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Poverhnostnaya fotometriya galaktik

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4.1 Zakon Vokulera

4.1.1 Obshie svedeniya

Znamenityi zakon byl opublikovan Zherarom de Vokulerom v 1948 godu v rabote pod nazvaniem ``Recherches sur les Nébuleuses Extragalactiques: I. Sur la technique de l'Analyse Microphotométrique des Nébuleuses Brillantes'' [58] i zapisyvaetsya tak: ${\rm lg}I(r)/I_e=-3.25~[(r/r_e)^{1/4}-1]$ . V privedennoi formule i -- eto effektivnaya poverhnostnaya yarkost' i effektivnyi radius (p. 3.2), a koefficient 3.25 byl empiricheski podobran tak, chtoby v predelah izluchalas' polovina polnoi svetimosti galaktiki. Pozdnee znachenie koefficienta bylo uvelicheno do tochnogo znacheniya 3.33 (sm. dalee).

V obshem vide zakon Vokulera mozhno zapisat' kak

$\begin{displaymath} {\rm lg}\frac{I(\alpha)}{I_e}=-\beta(\alpha^{1/4}-1), \end{displaymath}$

(8)

gde $\alpha=r/r_e$ i $\beta$ -- koefficient ( $\beta>0$ ). Predpolozhim dalee, chto izofoty galaktiki mogut byt' predstavlenny soosnymi gomocentrichnymi ellipsami s postoyannoi elliptichnost'yu $\epsilon=1-b/a$ . Togda polnaya asimptoticheskaya svetimost' galaktiki, opisyvaemoi zakonom Vokulera, budet ravna

$\begin{displaymath} L_T=2 \pi I_e r_e^2 (1-\epsilon) \int_0^{+\infty}{\rm exp}[-... ...alpha = 8! \pi \frac{e^{\nu}}{\nu^8} (1-\epsilon) I_e r_e^2, \end{displaymath}$

(9)

gde $\nu=\beta\,{\rm ln\,10}$ .

Krivaya otnositel'noi svetimosti (p. 3.2) takoi galaktiki soglasno [90] mozhet byt' predstavlena v vide:

$\begin{displaymath} k(\alpha)=L(\leq \alpha)/L_{T}=1-{\rm exp}(-\nu\alpha^{1/4})\Sigma_{n=0}^{n=7}\frac{\nu^n\alpha^{n/4}}{n!}. \end{displaymath}$

(10)

V predelah

( $\alpha=1$ ) dolzhna izluchat'sya polovina polnoi svetimosti galaktiki, to est'

. Otsyuda iz uravneniya (10) mozhno naiti, chto $\nu=7.66925$ i $\beta=\nu/{\rm ln}\,10=3.33071$ .

Sledovatel'no, v okonchatel'nom vide zakon Vokulera mozhno zapisat' kak

$\begin{displaymath} {\rm lg}\frac{I(r)}{I_e}=-3.33071\left[\left(\frac{r}{r_e}\right)^{1/4}-1\right] \end{displaymath}$

(11)

ili, v zvezdnyh velichinah na kvadratnuyu sekundu dugi,

$\begin{displaymath} \mu(r)=\mu_e+8.32678[(r/r_e)^{1/4}-1] \end{displaymath}$

(12)

Sootvetstvuyushaya etomu zakonu polnaya svetimost'

$\begin{displaymath} L_T=7.21457 \pi I_e r_e^2 (1-\epsilon)=22.66523 I_e r_e^2 b/a. \end{displaymath}$

(13)

Absolyutnaya zvezdnaya velichina galaktiki

$\begin{displaymath} M_{Vauc}=\mu_e-5\,{\rm lg}r_e-2.5\,{\rm lg}(1-\epsilon)-39.961, \end{displaymath}$

(14)

gde effektivnyi radius

vyrazhen v kiloparsekah.

Srednyaya poverhnostnaya yarkost' v predelah efektivnogo radiusa sostavlyaet ${\langle I \rangle}_e=3.61I_e$ ili ${\langle \mu \rangle}_e=\mu_e-1.39$ . Polnaya svetimost', vyrazhennaya cherez ${\langle I \rangle}_e$ , ravna $L_T = 2 \pi {\langle I \rangle}_e r_e^2\,b/a$ (sr. s (13)). Central'naya poverhnostnaya yarkost' galaktiki v modeli Vokulera $I_0^b = 10^{3.33}I_e = 2140I_e$ . Indeksy koncentracii (p. 3.2) dlya ob'ekta, opisyvaemogo zakonom Vokulera, sostavlyayut $C_{21}=2.75$ i $C_{32}=2.54$ .

