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Poverhnostnaya fotometriya galaktik

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4.2 Formula Sersika

Formulu Sersika [87], kotoraya yavlyaetsya obobsheniem zakona Vokulera, chasto zapisyvayut v sleduyushem vide [95]:

$\begin{displaymath} I(r)=I_0\,e^{-\nu_{n}\alpha^{1/n}}, \end{displaymath}$

(24)

gde

-- central'naya poverhnostnaya yarkost', $\alpha=r/r_e$ ,

-- polozhitel'noe deistvitel'noe chislo i $\nu_n$ -- konstanta, vybiraemaya tak, chtoby v predelah

izluchalas' polovina polnoi svetimosti. V bolee pohozhem na zakon Vokulera vide (formula (11)) ee mozhno zapisat' tak:

$\begin{displaymath} \frac{I(r)}{I_e}={\rm exp}\left[-\nu_{n}\left(\left[\frac{r}{r_e}\right]^{1/n}-1\right)\right], \end{displaymath}$

(25)

gde $I_e=I_0\,e^{-\nu_n}$ . Pri

$\nu_4$ =7.66925 i formula Sersika perehodit v formulu Vokulera.

Raspredelenie yarkosti, sootvetstvuyushee formulam (24-25), mozhno predstavit' tak:

$\begin{displaymath} \mu(r)=\mu_0+\frac{2.5\nu_n}{{\rm ln}\,10}\,\left(\frac{r}{r_e}\right)^{1/n}. \end{displaymath}$

(26)

Pri

formula (26) prevrashaetsya v formulu (12). Iz (26) sleduet, chto effektivnaya poverhnostnaya yarkost' ( $\mu_e=\mu(r_e)$ ) dlya etoi modeli zapisyvaetsya kak $\mu_e = \mu_0 + {2.5\nu_n}/{{\rm ln}\,10}$ .

Svetimost', izluchaemaya v predelah rasstoyaniya ot centra galaktiki, ravna

$\begin{displaymath} L(\leq r)=\frac{2 \pi n}{\nu_n^{2n}}\,\gamma(2n,\nu_n \alpha^{1/n})\,I_0 r_e^2, \end{displaymath}$

(27)

gde $\gamma(\eta,x)=\int_{0}^{x}e^{-t}t^{\eta-1}{\rm d}t$ -- nepolnaya gamma-funkciya. Polnaya svetimost':

$\begin{displaymath} L_T=\frac{2 \pi n}{\nu_n^{2n}}\Gamma(2n)\,I_0 r_e^2, \end{displaymath}$

(28)

gde $\Gamma(\eta)=\gamma(\eta,\infty)$ -- gamma-funkciya. Krivaya otnositel'noi svetimosti:

$\begin{displaymath} k(\alpha)=L(\leq \alpha)/L_{T}=\frac{\gamma(2n,\nu_n \alpha^{1/n})} {\Gamma(2n)}. \end{displaymath}$

(29)

Kak neslozhno ubedit'sya, pri

formuly (9) i (28) sovpadayut. (Edinstvennoe otlichie sostoit v tom, chto formula (9) zapisana dlya galaktiki s vidimoi elliptichnost'yu $\epsilon$ , a (28) -- dlya ob'ekta s kruglymi izofotami. V sluchae, esli izofoty galaktiki mogut byt' approksimirovany ellipsami, uravneniya (27) i (28) dolzhny byt' domnozheny na mnozhitel' (1- $\epsilon$ ).)

Iz opredeleniya effektivnogo radiusa sleduet, chto konstanta $\nu_n$ mozhet byt' naidena iz uravneniya $\gamma(2n,\nu_n)=\frac{1}{2}\Gamma(2n)$ . Dlya tochnye znacheniya $\nu_n$ privedeny v tablice 5. V literature opublikovano neskol'ko interpolyacionnyh zavisimostei $\nu_n$ ot . V rabote [95] pokazano, chto

$\begin{displaymath} \nu_n=2n-\frac{1}{3}+\frac{4}{405n}+\frac{46}{25515n^2}+{\rm O}(n^{-3}) \end{displaymath}$

(30)

s otnositel'noi pogreshnost'yu $\leq 10^{-6}$ . Formuly drugih avtorov, kak pravilo, svodyatsya k pervym dvum chlenam razlozheniya (30).

Tablica 5: Tochnye znacheniya koefficienta $\nu_n$ soglasno [95]

	$\nu_n$
1	1.67834699
2	3.67206075
3	5.67016119
4	7.66924944
5	9.66871461
6	11.6683632
7	13.6681146
8	15.6679295
9	17.6677864
10	19.6676724

Yarkie ellipticheskie galaktiki horosho opisyvayutsya zakonom Sersika pri $n\sim4$ , karlikovye ellipticheskie galaktiki i diski spiral'nyh galaktik -- pri $n\sim1$ , a baldzhi i ellipticheskie galaktiki promezhutochnyh svetimostei mogut byt' predstavleny formuloi (24) pri $1 \leq n \leq 4$ . Primery profilei yarkosti pri raznyh znacheniyah pokazany na ris. 14. Na risunke vidno, chto s rostom profili stanovyatsya bolee protyazhennymi i pologimi.

ris. 14: Raspredeleniya poverhnostnoi yarkosti dlya zakona $r^{1/n}$ pri

. Vse modeli imeyut odinakovoe znachenie

, effektivnaya poverhnostnaya yarkost' izmenyaetsya kak $\mu_e=22.0-0.25n$ . Samaya zhirnaya krivaya sootvetstvuet modeli s

, samaya tonkaya --

$\begin{figure}\centerline{\psfig{file=sersic.ps,angle=-90,width=11.0cm}}\end{figure}$

Dinamicheskie svoistva zakona $r^{1/n}$ obsuzhdayutsya v rabotah [96,97,95,98].

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