Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://lnfm1.sai.msu.ru/~rastor/Study/MaxLikelihood.pdf Äàòà èçìåíåíèÿ: Mon Apr 20 16:27:46 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 03:52:44 2016 Êîäèðîâêà: IBM-866

LOMONOSOV MOSCOW STATE UNIVERSITY FACULTY OF PHYSICS ASTROPHYSICS AND STELLAR ASTRONOMY DIVISION EXPERIMENTAL ASTRONOMY DIVISION

A.S. RASTORGUEV
USING MAXIMUM-LIKELIHOOD METHOD TO STUDY THE KINEMATICS OF GALACTIC POPULATIONS

..


Tutorial for 2nd í 3rd grade students on the Galactic Astronomy lectures course

Moscow, Sternberg Astronomical Institute MSU , 2002-2015


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1 2 3 4 5 6 7 8 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 9 10 11 12 13 R0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 6 9 11 13 16 18 20 21 21 23 24 25

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1

Abstract

Most detailed description of the maximum-likelihood technique applied to the study of galactic population is given. We consider 3D, 2D and 1D velocity field. Random errors in observational data, ellipsoidal velocity distribution and random and systematic errors in the distances are taking into account. For 3D velocity field, statistical parallax technique enables to improve the distance scale used. . , . , , . .

2



. -, . 1859 . . , . . 1887 . , (0.0041" / 0.0058" /), 100%. . 1920- . , , , , l = 270 . ., ., . .


..

5

, , . , . , ( ) . ( , , , 0 , .) , ( !) , . XX ( , ) , . (, , O-B- - , ). 1970- 1980- (, , HII). 1990- í HIPPARCOS [7], TRC [9], TYCHO-2 [10] í , . , . - ( ), ( ) ( ) . .


6



(, , ). , (, RR-) . , .

3



. , . , , , . 1980- í 1990- . , HIPPARCOS . 1997 . [7], , [8], - . , : (m - M ) = 5 § lg rphe () - 5 + A, m M í ( ), A í , r
phe

í

. M , , í , -


..

7

(., [6]). ( () , ) . , , ë í ¨ [2]: MV I = -1.01m - 2.87m § lg Ppls , Ppls í , M
V

í

V , I , , (, , lg P
pls

-0.15). ,

(B - V ) , ë í ¨ [2] B0 - V0 I = 0.24m + 0.47m § lg Ppls . RR , MV I = +0.60m [3]; , V [F e/H ] ([4], [11]) ( ë í ¨). RR- í RR-. [13]. , , ( ) (ë í ¨) [5]. , , . , .


8

, . -

í ( ) . , , . , , . , ( ) ( ), . , .. , lg Tef f í Lbol ( í ). , ( ) . , , . , . , -- , ( , ) [16]. , ( ), . ë¨ , . re í . (. ). r í ( ) , ï . ï M , . , rt í -


..

9

( ). r = rt - r ï ë¨ (.. ) . ( ) ï , . , r/r - M /2.17. ï p = re /r p ï re = (p + p) § rt . p rt r, .. ï ï ï M . p/p = - r/rt - r/r M /2.17. ï ï , M M ( , ï ). .
2 2 : p 0.21 § p2 § M . ï

. , p 1.

4



ë ( S0 ) í (GC) í S ¨, , . , , ( í ). l (GC - S0 - S ). . : , ; . , (); , S (. .). -


10
Vr Vl



GC S

R0

Sun - S 0

(x, y , z ) (u, v , w) (l = 0 , b = 0 ), (l = 90 , b = 0 ) (b = 90 ) . ( ) , ( ) , . gal í - e , l e
oc

í .

loc = G0 ½ gal , G0 e e


cos b cos l - sin l - sin b cos l

cos b sin l cos l - sin b sin l

sin b

G0 =

0 . cos b

(1)

-,


x

r cos b cos l

= y = r cos b sin l , r z r sin b

(Vr , Vl , Vb ) ,


.. (x, y , z ).


11

cos b cos - sin

cos b sin cos

sin b



GS =

0

(2)

- sin b cos - sin b sin cos b , S , (. .). , S ( ) (1), tg = R0 § sin l . R0 § cos l - r § cos b ï

, (0, 360 ). R0 í , r. ï : AT AN 2, .

5

: , ë¨

, : , , (). ë¨ , (. ). , .. z - ( R, z ). .


