A Formula Collection

In this formula collection, those formulae are summarized which are used in the programs of the Tübingen NLTE Model-Atmosphere Package (TMAP). The abbreviations for the different transitions (CBF- ...) refer to those of Sect. 1. The last numbers of the section titles are the formula numbers which have to be inserted in the atomic data file (ATOMS, Sect. 2, 2.2).

A.1 CBB Transitions

A.1.1 van-Regemorter Formula (Allowed Dipole Transitions)

------ |_ _ | 8k V ~ -- ( EH )2 Cij = pa02V ~ -----ne T |_ 14.5fij ---- _| u0e - u0Γ (u0 ) pme E0

u0 = hnij/kT , E0 = hnij, Γ (u0 ) = max [¯g, 0.276eu0E1 (u0 )]

= 0.2 for n’n else = 0.7. E_H is the ionization energy of the hydrogen ground state.

------ 8k pa02V~ ----- = 5.465 .10 -11 pme

2 input parameters: f_ij,

A.1.2 Forbidden Transitions

-6 8.631-.10---- -u0 Cij = g V~ T nee Ω i

NF sum IT i-1 Ω = ai .x i=1

NFIT+1 input parameters: NFIT, a₁, ... , a_NFIT
for the effective collision strength. Is only one parameter supplied (explicitly: 1), then Ω = 1 is set.

A.1.3 Hydrogen

following Mihalas

------ V ~ --( )2 Cij = 4pa02V~ -8k--ne T EH--- u0fij (E1 (u0) + 0.148u0E5 (u0)) g pme hnij

g = b + 2(a - b )/Δn for Δn > 1, and g = 1 else b = 3 - 1.2/n , a = 1.8- 0.4/n2 i i

no input parameter

A.1.4 He II

following Mihalas & Stone

------ 8k V ~ --( EH )2 ( ) Cij = pa02V ~ -----ne T ----- u0fije e- u0 ln 2 + E1 (u0 ) g pme hnij

[ ] n----1- g = min [ni, 1.1] .min Δni, ni - Δn

no input parameter

A.1.5 He I: Allowed Transitions from its Ground State

following Mihalas & Stone

------ -- ( )2 C = 4pa 2 V~ -8k--n V~ Tf EH--- u E (u ) 1j 0 pme e 1j hn1j 0 1 0

1 input parameter: f_1j

A.1.6 He I: Allowed Transitions but not from its Ground State

following Mihalas & Stone

------ 8k V ~ -- ( E )2 ( u ) Cij = 4pa02V~ -----ne T fij --H-- u0 E1 (u0 )- --0e- 0.2(E1 (u1 ) pme hn1j u1

u1 = u0 + 0.2

1 input parameter: f_ij

A.1.7 He I: Forbidden Transitions from its Ground State

following Mihalas & Stone

------ -- [ ] C = pa 2 V~ -8k--n V~ T--a---u --1--- a E (u ) + a u e-b1u0x + a u e- b2u0x ij 0 pme e neff 3 0pa02 0 1 0 1 0 1 2 0 2

( 1) bi u0 + c1 + 2 xi = ---(------1-)3---- u0 + ci

------ EH neff = ZV ~ ----- (Z = 1) hnth

8 input parameters: a, a_i, b_i, c_i (found in tabular form in M&S; Attention: their equation A16 is wrong!)

A.1.8 He I: Forbidden Transitions between its n = 2 Sublevels

following Mihalas & Stone

------ -- C = pa 2 V~ -8k--n V~ T e-E0/kT Γ (T ) ij 0 pme e ij

E0 = hnij

-2 log Γ = c0 + c1 log T + c- 2 (log T )

3 input parameters: c₀, c₁, c_-2 (in tabular form by M&S)

A.1.9 Unknown Collisional Cross-Sections

------ 8k V~ -- Cij = pa02V ~ ----- ne T e- u0 (1 + u0 ) pme

no input parameter

A.1.10 Ω-Fit of 3rd Degree

like A.1.2, but:

sum 4 Ω = ai (log T )i-1 i=1

4 input parameters: a₁, a₂, a₃, a₄

A.1.11 Mg II: Allowed Transitions from its Ground State

following Mihalas (1972)

