered

light

from

the

Moon we

find

that

air

masses based on

equation 3)

give

he better i t of

observed unar brightness

s .

model

brightness,

or

a

very large range

of B

m

oon

.

We might

cal l

equation (3) the

scattering

air

mass and equation

(14)

the

extinction

air

mass ,

and it

is clear from

our

data

that

equation

(14) leads

to a gross

underestimate

ofthe lunar sky-brightness effect

when the Moon is low on the horizon.

From

equations (7),

12),

and

(13),

he

model

surface

brightness will be

B

m

oon

= / p )

rlO-o· [1-10-°«] 15)

The

evaluation

of/(p)

will

be

given

in the

next

section.

T h i s

last

equation embodies a number ofcomplexities,

the

details

of

which

have

not been fu l ly elaborated.

The

most obvious are:

(1)

The volume extinction coefficient b will

be

a

sum

of

two exponential

functions

ofheight.

(2)

The extinction in equation (7) will

be

a function of

the height of

the

volume.

(3)

The e f fe c t s of a spherical Earth and refraction be-

come

important

near

the

horizon.

Better

calculations

can

and

have

been made (cf. Rozenberg 1966 and references

therein). However,

the results are of much greater com-

plexity.

(4)

The

scattered

light

h as

a

small

but

significant

frac-

tion

of multiply

scattered

light, whereas

he

use of the

scattering

function

n equation

12) mplicitly assumes

single scattering.

Equation (15) h as

also

ignored several

e f fe c t s ,

some of

them quite

minor:

(1)

The Moon has an asymmetric

distribution

of maria

so

hat

he Moon

and,

hence, he scattered moonlight

should

be

slightly

brighter before

u l l

Moon

han

after

(Stebbins

& Brown

1907;

Russell

1916; Rougier 1934).

However, this effect is very small and Lane

&

Irvine

1973

were unable to detect it .

(2) or

la l

<

7° the lunar opposition effect should be

taken nto account,

n

which

case

he

derived

value

of

B m o o n must be multiplied by a

factor

in between 1.00 and

1.35.

This

factor

only comes

into

play within

a day

of full

Moon.

(3)

Refraction

will change the altitudes and separation

of the Moon

and

the

sky

position

from

those

used

in

this

model.

However, even on the horizon, he altitude will

only

shift by

half a

degree

at

most and will

change

he

predicted brightness by only small amounts.

(4) The position of the

Moon

as used

in

this paper is

that

of the

Moon's

center,

whereas

t

would be more

accurate

to use

the position of the light centroid. How-

ever,

under

extreme

conditions,

he

difference

in

posi-

tion

is

a quarter

of

a degree

at

most,

o

that

this

effect

is

negligible.

(5)

The

Moon

is

sometimes closer

and,

hence, arger

and brighter than at other

times fo r the

same

phase angle.

The

apparent diameter

ofthe

Moon

will

vary

by

11.6%

on

average from

apogee to

perigee, so

that

the Moon's

mag-

nitude, as derived

by

equation (9), may be in error by as

much

as

±

0.12 magnitude.

A n improved

model

would

account

fo r al l of

the

above

e f fe c t s and more. Y e t the real

atmosphere

is never

as

well

behaved

as

in

any

model

ever

calculated.

In particular,

the concentration and

distribution

of the atmospheric

aerosols are

highly variable

with ime and

ocation

c f.

F i g .

56

ofRozenberg 1966). With such large variations, it

is

useless

to

refine

a model intended fo r general

applica-

tion.

The

real

justification

fo r

the

simplicity

of

the

model in

equation

(15)

is

that

it

reproduces

the observations with

remarkable accuracy. We will show

below that the model

reproduces

he Mauna

Kea data

(from

scattered

moon-

l i gh t ) with an

rms

uncertainty

of

23 % and the data (from

scattered

sunlight) from two Russian sites

(Pyaskovskaya-

Fesenkova 1957) to an

accuracy

of

8% to 11%.

This accu-

racy is

achieved

despite

the

hundredfold

variation

in

the

observed brightness

of

scattered

moonlight.

