Photometry, pole orientation and shape parameters of the minor planets 624 Hektor and 43 Ariadne

-'to7

:'

I I I I i

Photometrv. Pole Orientation

a11d

Shape

Parameters

of

the-ÉIinor

planets

624

Hektor

and 43

Ariadne:

A. Detar, P. Colletter, O. Haiuautl'2, A. Ibra,him'Denisr, A. Pospieszalska-Surdejr, P' Schilsr,

H.J. Sôobef & J. Surdejr.

I

_{hutitut d,Astrcphysique} dc Liègc, Atenuc dc Coink, 5, 8-1000 Liège, Belgium

z

_{Eu*p"on Southern Obsentatory,} LÂ Sillo, Cæiils 19001, Santiago, Chile

s

_hrstittt

_lûr

_{Aemnomie, Unioersitiitsplatz,} 5, A'8010 Gmz, Austria

1.

Introduction

Combining Dew measgrements (described elsewherelll) with preliously published photometric data for the minor plauets 624 Hektor and 43 Ariadae, we have applied the Rcvisited

Amplitude/Magnitude-Aspect relation (RAMA), the Free Albedo Map method (FAM), two iudependent variants of the Free Shape method (i'S)

."a

iU.

Photometric Astrometry

pA)

in order to derive their pole orientation

(À";po). These meihods are shortly described and their results a,re compared.

2.

The

methods

The nRevisited Amplitude/Maguitude-Aspect relation"l2] (RAMA) is derived from the uhistorical" Amplitude-Aspect dcbtioo. It

L**o

that the asteroid shape is a triarcial ellipsoid with no albedo

variations. The shadowing effects are neglected, and the diffusion law adopted is that of Bow-ell and Lumme[3].

It

is eIsy

to

show[2] that the expressiou of the lightcurve can be reduced to F2

₌

B.(cos, V)

+ C,

wherà

F

is the total observed flux and

!I

the rotational phase-

B

and C

are functions of the unkuowus:

\,pe

(pole coordiuates), a,b,c (ellipsoid semi-a:<es), _{Q (multiple}

scattering coefrcientl3]). A least square nt

ot.""n

[ghtcurve allows one to determine the numerical .ralues

oiB

and C, pioviding two equations for the uaknowns.

lo

practice' the 6 unknowns are

obtained by scanning the celÀtial sphlre (wiih a typical 1-10 degrees step in b-oth Ào aud ps), while the four others are fitted for each trial pole usiug a least squa,les method' The Xz of these fits a're theo plotted as a contour map (cf Fig. 1).

the

besi pole corresponds to the lowest point.

this

kind of

plot i,

,""y

useful to ,ho* .""ooJary minima, which would uot appear in ao automatic minimization' Russeill{l showed, in a very geoeral way that the lightcurves of any convex body, with any albedo distribution, cau be

"*."tiyi"produceà Uy a sphercwith tbe origiual axis orieutation,

covered with

a suitable albedo distriburion. This idea is u""d in the Free Albedo Map methodtsl'tGl (FAM): the asteroid is modeled by a sphere covered with many facets which albedoes are adjusted to

fit

at best the observations. As the Russell law is valid only

if

the sphere has the same orientation of the

rotation a:<is as the asteroid, lower is the X2, better is the determination of the pole orientation' A x2 map is generated by scauning the celestial sphere and fitting an albedo distribution for each

.on.id"rà

tÀ

pole. Tuà dueao distributiou obtained for the best poles has to be considered very

carefully: d/

it

is not unique (i.e. addiag any combination of odd spherical harmonics will not change

the resultiug lightcurves, roà the X2

;in

remaiu exactly the sameFl),

tt

J!"

real shape of the as-teroid is proU"tty not siherical at

all'

So, _{.the pseudoalbedo} of the model is related to the ratio between real albedo

"oi

th"

local curvatuteltl. Àaditional pa,rarneters are simultaneously obtained:

a normalization factor (corresponding to the radius of the sphere), and the multiple scatteriug factor'

In the "Free Shape metho6"tsl (FS), the asteroid is modeled by an irregular polyhedron with a

_30t_

center, and the multiple scattering coefficient.

A

constraint has

to

be applied

in

order to ensure the convexity

of the

rh"j".