------ 8k V ~ -- EH 2 ( ) Cij = pa02V ~ -----ne T 4fij---- u0 aE1 (u0) + be-u0 pme E0

2 input parameters: a, b

A.1.12 van-Regemorter: Combined Levels of Complex Ions

------ 8k V~ -- Cij = pa02V ~ -----ne Te -u0Γ ij(T ) pme

log Γ (T ) = a + a x + a x2 + a x3, x = log kT [in eV ] ij 0 1 2 3

4 input parameters: a₀, a₁, a₂, a₃

A.1.21 He I: Transition between its Levels with s < 4

D. Hummer, priv. comm.
no input parameter

A.1.25 Ω-Fit in Temperature, General Case

equal to DETAIL Formula No. 25,
8 input parameters

A.1.26 Ω-Fit in log T, General Case

equal to DETAIL Formula No. 26,

NF sum IT Ω = ai (log T - T1)i-1 i=1

NFIT+2 input parameters: T1, NFIT, a₁, ..., a_NFIT

A.2 CBF Transitions

A.2.1 Hydrogen, n = 1,..., 10

following Mihalas

------ 8k V ~ -- Cik = pa02V ~ -----ne T e-hnth/kTΓ i(T) pme

own fit formulae for Γ, because Mihalas is restricted in temperature,
no input parameter

A.2.2 He II, n = 1,..., 10

following Mihalas, like CBF1, own fit formulae for Γ, because Mihalas is restricted in temperature,
no input parameter

A.2.3 He I, n < 15

------ { 2 ( )} 2 V~ -8k-- V ~ -- 0.728u0--- 2 -u0 2.0-+-u2-- Cik = pa0 pme ne T s0 u0E1 (u0) - u1 E1(u1 ) - 0.189u0 e u23

u0 = hnth/kT , u1 = u0 + 0.27, u2 = u0 + 1.43

1 input parameter:

₀ (in tabular form by Mihalas & Stone)

A.2.4 Seaton Formula

13 -1---- u0 - 1 Cik = 1.55 .10 ¯gs0ne V~ --e u0 ¯g = [0.1,0.2, 0.3] for Z = [1,2, > 3] T

Z = charge of the ion

₀ = threshold photoionization cross-section
2 input parameters:

₀,

In case that formula 4 is requested and a cross-section of 0.0 inserted, a mean cross-section (at the threshold energy) of the Opacity Project data is calculated and used as threshold cross-section (Sect. A.4).

A.2.5 Lotz Formula

------ ( )2 { } 2 V~ -8k-- V~ -- EH-- au0-- Cik = pa0 pm ne TP E u0 E1 (u0) - u E1 (u1 ) e 0 1

3 input parameters: P,

, c
u₁ = u₀ + c

A.2.6 Mg II 3s Mg III 2p⁶ +

following Mihalas (1972)

------ ( )2 C = pa 2 V~ -8k--n V~ T- EH-- {au E (u ) + b (u /u )2[E (u ) + e-u1]} ik 0 pme e E0 0 1 0 0 1 1 1

u1 = u0 + c

3 input parameters: a, b, c

A.3 RBB Transitions

A.3.1 Doppler Profiles

V~ -- 2 sij = ---pe---fij-e -(Δn/ΔnD )2 mec ΔnD

---------- Δn = n0 V~ --2kT---- D c mAT OM

1 input parameter: f_ij

A.3.2 Voigt Profiles, only Radiative Damping

V~ -- pe2 fij sij = -----------H (a, v) mec ΔnD

Γ Δn a = --------, v = -----, Γ = Γ low + Γ up 4p ΔnD ΔnD

2 input parameters: f_ij, Γ

A.3.3 Voigt Profiles, Radiative and Collisional Damping (Electrons)