4. Scattering Function

Equation (15) g i v e s a simple formula fo r the moonlight

brightness,

where

the

functional

forms

for

I

fc,

X

m

,

and

X

are derived from theory. However, the scattering

func-

tion,

(p), as

not

yet

been evaluated. The scattering

function,

/(p),

s

proportional

to

the

fraction

of incident

light

scattered

nto

a unit

solid angle

with

a

scattering

angle

. The scattering

angle,

,

will

be

equal o he

angular

separation between

the

Moon

and the

sky

posi-

tion for single scattering.

The

scattering function

will

be

composed

of

additive

terms

associated with

the

two

types

of

scattering

in

our

atmosphere. The

first

ype

s

Rayleigh scattering from

atmospheric gase s , which will

contribute/

R

(p). The s e c -

ond

type

is Mie scattering by atmospheric aerosols, which

will contribute/

M

(

p). The two

terms

will add ( s e e eq. II.

2,

38 ofRozenberg 1966),

so that

/(PH/r(P)+/M(P)

16)

Remember that the scattering

functions

have

absorbed

constant

factors

relating

to unit

conversions and

normal-

iza t ions .

©


of the

Pacific Provided

by

the

NASA Astrophysics

Data

System




MODEL

1037

For Rayleigh scattering

from atmospheric gases, he

contribution

to the

scattering

function will

be

/r(p) =

C

R

[1.06

+

c os

2

(p)]

(17)

(Rozenberg

1966).

C

R

is

a

proportionality constant yet

to

be

determined.

For

Mie

scattering

of

aerosols,

he

contribution

to

the

scattering function will be /m(p)·

Explicit

equations

or

(

p) have been compiled in the

treatise

of van de

Hülst

1957 fo r

many

particle

shapes,

i z e s , and

optical

proper-

ties. Unfortunately,

hese

idealized

cases have no utility

for

the

real

atmosphere where

the

aerosol

size

distribu-

tion is

broad, and

the

particle

shape

and

composition

are

widely

variable. In principle,

(

p ) can be found by inte-

grating

the single particle

functions

over the s ize ,

shape,

and optical

properties

of the

aerosols,

but

in practice

this

knowledge

s never

known

with

sufficient

accuracy

o

justify

the

integration.

Henyey

&

Greenstein

1941

offer

a

scattering function that

is

merely a convenient

functional

form

with

no

basis

in

data. Thus,

he

only

alternative

is

that

of

empirical measurement.

Therefore, n practice, we will use

he

observations

from

the previous section

to

define

the

scattering

func-

t ion . T h i s can

be done

because B

moon

,

I

k, X

m

, and X

are

known

fo r al l

observations

from

Mauna K e a ,

o

that

equation

(15)

can be

solved

for/(p).

Figure 1 s h o w s a plot

of

the

scattering function

as

a

function

of the Moon/sky

separation.

Pyaskovskaya-Fesenkova 1957 presents smoothed data

for

the daytime ky

brightness

hat

can also

be used to

evaluate

/(p).

The

eason s

hat scattered

sunlight s

identical in everything but intensity

with scattered moon-

light.

The

difference

in

intensity

can

be easily accounted

6.4

Observed cattering

unction

D )

O

6.2

6

5.8

5.6

5.4

5.2

30

0

0

20

Scattering Angle degrees)

150

Fig.

-The

scattering

function, /(p), a s d ed u c ed

from

th e moonlight

observations repor ted in th i s paper and in Krisciunas 1 9 9 0 . That i s , for

th e

observations,

all

th e

quanti ties

in

equation

( 1 5 ) are known

s o

that

th e

scattering

may

be

solved

f o r .

Th e scat ter ing function i s a function of th e

scattering angle, p . Th e scat ter ing

function from equation

( 2 1 )

i s drawn

a s th e

smooth

curve .

fo r

by

using

the

Sun's

magnitude (m

su n

= —26.74) instead

of

the Moon's magnitude in equation (8). The Russian data

are

fo r the

whole

sky

from

sites

with

A : =

0.15 and k

=

0.24

and fo r solar zenith

distances

of0, 30 , 60 , and 8 0 degrees.

We

have extracted

many

representative

points, alcu-

lated/(p) from

equation

(15),

and plotted

the results

n

Figure

2.

The

first

item to

notice

is that the scattering

function

is

identical

to

within

he

scatter) or

the

hree

sites

with

extinction

coefficients

of

0.15, 0.172, and 0.24.

