Two-parameter Magnitude System for Small Bodies

Kuliah AS8140 & AS3141(Fisika) Benda Kecil [dalam] Tata Surya

Prodi Astronomi 2006/2007

Observing Plane

• The plane Sun-Object-Observer is the plane of light scattering of the radiating reaching us from the Sun via the object.

• It is a symmetry-breaking plane, and because of this, makes the light from the object polarized

Karttunen et al. 1987

Muinonen et al. 2002

Photometric & Polarimetric

Phase Effects

Albedo & Phase Function (1)

phase angle, = (2 ) scattering angle, (solar) elongation

Sun

de Pater & Lissauer 2001

Albedo & Phase Function (2)

Karttunen et al. 1987

Albedo & Phase Function (3)

A solar system body radiated by the Sun: 4

2)1()1(

4

1TA

r

FA IR

SunV

FSun = 1.36103 W m-2 is the solar constant at 1 AU,r is heliocentric distance in AU,For a black body albedo AV = AIR = 0

Assuming isotropic thermal emission (not quite true!)

when observing a target …

Data: Vobs V(r ;, ), dVobs

Brightness at unique distances [r,] and phase angle ()

Vobs should be a magnitude at base-level lightcurve

Sometimes V(r ;, ) V() Vobs()

Data set: exp. per dayTime: ti ; i = 1,…,n

Magnitude at base-level: V(ri ;i, i) ; i = 1,…,n

Magnitude error: i ; i = 1,…,n

The Two Parameters: H & G (1)

Reduced magnitude ) log( 5) ,; (),1( rrVV obs

Vobs is observed magnitude

Absolute magnitude is the magnitude of a body if it is at a distance 1 AU from Earth and Sun at phase angle = 0

HHV V )0 ,1( Standard visual

Two-parameter (HG) magnitude system:

)()()1( log 5.2)(),1( 21 GGHHV G is the slope parameter the gradient of the phase curve

Bowell et al. 1989 (Asteroids II)

The Two Parameters: H & G (2)

Phase function l(); for 0 120, 0 G 1

2tan 56.90exp)(

2 ,1 );( )(1)()()(

2

W

lWW lLlSl

lB

llL

llS

A

C

2 tan exp)(

sin 754.0sin 341.1119.0

sin1)(

2

238.0 ,986.0

,218.1 ,631.0

,862.1 ,332.3

21

21

21

CC

BB

AA

Simpler, more symmetric, but slightly less accurate expression

2 ,1 ;2

tan exp)(

lA

lB

ll

22.1 ,63.0

,87.1 ,33.3

21

21

BB

AA

Bowell et al. 1989 (Asteroids II)

Obtaining H & G (1)

Observation ( reduced mag) data Vi(i) and errors i)()(10 2211

)(4.0ii

V aaii

Least-squares solution

2122211

2

2

)(4.0

2 ,1 ;,,1 ; )(

2 ,1, ;,,1 ;)( )(

,,1 ;10

hhhD

jniI

g

kjnih

niI

i

iijj

i

ikijjk

Vi

ii

Dghgha

Dghgha

/)(

/)(

1122112

2121221

Buktikan…

Bowell et al. 1989 (Asteroids II)

Obtaining H & G (2)

a1 and a2 are of order 10-0.4H, which may be computationally inconvenient. If so, they may be scaled to order unity by setting

)()( log 5.2)( 2211 aamH m is one of the reduced magnitude Vi(i) (for instance at smallest )

21

2

21 log 5.2

aa

aG

aaH

Thus,

albedo high for 400

albedo moderate for 25.0

albedolow for 15.0

.

G

Bowell et al. 1989 (Asteroids II)

Phase integral q = 0.290 + 0.684 G

H & G Error Analysis

Magnitude residuals )()()1( log 5.2)(

)()(

21 iii

iiii

GGm

mHVr

m(i) is the calculated magnitude drop from zero phase angle

Then,

;/2

1 ;

/1/

1

;/1

1 ;

/1

/

2222

022

2

22

)(2

2

0 0

iiiii

iH

i

ii

rn

s

20

20

2

2

)(0

)()( ),1()(

; 0615.0 1132.00673.0

; ; )(0

HVH

GGG

ssH H

Bowell et al. 1989 (Asteroids II)

Drawbacks…

The H,G-magnitude system fails to fit the narrow opposition effects of E-class asteroids (Harris et al. 1986)

It shows poor fits to the phase curves of certain dark asteroids (e.g., Piiroen et al. 1994, Shevchenko et al. 1996)

Hapke’s photometric model (5 parameters) has photometric fits as good as the H,G-magnitude system (Verbiscer & Veverka 1995)

Lightcurve Amplitude – Phase Angle Relation

A(0°) and A() are, respectively, the lightcurve amplitude at zero phase angle and that at a phase angle

Amplitude at zero phase angle is smaller than that at a phase angle

1

)()0(

AA

type-M for deg 0130

type-C for deg 015.0

type-S for deg 030.0

1-

1-

-1

.

Zappala et al. (1990)

a b

c

)0( 4.0 log10A

b

a

Examples

Asteroid 1990 FV

11.5

11.7

11.9

12.1

12.3

12.5

12.70 4 8 12 16 20

Phase Angle (deg)

Redu

ced V

-M

agnitude

Karin cluster interloper Cometary asteroid