12



, , , [12]. , .. í , , - -. , , 20 í 30 /, ë¨ . , , . , , , , . = r § cos b í (.. ï ) S , Vr Vl í ( , , ). "/, ( ) /, , //. k = 4738§ (//)("/)-1 , , , Vt = k § ² § re , Vt í ( ). () S0 .


u0

V0 = v0 ,

w

0

( x , y í , z í ). , R r ï
2 R2 = R0 + r2 § cos2 b - 2 § r § R0 § cos b § cos l. ï ï

(3)


..

13

, (l, b); ( ), ( ). re ,


Vr

Vr k re § ²b

Vloc (re ) = Vl = k re § ²l V
b

, [1], r ï


R0 § ( - 0 ) § sin l § cos b -R0 § ( - 0 ) § sin l § sin b

ï Vrot (r) = (R0 § cos l - r § cos b) § ( - 0 ) - r § 0 § cos b , ï ï

(4)

(R) 0 = (R0 ) - R R0 .
( - 0 ) 0 § (R - R0 ) +

1 1 § 0 § (R - R0 )2 + § 0 § (R - R0 )3 + ..., 2! 3!

(5)

, , . 5 í 6 . , S , ë¨ .

6



[14], . . : , , (..,


14



, ). , , , . , . , . , , . , [14]. , , ë¨ . , , , .


100





000



~ R = 0 0 0,

~ M = 0 1 0

(6)

000


001

100



~ ~ ~ P = 0 p 0 = R + p § M,

00p , , , r , ï re ~ï Vloc (re ) = P ½ Vloc (r). (7)

~ P 1, .. . ( ) Vr = Vr, t + Vr , ²l = ²l, t + ²l , ²b = ²b, t + ²b ,


..

15

t , Vr , ²l ²b í . , Vr, t , ²l, t , ²

t ,t t b, t



, :
k rt § ²l

Vr,

= Vloc (rt ) = G0 ½ V0 + Vsys (rt ) + ,

(8)

k rt § ²b,

Vsys (rt ) í , , . , (6) ~ ~ Vloc (rt ) = R ½ Vloc (rt ) + M ½ Vloc (rt ). , . 2 re /rt = (p + p), Vloc (re )
~ ~ Vloc (re ) = k re § ²l + k re § ²l,t = Vloc (re ) + R ½ Vloc (rt ) + (p + p) § M ½ Vloc (rt ), (9)

Vr

Vr,t

k re § ²

b

k re § ²

b,t



V


r

Vloc (re ) = k re § ²l .



k re § ²

b

, ( , (8), ë¨ !), r: ï V
loc,mod

(r) = G0 ½ V0 + V ï

sy s

(r). ï

(10)

re , (7), Vl Vloc (re ) = Vloc (re ) - V
loc, mod oc, mod

~ (re ) = R ½ V

loc,mod

~ (r) + p § M ½ Vl ï

oc, mod

~ (r) = P ½ V ï

loc, mod

(r). ï

(11)

(re )

(12)

, , . ,


16



(12) (8) í (11), ( ) r p: ~ Vloc (re ) = Vloc (re ) + P ½ [Vloc (rt ) - Vl
oc, mod

~ (r)] + p § M ½ Vloc (rt ) ï

~ ~ = Vloc (re ) + P ½ [Vsys (rt ) - Vsys (r) + ] + p § M ½ Vloc (rt ) ï ~ ~ Vloc (re ) + P ½ [ r § Vsys (r)/ r + ] + p § M ½ [G0 ½ V0 + Vsys (rt ) + ] ïï ~ Vloc (re ) + r § P ½ V
sy s

~ ~ ~ (r)/ r + (P + p § M ) ½ + p § M ½ [G0 ½ V0 + Vsys (r)] (13) ïï ï
sy s

~ Vloc (re ) - r § p/p § P ½ V ï

~ ~ (r)/ r + P ½ + p § M ½ + p § M ½ Vl ïï~
loc, mod

oc, mod

(r) ï

~ ~ ~ Vloc (re ) + P ½ + p § M ½ + p § [M ½ V

~ (r) - r/p § P ½ Vsys (r)/ r] ï ï ïï

~ ~ Vloc (re ) + P ½ + p § M ½ + p § ,

~ =M ½V
loc,mod

~ (r) - r/p § P ½ Vsys (r)/ r ï ï ïï

~ ~ = M ½ [G0 ½ V0 + V sy s(r)] - r/p § P ½ Vsys (r)/ r ï ï ïï .