Second,

there is

no systematic difference between solar and

lunar

data.

The

third

and

fourth

items

are

that the data

with the

zenith

distance of

the

source

nd of

the

ky

position

greater

than 80 degrees

are

not

systematically

different

from

the

average.

These

four items

show

that

the

func-

tional

dependence on k,

I

X

m

,

and X

is correct.

This is

the empirical

justification

that

the

approximations

leading

to equation (15) are good.

The

Mie

scattering

by

aerosols

is

highly

forward s ca t -

tering.

Thus,

he

Rayleigh

scattering

will

dominate

or

scattering angles greater

than 90 degrees.

Therefore,

the

empirical measures of/(p) fo r

large

angles can be used to

measure C

R

.

The value

will

be

he average

over

data

values with a scattering angle

of

greater

than 90 degrees)

of

the empirical/(p)

divided by 1.06

+

c os

2

(p).

We find

a

value

of2.27

X

10

5

forC

R

.

The Rayleigh

component

of/(p) can then

be

subtracted

out,

leaving

only/

M

(p

).

For scattering

angles from 10

to 80

degrees, the empirical/

M

(

p) is roughly

a

straight line on

a

Observed

cattering unction

30

0

0

20

Scattering

Angle

degrees)

150

Fig.

2-The

cattering

function, /(p),

s

ed u c ed

from

th e

unlight

observations repor ted

in

Pyaskovskaya-Fesenkova

1 9 5 7 . A s

in Figure 1 ,

a l l

th e

quanti ties

n equation

1 5 ) are known

o

that

th e scattering

function

may

be

de te rmined .

Th e scattering

function

i s

plotted

versus

th e

cattering

angle,

, or

si te s

with extinction coefficients

of 0.15

(crosses)

and 0.24

( d o t s ) .

Th e scattering function from equation ( 2 1 ) i s

drawn

a s

th e

smooth

curve . Note that

th e

scat ter

about

th e model

i s

small and that

the r e

i s no

significant difference

be tween

th e

two sites, a s

th is

i s th e

empirical

justification that

th e functional

form

of

equation

( 1 5 )

i s

valid.

©


of the

Pacific Provided

by

the

NASA Astrophysics

Data

System



1038 KRISCIUNAS

AND

SCHAEFER

log-linear

plot.

From

this

plot

we

will

adopt a

formula

fo r

/M(p)

=

10

ei5

~

p/40

18 )

N ow

the

total scattering function

can

be

found as

a

func-

tion of ρ from

equations 16),

17), and

18),

and

is dis

played

as

a

curve in

Figures

1

and 2.

For

angular

separations

less

than

10

degrees,

he

data

in K i n g 1971 show

that

equation

(18)

is

not

va l i d .

Schaefer

1991 h as

shown that

/m(p)

=

6.2 X 10

7

p~

(19)

The

cattering

angle

s measured n degrees or both

equations

(18) and

(19).

This

small

angle

scattering forms

the aureole.

The reader may wish to compare

our

scattering func-

tion,

sing

appropriate

cal ing,

o hat

of

McClatchey

et al .

1978

(Fig .

26

in Section 14) and

that

of Rozenberg

1966 (Fig .

72).

5.

Discussion

The ina l quation or

moon

s

erived rom qua-

tions

3),

8) , 9),

nd

15)-(19).

or

Moon/sky

separa-

tions greater than 10 degrees,

hese

results can be

sum-

marized

as

j*

=

20

- 0 . 4 ( 3 . 8 4 +

0 . 0 2 6

lal+4

X

1 0 - 9

a 4 )

/(p)

Ι Ο

536

[1.06

+ c os

2

(p)] +

10

615

-

p

/40

X(Z)

=

(1-0.96

s in

2

Z )~

0

·

5

oon riO il lO·

4

(20)

(21)

(3)

(15)

The

required nput or his general formulation

s he

lunar phase

angle

a,

he

Moon/sky eparation

,

he

extinction coefficient k, he

zenith

distance of

the Moon

Z

m

,

and

the

zenith

distance of the

sky

position

Z. These

equations are not

as

formidable

as

hey might ook.

pocket calculator

can evaluate B

moon

in

about

a minute

and

only

a

few

lines

of

code

are needed

in a computer pro-

gram.

