Standard NH 3 (1, 1) and (2, 2) analysis: parameter and error estimation

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1 Standard NH 3 (1, 1) and (2, 2) analysis: parameter and error estimation Robert Estalella 2015 May 1 Fit parameters The procedure HfS_NH3 ts simultaneously the hyperne structure (see Table 1) of a (1, 1) and a (2, 2) NH 3 spectrum. The tting procedure is that of HfS, described in Sánchez-Monge et al. (2013). The assumptions made are that the emitting region is homogeneous along the line of sight, and that the lling factor f, the excitation temperature T ex, the hyperne components linewidth V, and the central velocity V LSR are the same for all the hyperne components of the (1, 1) and (2, 2) transitions. However, since there is enough information, a dierent V LSR can be tted for the (1, 1) and the (2, 2) spectra. In general, dierences in central velocity of a few tenths of km s 1 are usually found between the (1, 1) and (2, 2) emissions (see for instance Sepúlveda et al. 2011). The parameters tted to a pair of NH 3 (1, 1) and (2, 2) spectra are six: V Hyperne components linewidth, assumed to be the same for all the hypernes of the (1, 1) and (2, 2) transitions; V 1 Central LSR velocity of the (1, 1) transition; A 1m A(1 exp{ }) Peak intensity of the (1, 1) main component (for hyperne components wider than the hyperne separation and the channel width); τ 1m 1 exp{ } where is the optical depth of the (1, 1) main component; V 2 Central LSR velocity of the (2, 2) transition; and A 2m A(1 exp{ τ 2m }) Peak intensity of the (2, 2) main component (for hyperne components wider than the hyperne separation and the channel width). For each set of t parameters, the optical depth of the (2, 2) main component is taken from the relation τ 2m 1 exp{ τ 2m } = τ 1m A 2m A 1m. (1) The tting procedure ends with a set of six values of the t parameters (plus τ 2m), which minimise the rms t residual of the (1, 2) and (2, 2) spectra, and an estimation of the uncertainty of the t parameters, ɛ( V ), ɛ(v 1 ), ɛ(a 1m), ɛ(τ 1m), ɛ(v 2 ), and ɛ(a 2m). 2 Derived line parameters From the values of the t parameters, ve derived parameters are calculated, which are necessary for the estimation of parameters with physical interest. 1

2 Table 1: Relative velocity and optical depth of the hyperne lines of the NH 3 (J, K) = (1, 1) and (2, 2) inversion transitions (Mangum & Shirley 2015). (i.o.: outer satellite, i.s.: inner satellite, m: main) (J, K) = (1, 1) (J, K) = (2, 2) V V (km s 1 ) τ/τ tot (km s 1 ) τ/τ tot o.s i.s m i.s o.s A Amplitude, assumed to be the same for the (1, 1) and (2, 2) transitions, A = f[j ν (T ex ) J ν (T bg )], (2) where f is the lling factor, T ex the excitation temperature, T bg the background temperature, and hν/k J ν (T ) = e hν/kt 1. (3) A is calculated from A = A 1m. (4) Optical depth of the (1, 1) main component, calculated from = 1 ln(1 τ 1m). (5) Special care has to be taken when 1, since the last expression involves the dierence of 1 and a number near 1. In this case, a good approximation is the Taylor expansion ; (6) A Amplitude times the (1, 1) main component optical depth, calculated from A = A 1m. (7) Note that for τ 1m 1, A A 1m; 2