Two completely indepeudent programs have been developed, one using a

minimization of the Surface/Volume ratio, the other maximizing the radii entropy. Again, the shape model corresponding to the best pole orientation is not unique

but

this does

not

affect the resulting pole orientation.

While

the

three previous methods use the photometric information

of

the lightcurves,

the

uEpoch

Method", traditionally called "Photometric Astrometry" (PA)14, only takes into account the

chrono-metric

information.

in"

relative motion and positions between

the

asteroid and the Earth lead to

slight variations

in

the observed rotational period of the asteroid. As these variations are involving the geometry of the rotation,

it

is possible to derive the pole position. This method has the advantage

to make no assumption on the asteroid (shape nor albedo),

but

it

is very demanding concerning the

quality of the data.

3. The

Asteroids

We have applied the above methods

to

previously published and newly recorded lightcurves of the asteroids 624 Hektor and 43 Ariadne. Table

I

lists the corresponding observational parameters. Here under, Figures 1 to 4 illustrate for each asteroid the 12 maps obtained by each method. Note

that

the X2 scaling is not normalized

for

the different methods. From these maps' one can find the

best pole orientati,ons by locating the lowest X2 points. The best and secondary solutions, as well as additional parameters are given for each method.

The shape models can be visualized using

a

PC program, and the resulting lightcurves compared

with

the observed ones. Such simulations were shown during the conference.

4.

Discussion

One should compare the morphology

of

the _{12 maps obtained}

with

the

PA on one side, and with

the 3 other methods on the other side. The main rralleys are nearly perpendicular. This corresponds

to the fact

that

completely independent information is being used: the chronometric information for

the PA

and

the

photometric one

for

the

others.

Usually,

for

asteroids which

orbit

inclination are low,

the

precision of

the

PA is good

in

latitude

and poor

in

longitude, while

it

is the opposite for

the photometric methods. This stresses how important

it

is

to

use

botb

types of methods

in

order

to derive reliable pole orientations. We plan to implement a ne\il FS-like method using directly both types of complementary information.

One has

to

remember

that

the shape

or

albedo models obtained

by

the

FAM

and FS methods are

only

one among

the

infinity of

possible models reproducing

exactly the

same lightcurvesFl.

Th"

usg' of moderating procedures like the ma:cimization of the

radii

entropy or the minimization of the surface/volume ratio ensures

that

the calculated model is the simplest one compatible with the data. Consequently,

it

is one of the most probable. The only way to determine the actual shape and albedo

distribution of an asteroid consists in also making use of some thermal

IR

lightcurves. This

will

also

-309-5.

References

t. Dcrel, 4., H.iaar, O., SchiL, P., Surdej, J.: æbnitrod ro À&A; 2. Pclri+t'L- Su!d.j,

^. & Surdcj' J. (f965)

A&A llle pr66; t. Boæ1,E.&Luruc,K.(r9?9)l!"Artcrcidr",G& CGTEL'Uaiv. ofArPær,pl32; a. Rwll"H.N.(f0O6)ApJ2rapl; E.

Schilr, P. (199f ) Mgtcr thair, Univcrrity of Uège; O. H.io.!rr, O., D€ad, A', Ibr.àiD-D€d., Â., PcpicrcrLù>SutdÊj, A. & Sut'dqi' J.

(19t9) iD thc prcædingr of .Artæidr Comcr Mctcorr tltr' Uppsll; ?. Trylor, R (19?9) ia "Artæiù', cA GchnL' Univ. of A: prcs, p{eo; !. DuLp & Gchnlr (f 969) tutm.J.7l p796; g. Vra Housca Groacnld ca d. (f9?9) AASS ff p223; 10. 8utùi

& Milano (tg7{) AASS 15 pl73; ll. Di Mstiao & Crcci,rrori (19&l} Icrnu 6O p75; 12. Di Merliao cl el. (f9t7) Icrn|| 69 p33t ;

r3. Weidearcùi[iag ct d. (f 987) Icro ?0 pl9l.

Table

1:

Observational parameters:

I

and p are the geocentric ecliptic coordinates of the asteroids;

A

and R are their geocentric and heliocentric distanc€xt;

a

is the solar phase angle.