The change in the V-band sky brightness caused by the

moonlight

will be

A V

= -2.5 log [ ( ß m o o n

+

B

0

{Z))/B

0

{Z)]

(22)

A s

a guide, we

have

tabulated

typical

values

of B

m

oon

(i n

n L )

and

A V

(i n

magnitudes

per

square

arc

second)

fo r

a

range of α and ρ ( s e e

Table

2) . This table was constructed

with

he

assumption

hat k

=

0.172 mag/air

mass the

median V-band extinction at the 2800-m level of Mauna

K e a ) ,

and

that

V

ze n

=

21.587

mag

sec

(the

mean

dark-sky

value

at

he

same

site,

quivalent o B

ze n

=

9.0

n L ) .

Strictly speaking,

he values

of B

moon

and

A V

in Table

2

only

apply

o a

site

with he

same

median

extinction

coefficient and mean dark-sky

brightness

as

the 2800-m

level

of

Mauna

Kea (e.g., McDonald

Observatory).

Given the two extinction

terms

in

equation

(15),

a

table

Lunar ky

r igh t n e s s

ffect rom odel

a

Phase

ngular

i s tance

between

angle o o n

and sk y

posi t ion

p )

( a )

°

0°

0°

0°

20°

30°

60°

7 2 1 6

1 6 0

3 0

3 7

1 8

{-4.48)

-2.87)

-2.22)

-1.93)

-2.16)

3 3 6 4

5 4 1 2 4 7

2 0 4

3 8 1

1 2 0

(-3.67)

-2.13)

-1.54) -1.30) -1.49)

1 3 5 1 1 7

9

2

5 3

(-2.73)

-1.35)

-0.88)

-0.71)

-0.85)

3 9 1

3

9

4

4

(-1.58)

-0.58)

-0.34)

-0.26)

-0.32)

a

Values

of

moon/

measured

n nanoLamberts. or unar

zeni th angle

of

60°

and

V-band

e x t i n c t i on

of 0.172

mag/

ainnass.

or this able is

measured

along

h e

grea t

circle on

th e ky

pass ing

hrough

t h e m o o n

and th e

zen i t h .

Therefore , h e olumn

=

60°

orresponds

o h e zen i t h .

Values

n

pa r e n t h e s e s

ar e

ν

in

mag/sec

2

,

using

V

=

21.587

mag/sec

2

B

zen

=

79.0 L ) s h e nominal zeni th ky

br igh tne s s ,

nd

sca l ing

h e zeni th ky

br igh tne s s

o

h e

nominal

alue

at h e

zeni th

angle

orresponding

o

.

comparable to Table

2

fo r the

Mauna Kea summit (with

median V-band

extinction

of

0.113

mag/air

mass;

Krisciu-

nas 990)

g i v e s B

moon

values

which are al l ower. The

extreme

example

would

be fo r a satellite

in

Earth orbit,

fo r

which

k

=

0 and B

m

oon

always equals 0 (i.e., the sky is

black) . For

a

dark

sky

site

with k

=

0.30

the sky-bright-

ness

effect due to the Moon is larger than those

values

tabulated

in

Table 2.

Because the

moonlight

contribution

comes

from scattering

of

light,

where

one

s observing

through

more

atmosphere

(i.e.,

sea level v s . on

a

moun-

taintop),

there will be

more

scattering.

For

Palomar and Mount

Wilson,

which have

V-band

extinction

omparable o

hat t he 800-m

evel

f

Mauna K e a ,

but

different dark-sky

brightnesses

(due to

art i f i c ia l light), one

can

use the

values

ofB

m

oon

from

Table

2 and equations (1),

(2),

(3), and (22), with an

appropriate

value

of

V

ze n

for equation (1),

o calculate the sky-bright-

ness

effect

( A V )

in

mag

sec

.

One of the principal

uncertainties

n using equation

(22)

on any

given

night

comes

from

the

uncertainty

ofthe

adopted

dark-sky

zenith brightness.

Even

f it s mea-

sured,

one

does

not

know

the exact contribution of

faint

stars in the beam or what the effect is of the solar wind on

the

air

glow

on

that

specific

day .

See

Krisciunas

1990

fo r

further

discussion.