Na ris. 11 pokazan fotometricheskii razrez NGC 3379 i ego approksimaciya formuloi (12). Kak vidno na etom risunke, zakon Vokulera yavlyaetsya horoshim opisaniem nablyudaemogo raspredeleniya galaktiki v ochen' shirokom diapazone poverhnostnoi yarkosti $\mu(B)$ (ot 17 do 29) i (ot $\sim1''$ do $\sim10'$ )³. V dolyah effektivnyh parametrov eti intervaly sostavlyayut $0.002I_e \leq I \leq 125I_e$ i $0.02r_e \leq r \leq 11r_e$ . V [91] bylo pokazano, chto zakon $r^{1/4}$ opisyvaet raspredelenie yarkosti v NGC 3379 s oshibkoi, ne prevyshayushei $\pm$ 0.08, v intervale poverhnostnyh yarkostei $\Delta\mu\sim10$ .

ris. 11: Nepreryvnoi liniei pokazano raspredelenie poverhnostnoi yarkosti vdol' bol'shoi osi ellipticheskoi galaktiki NGC 3379 soglasno [38]. Shtrihovaya pryamaya -- approksimaciya nablyudaemogo raspredeleniya yarkosti zakonom Vokulera s parametrami $\mu_e(B)=22.24$ i

[38]. Sleva risunok priveden v koordinatah $\mu$ -

, sprava -- $\mu$ - $r^{1/4}$ .

$\begin{figure}\centerline{\psfig{file=n3379vauc.ps,angle=-90,width=16.5cm}}\end{figure}$

Izobraziv razrez v koordinatah $\mu$ - $r^{1/4}$ , mozhno bystro ustanovit' sootvetstvie raspredeleniya yarkosti galaktiki zakonu Vokulera (sm. ris. 11). Kak sleduet iz formuly (12), v takih koordinatah zakon Vokulera -- eto pryamaya liniya $\mu(r)=u+vr^{1/4}$ , gde $u=\mu^b_0=\mu_e-8.3268$ (central'naya poverhnostnaya yarkost') i $v=8.3268/r_e^{1/4}$ .

V tablice 4 privedena normirovannaya krivaya otnositel'noi svetimosti, postroennaya po uravneniyu (10) [90]. Na ris. 12 pokazany rezul'taty aperturnoi fotometrii NGC 3379 soglasno [91]. Kak vidno na risunke, nablyudatel'nye dannye horosho soglasuyutsya s model'noi ``krivoi rosta''. Izmereniyu v maksimal'nuyu diafragmu () sootvetstvuet znachenie ``asimptoticheskoi popravki'' $\Delta\,m=0.21$ .

Tablica 4: Normirovannaya krivaya otnositel'noi svetimosti dlya zakona Vokulera po [90]


0.01	0.00355	6.124
0.05	0.03193	3.739
0.10	0.07197	2.857
0.20	0.14716	2.081
0.30	0.21273	1.680
0.50	0.31981	1.238
1.00	0.50000	0.753
2.00	0.69001	0.403
3.00	0.78807	0.259
4.00	0.84658	0.181
5.00	0.88455	0.133
10.0	0.96149	0.043
30.0	0.99701	0.003
80.0	0.99990	0.0001

ris. 12: Aperturnaya fotometriya ellipticheskoi galaktiki NGC 3379 v cvetovoi polose

soglasno [91] (kruzhki). Nepreryvnoi liniei pokazana krivaya otnositel'noi svetimosti dlya zakona Vokulera (uravnenie (10), tablica 4) pri

$\begin{figure}\centerline{\psfig{file=n3379mag.ps,angle=-90,width=9.0cm}}\end{figure}$

Raspredelenie poverhnostnoi yarkosti u normal'nyh ellipticheskih galaktik, kak pravilo, horosho opisyvaetsya zakonom (11). Sistematicheskie otkloneniya ot etogo zakona nablyudayutsya obychno tol'ko v central'nyh i periferiinyh oblastyah galaktik. V promezhutochnoi oblasti (naprimer, pri $0.1r_e \leq r \leq 1.5r_e$ ) zakon Vokulera yavlyaetsya ochen' horoshim priblizheniem dlya galaktik rannih morfologicheskih tipov [92].