(14)

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Vloc (re ). , Vr ²l , ²b r (, , c p) , ²l ²b (, HIPPARCOS [7] ), . , - Y L(Y ) = Y § Y T , T í , ~ ~ Y . Y = A ½ X , A í , , , : ~ ~ ~ ~ ~ ~ ~ ~ L(Y ) = (A ½ X ) § (A ½ X )T = (A ½ X ) § (X T ½ AT ) = A § X § X T § AT = A ½ L(X ) ½ AT .


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17

, . , (ë¨) L( ). , ,

2 u



0
2 v

0



L0 = 0 0

0 ,
2 w

0

u , v , w í . GS (2), Lloc ( ) = GS ½ L0 ½ GT . S L
obs

oc (re ) = Vloc (re ) § VlT (re )

(13) , , , : p, Vloc (re ) , . L
obs

(re ) = L

err

(re ) + Lr

esid

(re ) + L(re ),


(15)





2 Vr

0
2 2 k 2 re § ²
l

0
2 2 k 2 re § ²

Lerr (re ) = Vloc (re ) § VlT (re ) = 0 oc 0 ² ²
b l

2 k 2 re § ²b ²l ²b , ²l
b

(16)

2 k 2 re § ²b ²l ²b ²l

í , Vr , ²l

²b í . (15) ë¨ Lr
esid

~ ~ ~ ~ (re ) = P ½ Lloc ( ) ½ P T = P ½ GS ½ L0 ½ GT ½ P T . S

(17)

, , (15) ; ë¨ (. . 2):
2 ~ ~ L(re ) p § [M ½ GS ½ L0 ½ GT ½ M + § T ] S 2 ~ ~ = 0.21p2 § M § [M ½ GS ½ L0 ½ GT ½ M + § T ]. ï S

(18)


18



(16)í(18) , r ï (.. ) re p r = re /p, (15) re . ï
ï M

.

, - (18) (15) M . ï

8



, . Vloc (re ), (10)í(12), [14] f (Vloc (i)) = (2 )-3/2 § |L
obs

(i)|-

1/2

1 § exp{- § VlT (i) ½ L oc 2

-1 obs

(i) ½ Vloc (i)},

(19)

, , |Lobs | L
-1 obs

í obs

L

(15).

. , N- () (19) ( ): F (Vloc (1), § § § , Vloc (N )) =
N i=1

f (Vloc (i)),

(20)

N í . , (.. ) . , ( , p í ), (19), , F .


..

19

, N- (20), .. . LF = - ln F (Vloc (1), § § § , Vloc (N )) = -
N i=1

ln f (Vloc (i))

(21)

LF - . (21) (19), (21) :
N 3 1 LF = N § ln 2 + {ln |L 2 2 i=1

obs

(i)| + VlT (i) ½ L oc

obs

(i)

-1

½ Vloc (i)},

(22)

i, . : Vloc (i) L
obs

(i)

, , LF , . , (22), , .. ( ). . LF , , : § (u0 , v0 , w0 ) í ; § (u , v , w ) í (, ); § (0 , 0 , 0 , § § §) í ; § p í ; § , , . , p 1, .



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8.1

R0

R0 . . , 7 9 [18]. , 1985 . í R0 = 8.5 , , . , . R0 , , , , . ( ), , . -, ( ), . /, , . R0 , . , R0 (< 8 ). ( ) , R0 . R0 (22) , . , - R0 R0 .


..

21

, , p, R0 . < R0 > 7.5 - 8.0(‘0.5) . . R0 .

9

:

( (8, 10) Vsys (r) , ). ï , Vsys (r) Vrot (r), ï (14, 18) , (4). , (4),
ï Vrot (r) = (R0 § cos l - r § cos b) § r ( - 0 ) - § cos b , ï ï r ï -R0 § r ( - 0 ) § sin l § sin b ï

R0 §

r ï

( - 0 ) § sin l § cos b

(23)

(R - R0 ) (. (3) (5)) ( - 0 ) 1 § (r cos b - R0 cos l) § [0 + 0 § (R - R0 ) + 1/2! § 0 § (R - R0 )2 + § § §] § cos b. ï r ï R

10



. , .. . , [17] HIPPARCOS. , .. . , (


22



) . , , , - ( , ..). , . (, , ; ..). , . , , , , , . , , . , , . , , , p 1, r ï re . , r 0.46 § r §
ï M




. : Vr 0, Vr 0, , (13) , , . (15) , . (19) :

f (Vloc ) = (2 )-1 § |Lobs |-

1/2

1 § exp{- § VlT ½ L oc 2

-1 obs

½ Vloc },

Vloc , L

obs

í

. .