The accuracy

of

our general

formulation

can

be

deter-

mined rom he catter of he bserved rightnesses

about he predictions. n

Figure

3

we plot B

moon

ob -

©


of the

Pacific Provided

by

the

NASA Astrophysics

Data

System




MODEL

1039

M o d e l Brigh tnes s

n L )

Fig.

3-Obse rved

values

of

s k y brightness

d ue to

scat tered moonlight

( i n nanoLamberts) v s . model values from equation ( 1 5 ) .

Open

circles:

data for which

B

moon

<

B

Q

{ Z ) . Squares:

data of

1 9 9 1

January 4 .

Large

dots:

a l l

other

data. A

line of

slope

1 i s indicated.

served)

v s .

B

moon

(mode l )

fo r the

data

of

Krisciunas

1990

and

from

Table

1 of

this

paper. If we take the

ratios

[ B m o o n

(observed)

ß

moon

(mode l) ] /B

moon

(observed)

we find

an

rms

variation ofthis ratio

of

23 %

fo r the

Mauna

K e a data (with B

moon

> B

0

{Z)),

8%

fo r the k =

0.15

Russian

site,

and

11 %

for the

k

=

0.24

Russian site.

Hence, we

conclude

that

our

formula

has an accuracy of

8%

to 23 % in

the prediction

of

the sky brightness from

moonlight.

The

accuracy will

be

poorer

when

the Moon or

sky

position is

c los e

o

he

horizon

o

hat

small

uncertainties

n

he

extinction

coefficient

will

be

magnified.

To the

best

ofour

knowledge,

no

formula

fo r

moonlight

brightness has

ever

appeared

in the

literature.

Our

for-

mula

h as

the

advantages

of having

the

correct

functional

dependencies, et being

easy

o

use

nd

accurate o

better than 23%. A s

such, he equations can

be

used to

optimize exposure times

fo r photography

and

CCD

im-

age s , o predict limiting magnitudes

fo r visual

observa-

tions, as

well

as

to predict

the

visibility

of

stars,

planets,

or

comets

near

the Sun

or

Moon fo r historical purposes.

We hank Ewen Whitaker or

useful

comments and

references to

other

work.

R EF ER EN C ES

Allen,

C .

W . 1 9 7 6 ,

Astrophysical

Quanti ties

(London,

Athlone)

Garstang,

R . H. 1 9 8 9 , PASP, 1 0 1 , 306

Hardie , R . H. 1 9 6 2 , in Stars and Stellar

Systems,

V o l . 2 ,

Astronomical

Techniques , d .

W .

. Hi l tne r

(Chicago, Univ.

Chicago Press),

p.

1 7 8

H e c h t , S . 1 9 4 7 , J . Opt. S o c . Am.,

3 7 ,

59

Henyey,

L .

G .,

Greens te in , J .

L .

1 9 4 1 , ApJ, 9 3 , 7 0

King,

I .

R . 1 9 7 1 ,

PASP,

8 3 ,

1 9 9

Knoll,

Η

. .

, Tousey,

R ., Hulbur t , E .

O.

9 4 7 , J . Opt. S o c .

Am.,

3 6 ,

480

Krisciunas, K . 1 9 9 0 , PASP, 1 0 2 , 1 0 5 2

Lane,

A .

P., Irvine, W . M . 1 9 7 3 , A J ,

7 8 ,

267

McClatchey,

R . ., Fenn, R . W ., Selby, J . Ε . .,

V o l z ,

F.

E ., &Garing,

J .

S .

1 9 7 8 ,

in

Handbook

of Optics ,

e d .

W .

G .

Dri scol l&W .

Vaughan

(New Y o r k ,

McGraw-Hill),

Chap t . 1 4

M

ecus , . 9 8 0 ,

Astronomical

Formulae for

Calculators

(Hove, Bel-

gium, Volkssterrenwacht Urania)

Pyaskovskaya-Fesenkova, . V . 9 5 7 , Investigation of th e Scattering

of

Light in

th e

Earth 's

Atmosphere

(Moscow, U.S .S .R. Acad .

S e i .

Press)

Rougier, G . 1 9 3 4 , L'Astronomie,

4 3 ,

2 2 0 ;

43, 2 8 1

Rozenberg, G . V . 1 9 6 6 , Twilight (New

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