_ij like in A.3.2, but:

Γ = Γ rad + Γ St

( up 2)2 Γ St = 6.11 .10 - 5 n V~ e-- n-eff- T z

(Cowley 1970, 1971) 3 input parameters: f_ij, Γ_rad, ( up 2)
neff-
z+1

A.3.4 Voigt Profiles and “Stark Wings” (Linear Stark Effect)

[ Formel 3 St] sij = max sij ,sij

( ) St 0.0368 .Zfij ΔnZ .1.385 sij = --------------U -------------- snzMikro snzMikro

srmn = {nup (nup - 1) + nlow (nlow - 1 )}

[NION ]23 sum 32 zMikro = Zi .ni i=1

6 input parameters: f_ij, Γ_rad, (nupeff2)
z+1

², Z, n_low, n_up

A.3.5 Stark Line Broadening following Dimitrijévic

like formula 3

A.4 RBF Transitions

A.4.1 Seaton Formula

( )s [ ] s = s nth- a + (1 - a )nth- n 0 n n

4 hydrogen - like : s = 2.815 .1029 z--gII (nth)- 0 n5 nth3 eff

with the effective principal quantum number

---- V~ R--- neff = z nth

g_II is the bound-free gaunt factor

z core charge of the ion
3 input parameters: ₀,,s

A.4.2 Seaton Formula with Gaunt Factor

( ) [ ] nth- s nth- sn = s0 n a + (1 - a ) n gII (x, y,z )

6 input parameters:

₀,

,s,x,y,z

A.4.3 Koester Formula for He I

A&A,

ln (gsn ) = a0 + a1 ln c + a2 ln2 c, c[˚A ]

3 input parameters: a₀,a₁,a₂

A.4.4 Opacity-Project Photoionization Cross-Sections, Seaton tails

See Sect. A.4.10.

A.4.5 Karzas & Latter data with Gaunt Factor

Tables taken from Karzas & Latter (1961) are used to calculate the photoionization cross-sections.

3 input parameters: z_eff, n (principal quantum number), l (azimuthal quantum number)

A.4.10 Opacity-Project Photoionization Cross-Sections

The Opacity Project data for a level of the ion XXXX are read from the file OP_RBF_XXXX (Sect. 8.1). The programs recognize an A10 label at the begin of the data set which represents the level name in TMAP code (Sect. 2.1). For the actual frequency grid FGRID (Sect. 3), the cross-sections are interpolated or extrapolated (including possible resonances etc.). Is formula 4 requested and a cross-section of 0.0 inserted, a mean cross-section (at the threshold energy) of the Opacity Project data is calculated and used as threshold cross-section.
no input parameter

A.4.12 DETAIL Fit Formula

sum 5 [ (nth )]i ln sn = ai . ln ---- i=0 n

6 input parameters: a₀,...,a₅

A.5 RFF Transitions

None of the RFF formulae needs an input parameter.

A.5.1 Including Contributions of LTE Levels (Unsöld)

8 hn /kT Z2 skk(n, T) = 3.694 .10 e min --3 V~ -- n T

n = min [n,n ] min LTE

_LTE is the ionization threshold energy of the lowest LTE level.

(cf. Unsöld 1968)

A.5.2 Including Contributions of LTE Levels with Gaunt Factors

8 [ hn /kT ] Z2 skk (n,T ) = 3.694 .10 gff (n,T )(e min - 1) -3 V~ --- n T

A.5.3 With Gaunt Factors, no LTE Contributions

Z2 skk (n, T) = 3.694 .108gff (n,T )--V ~ -- n3 T

For the free-free Gaunt factors, the default is a calculation following Mihalas (1967, ApJ 149, 169). Since these values are calculated from a fit formula within 100 < < 10000 Å and an extension to longer wavelengths, data (valid from submillimetre to hard X-ray wavelengths and for temperatures from 10 - 10⁹ K) provided by Sutherland (1998) can be chosen by input card (Sect. 5.4).

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