A 195 1957 05 30 1965 02 (x 1967 (xt 07 l96t 05 0l 1984 r0 (Il 3t6.Oc ll9.lo 192.90 22S.to -22.r -22.60 l,l.60 -g.gc -20.3" l0.lo I 1.t12 {.1l{ {.ll{ {.109 {.28t 5.0t4 s.093 5.02E 5.æ7 s.ftxt 5.276 1965 l96lt 05 (xt rgn ot 09 1972 06 13 t9t2 lo r5 l98a 02 0l lgt{ 02 20 r9t5 08 16 22t3c 3{{.60 3{3.9c 13.t" 12E.60 r2:tJc 3l9.tc 0.9s7 0.s82 o.972 l.2m 1.52{ 1.561 o.&lr l.(xl?l l'qtE{ r.ol37 l.or30 03869 0.9&t{ 0.9086 1.0r26 -1

4t

630 6.{o 5.t. -{.to 1.20 6.20 2.6c l{3c l2.lo i.5. 2.0c 1030 3.70 2 1.10 l.lc 5.5c {.lo 2.O" 16l l8l 16l 16l tu

Table

2:

Results of the different methodg.

Às and Fo æe the ecliptic coordinat$ of the pole; a, b, c are the ellipsoid semi-axes; _{Q is the} multiple scattering pa,rameter;

T

is the sidereal Period.

Method

Solution Pole Additional

824

AMA

)o ₌ 315o

- -160 {b

₌

2.27 1 c

=

1.41

ç

₌

1.32 Secondary Solution Às

₌

l52o

=

+270

=

2.26;

Solution (721æets

₌

1450

={

₌ 0.717

lnlm

of

S/V

80 facets)

Best Solution

Às=149o

ps=1220

Q=0.38

Maximisation

of

Entropy

(80 facets)

Best

Solution

Àq

₌

1440 Êo

- *Llo

Q = 0'35

Secondary

Solutiou

@

? =

6.9205109h

*1.3x

10-6 (Retrograde) FA

43

A

Best

A

= *13o Ào ₌ 250o Êo

=

*8o

a,lb ₌ 4lb

-1.84 ;

1.84 ;

Solution

=

1.50

A est Solution (72 facets

₌

250o Bo

₌

*l

I

FS

inimization

of

S/V

(80 facets)

Best

Solutiou

Ào

₌

2480 ps ₌

{20o

Q = 0'098

Secondary

Solution

Ào

=

73o ps

=

j26o

Q = 0'092

Maximisation

of

Entropy

(60 facets)

Best

Solution

Ào

₌

70o Éo

=

*5o

Q =

0.19

_"

,=

_.

,

+22"

1=

5'7619820h+0'6x l0-b (Retrograde) Secondary

Solution

À0

₌

?4o po

₌

+24'

1

₌

5'7619819 h

t0'9 x

10-6 (Retrograde)

-3t0-3o

.5

ô

Figure

1:

The

RAMA

x2 maPs.

For 624 Hektor

(left),

43 Ariadne

(right).

The best pole solutions lie

in

the shaded regions

-o {5 g o .3 K o i5

?s

Às

o 0.t

I

for the FAM method.

Figure 2:

Same as Fig.

/.-'

"--\

(

''a-,

i -tj''J+

\,--;F-/:

<)*.,

f---"' /'---r\

,

.-_--.-,,,' i,-_.i,

'i___.'-rl i....,',\1.

.

_-)

,r.

,\l'

Ii,---,':^\

--"'i"',

.ii''.,';2;âi-.

l!,

_-";oli

ij,1

116.i+ii

i

l

ii

(.(0i

11

ii

.\\.\Yi

'-;..1,

_{;irr'-.q==,,,;jj'"'t,'\,}

i-:Yt

---'^ii..

t

li

'

"-i

:',)

\i.-..

i"

\,

!--_--.;

i.---,i '

_"-..,-)

\j

'

rro

270

_Ào*

o

s

Figure 3:

Same as Fig. 1 for the FS method.

R vo .:' tùO 710 ) ltâ e o

Figure 4:

Same as Fig.

I

for the PA method.

t /. o

é-'\

t:/

,l(

'\e\

.à

r-/'

Ki

{o

.-)

I n

t''

I

è,

)l

I

)