4.1.2 Raspredelenie plotnosti i krivaya vrasheniya

Esli galaktika sfericheski-simmetrichna i ''prozrachna'', to svyaz' trehmernogo raspredeleniya plotnosti svetimosti i sproecirovannogo raspredeleniya yarkosti daetsya sleduyushei formuloi (naprimer, [26]):

$\begin{displaymath} I(r)=2\int_{r}^{\infty}\frac{j(R)R{\rm d}R}{\sqrt{R^2-r^2}}. \end{displaymath}$

(15)

Reshenie etogo integral'nogo uravneniya horosho izvestno:

$\begin{displaymath} j(R)=-\frac{1}{\pi}\int_{R}^{\infty}\frac{{\rm d}I}{{\rm d}r}\frac{{\rm d}r}{\sqrt{r^2-R^2}}. \end{displaymath}$

(16)

Esli galaktika imeet postoyannoe otnoshenie massa-svetimost', to uravnenie (16) daet iskomoe trehmernoe raspredelenie plotnosti $\rho(R)$ .

Prostranstvennoe raspredelenie plotnosti, kotoroe v proekcii privodit k zakonu Vokulera, opredelyalos' v ryade rabot (naprimer, [90,93]). V [90] bylo naideno, chto

$\begin{displaymath} \rho(R\rightarrow0)=0.24099\,R^{-3/4}{\rm exp}(-\nu R^{1/4}), \end{displaymath}$

(17)

$\begin{displaymath} \rho(R\rightarrow\infty)=\frac{1}{2}\sqrt{\frac{\pi}{8 \nu R... ...xp}(-\nu R^{1/4})\left[1-\frac{7}{8\nu R^{1/4}}+\ldots\right]. \end{displaymath}$

(18)

Esli ostavit' v razlozhenii (18) tol'ko pervyi chlen, to polnaya massa galaktiki v predelah rasstoyaniya ot yadra budet ravna

$\begin{displaymath} {\rm M}(\leq R)=\frac{\sqrt{8\pi^3}}{\nu^9}\,\gamma(8.5,R^{1/4}), \end{displaymath}$

(19)

gde $\gamma(\eta,x)$ -- nepolnaya gamma-funkciya. Tochnost' formuly (19) sostavlyaet $\sim10\%$ pri $R\rightarrow\infty$ .

V [93] bylo predlozheno bolee prostoe analiticheskoe priblizhenie dlya raspredeleniya plotnosti:

$\begin{displaymath} \rho(R)=\rho_0\,R^{-0.855}{\rm exp}(-R^{1/4}). \end{displaymath}$

(20)

Iz etogo vyrazheniya sleduet, chto

$\begin{displaymath} {\rm M}(\leq R)={\rm M_0}\gamma(8.58,R^{1/4}), \end{displaymath}$

(21)

gde ${\rm M_0}=16 \pi \rho_0 (r_e/\nu^4)^3$ i

$\begin{displaymath} {\rm M_{tot}}=1.65 \cdot 10^4\,{\rm M_0}. \end{displaymath}$

(22)

V rabote [93] privedeny sootvetstvuyushie tablicy dlya raschetov po formulam (20-21), a takzhe sravnenie s rezul'tatami [90].

V stat'e [94] predlozhen prostoi sposob postroeniya krivoi vrasheniya dlya opisyvaemoi zakonom Vokulera galaktiki. Skorost' vrasheniya splyusnutoi galaktiki v ekvatorial'noi ploskosti mozhet byt' predstavlena kak

$\begin{displaymath} {\rm V}^2(r)=0.0172\,f\,I_e\,r_e\,q\,C(\alpha,q_0), \end{displaymath}$

(23)

gde V(

) vyrazhena v km/s,

-- otnoshenie massa-svetimost' v solnechnyh edinicah,

-- effektivnye poverhnostnaya yarkost' (v $L_{\odot}$ /pk

) i radius (v pk),

-- vidimoe szhatie galaktiki. Bezrazmernaya funkciya $C(\alpha,q_0)$ , gde $\alpha=r/r_e$ i

- istinnoe szhatie, opredelyaet formu krivoi vrasheniya. Funkciya $C(\alpha,q_0)$ (bezrazmernaya krivaya vrasheniya) dlya $q_0=0.4,\,0.6,\,0.8$ pokazana na ris. 13 (sootvetstvuyushie tablicy privedeny v [94]). Maksimum krivoi vrasheniya dostigaetsya pri $r/r_e \approx 0.27$ . Pri etom maksimal'naya skorost' vrasheniya sostavlyaet V $_{max} \approx \sqrt{0.06 f I_e r_e q}$ (km/s). Pri izmenenii istinnogo szhatiya galaktiki ot 0.4 do 0.8 maksimal'naya skorost' vrasheniya umen'shaetsya primerno na 15%.

ris. 13: Funkciya $C(\alpha,q_0)$ pri

(nepreryvnaya krivaya),

(shtrihovaya krivaya) i

(krivaya iz tochek).

$\begin{figure}\centerline{\psfig{file=rcv.ps,angle=-90,width=9.0cm}}\end{figure}$

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