..

23

11



( , ) . , , , . . r, rt r í ; Vr Vr í . (10), Vr,
mod

= U T § (G0 ½ V0 ) + V
U =

r, rot

(r),

1

0 (r) í



0 , V

r, rot

(4). V V
r, t r, obs

= Vr, t + Vr ,

= U T § [G0 ½ V0 + ] + Vr,

r ot

(rt ),

(24)

í ( ), . (24) Vr,
rot

(rt ) Vr,

r ot

(r) + r § V

r, rot

(r)/ r,
r, rot

, V (23): Vr,
obs

(r)/ r

= Vr,

obs

-V

r, mod

Vr + r § V

r, r ot

(r)/ r + U T § .

(25)

, Vr , r , (25) (.. ë ¨): L
obs 2 2 = Vr + r § ( V r, r ot

(r)/ r)2 + U T § (GL ½ L0 ½ GT ) § U . L

Lobs , .


24



, r M , (. . 5). ï V f (Vr, ) = (2 )-
1/2 r, obs



obs

§L

-1/2 obs

1 § exp{- § (V 2

r, obs

)2 § L-1 }. obs

12



LF . ( ). , (.. ) LF () = LF0 + 1 ( LF0 í , ) [19] (. 532). ( ) .
0 , [15]. i í - -

. :
0 0 i = i + i , i ( i ).

i , LFi , ( ). i
2 i 2 i 2(LFi - LF0 )

, LFi > LF0 , . , .


..

25

13



- (20 í 22) , (15), ë¨. , , . , ë¨, . ( 3 ), . ( ) . , , ë¨ í , . (, , ), , . (22) , , , [19]. () (. 294) ( í , í ) (. 301), ( ) (DFP í , , ; BFGS í , , , ) (. 307), í (. 523), í ( . -, . 289), , . , FORTRAN, Pascal, C [19], MATLAB, MATHEMATICA IDL.


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[1] .. . .: ë¨. 1985. [2] .., .., .. í BVRIJHK. . . .22. N.12. .936-944. 1996. [3] .. . . .9. N.6. C.349-370. 1953. [4] Carney B.V., Storm J., Jones R.V. The Baade-Wesselink method and the distances to RR Lyrae Stars. VIII - Comparisons with other techniques and implications for globular cluster distances and ages. Astrophys. J. V.386. P.663. 1992. [5] .. . .: ë¨. 1982. [6] . . : ë¨. 1977. [7] The HIPPARCOS catalogue. ESA SP-1200. 1997. [8] Proceedings of the ESA Symposium `Hipparcos - Venice '97', 13-16 May, Venice, Italy, ESA SP-402 (July 1997). [9] Hog E., Kuzmin A., Bastian U., Fabricius C., Kuimov K., Lindegren L., Makarov V.V., Roeser S. The Tycho Reference Catalogue. Astron. Astrophys. V.335. P.L65-68. 1998. [10] Hog E. et al. The TYCHO-2 catalogue of 2.5 million brightest stars. Astron. Astrophys. V.355. P.L27. 2000. [11] .., .. RR . . . .27. N.2. .132-143. 2001. 27


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[12] . . .: ë¨. 1982. [13] Burstein D., Heiles K. Reddenings derived from HI and galaxy counts: accuracy and maps. Astron. Journ. V.87. N.8. P.1165-1189. 1982. [14] .. . : ë ¨. 1986. [15] Hawley S.L., Jeffreys W.H., Barnes T.G. III, Wan Lai. Absolute magnitudes and kinematic properties of RR Lyrae stars. Astrophys. Journ. V.302. P.626-631. 1986. [16] .., .. . ë ¨. .35. N.1. .37. 2000. [17] Dehnen W., Binney J. Local stellar kinematics from HIPPARCOS data. Mon. Not. Roy. Astron. Soc. V.298. P.387-394. 1998. [18] Reid M.J. The distance to the center of the Galaxy. Annual Rev. Astron. Astrophys. V.31. P.345-372. 1993. [19] Press W.H., Flannery B.P., Teukolski S.A., Vetterling W.T. Numerical Recipes. The Art of Scientific Computing. Cambridge: Cambridge University Press. 1987.