3 τ 2m Optical depth of the (2, 2) main component, calculated as for the (1, 1) transition; and Aτ 2m Amplitude times the (2, 2) main component optical depth, calculated as for the (1, 1) transition. 3 Derived physical parameters The physical parameters that can be derived from the standard analysis of NH 3 (1, 1) and (2, 2) observations include the excitation temperature T ex, the NH 3 (1, 1) and (2, 2) beam-averaged column densities N(1, 1) and N(2, 2), the rotational temperature T rot, the NH 3 beam-averaged column density N(NH 3 ), and the kinetic temperature T k. T ex The excitation temperature T ex is obtained from the amplitude A, T ex = T ν { }, (8) T ν ln 1 + [A/f + J ν (T bg )] where T ν is the frequency of the (1, 1) transition in temperature units (T ν = hν 11 /k = 1.14 K, see Table 2). For values of T ex T bg > T ν, this expression simplies to T ex A/f. (9) The value of the excitation temperature depends on the value assumed for the lling factor f. The usual assumption is that the lling factor f = 1. The value obtained with this assumption is a lower limit for the value of T ex. On the contrary, if we assume that f 1, T ex. N(1, 1), N(2, 2) The column density of the (J, K) level can be given as (Anglada et al. 1995, Estalella & Anglada 1997) N(J, K) = 8πν jk 3 exp(t ν /T ex ) + 1 c 3 R m A jk exp(t ν /T ex ) 1 τ m V, (10) where A jk is the Einstein coecient of the inversion transition of the rotational level (J, K), and R m = τ tot /τ m is the ratio of total and main component optical depths of the inversion transition (see Table 2). Table 2: Values of the NH 3 (1, 1) and (2, 2) inversion transition frequencies (Kukolich 1967), spontaneous emission Einstein coecients (Osorio et al. 2009), ratios of total to main component optical depths (Mangum & Shirley 2015), and B(J, K) and C(J, K) coecients of Anglada et al. (1995) recalculated with the improved values of the constants in this table. ν jk T ν = hν jk /k A jk R m = (J, K) (GHz) (K) (10 7 s 1 ) τ tot /τ m B(J, K) C(J, K) (1, 1) (2, 2) The last expression depends on the value of the lling factor f assumed to derive T ex. The explicit dependence on f is exp(t ν /T ex ) + 1 exp(t ν /T ex ) 1 = 2 (A + f[j ν (T bg ) + T ν /2]), (11) ft ν 3

4 so that the beam-averaged column density can be expressed as fn(j, K) = 16πkν jk 2 hc 3 A jk R m (A + f[j ν (T bg ) + T ν /2])τ m V, (12) The maximum value of fn(j, K) is obtained for f = 1 (the usual assumption to derive T ex ), while the minimum value is obtained for f 1. In the latter case (or for A T ex T bg > T ν ), the expression simplies to fn(j, K) = 16πkν jk 2 hc 3 A jk R m Aτ m V, (13) The values of the constants appearing in these equations, ν jk, T ν, A jk, R m, are given in Table 2. In practical units the two equations become (Anglada et al. 1995) [ ] N(J, K) cm 2 = B(J, K) exp(t [ ] ν/t ex ) + 1 V exp(t ν /T ex ) 1 τ m km s 1 and, for f 1, or T ex T bg > T ν, [ ] fn(j, K) cm 2 (14) [ ] V = C(J, K) Aτ m km s 1, (15) The values of the constants B(J, K) and C(J, K) appearing in these equations for the (1, 1) and (2, 2) transitions are given in Table 2. The expression equivalent to these two equations, with an explicit dependence on f, is, [ ] [ ] fn(j, K) V cm 2 = C(J, K) (A + f[j ν (T bg ) + T ν /2])τ m km s 1. (16) T rot The rotational temperature T rot is obtained from the ratio of (1, 1) and (2, 2) column densities, or, in practical units (see Table 3), T rot = (E 22 E 11 )/k ( ), (17) g22 N(1, 1) ln g 11 N(2, 2) [ Trot K ] = ln ( N(1, 1) N(2, 2) ). (18) Table 3: Degeneracies and energies above the (1, 1) level of the lower metastables levels of NH 3 (Poynter & Kakar 1975, Mangum & Shirley 2015). Note that the values of the energies are slightly dierent from those given in Ho & Townes 1983). (E JK E 11 )/k (J, K) g JK (K) (0, 0) 1/ (1, 1) (2, 2) 5/ (3, 3) 14/

5 N(NH 3 ) The ammonia total column density N(NH 3 ) is usually estimated with the assumption that only the metastable levels J = K (up to (3, 3)), are populated, with the same rotational temperature giving the population ratios. With these assumptions, with the partition function Q given by In practical units (see Table 3), N(NH 3 ) = N(1, 1) Q = N(NH 3 ) = N(1, 1) Q, (19) 3 J,K=0 g JK g 11 e (E11 E JK)/kT rot. (20) [ 1 3 e22.64/trot e 40.99/Trot + 14 ] 3 e 99.76/Trot. (21) T k The kinetic temperature T k can be taken to be equal to the rotational temperature T rot, but a better estimation is given by the relation (Rosolowsky et al. 2008, Mangum & Shirley 2015) T k T rot =. (22) kt k 1 + ln[ e E 22 E 15.7/T k ] 11 Note that this relation implies that T rot is always below a value T rot = 40.99/ ln 1.6 = 87.2 K. Given a value of T rot, the implicit equation must be solved to nd T k. A possible iterative algorithm to solve the equation is starting with T (0) k = T rot. T (n+1) k = T rot ( 1 + T (n) [ k ln e (n) 15.7/T k ] ), (23) Another approach is to use a polynomial approximation for T k. A good approximation with a 4th degree polynomial is (see Fig.1) T k T rot T rot T rot T rot 4. (24) The dierence between both methods is less than 0.1 K for kinetic temperatures below 100 K. 4 Error estimation (analytical approach) We assume that the errors in the t parameters are statistically independent. Thus, in general, for a parameter d derived from m t parameters (m = 6), the error ɛ(d) is ɛ(d) = [ m k=1 ( ) ] 2 1/2 d ɛ(p k ). (25) p k 5

6 T k (K) T rot = T k {1 + (T k /40.99) ln [ exp (-15.7/T k )]} -1 T k ~ T rot T rot Trot T rot (K) Figure 1: T k as a function of T rot. Blue line: Mangum & Shirley (2015); dashed blue line: vertical asymptote, T rot = 40.99/ ln 1.6 = 87.2 K; blue circles: polynomial approximation; red line: T k = T rot line; τ 2m 1 e τ 2m The error in τ 2m is given by [ (ɛ(a ) 2 ( ) 1m ) ɛ(τ 2 ( ) ] + 1m ) ɛ(a 2 1/2 + 2m ). (26) ɛ(τ 2m) = τ 2m A 1m τ 1m A 2m A The error in A is given by [ (ɛ(a ) 2 ( ) ] ɛ(a) = A 1m ) ɛ(τ 2 1/2 + 1m ). (27) A 1m τ 1m τ m The error in is estimated through the derivative d d = 1 1 τ 1m = e τ1m, (28) so that the error is ɛ( ) = e τ1m ɛ(τ 1m). (29) Note that for τ 1m 1, we have ɛ( ) ɛ(τ 1m), while for τ 1m 1 the error ɛ( ) can be very large, i.e. the value of is not well constrained for 1. The error in τ 2m is evaluated in the same way. 6

7 Aτ m In order to evaluate the error in A we note that A depends only on two t parameters, A 1m and. Let us call R = /. We need the derivative dr d = 1 [ ln(1 ) ] = eτ1m R. (30) For the case 1 a better estimation can be made by the derivative of the Taylor expansion of R, dr (31) The error can be expressed as ɛ(a ) = [ (τ1m τ 1m dτ 1m 2 ( ɛ(a 1m)) + dr A 1m d ɛ(τ 1m)) 2 ] 1/2. (32) The error in Aτ 2m is evaluated in the same way. T ex For f = 1, the error in T ex can be estimated from the derivative 2 dt ex da = T ex [A + J ν (T bg )][A + J ν (T bg ) + T ν ], (33) resulting in an error ɛ(t ex ) = dt ex da ɛ(a). (34) N(1, 1), N(2, 2) The error in N(J, K) has to be estimated through the derivatives of the column density over A m, τ m, and V. After some algebra, the error can be expressed as [ ] 2 [ ] 2 [ ] ɛ(fn(j, K)) A ɛ(a 2 = m ) fn(j, K) A + f[j ν (T bg ) + T ν /2] A + (35) m [ ] 2 [ ] τ m e τm A ɛ(τ 2 [ ] 2 m ) ɛ( V ) +. τ m A + f[j ν (T bg ) + T ν /2] V This expression gives the correct value of ɛ(fn(j, K)) for any value of f, i.e. for f = 1, or f 1, when the column density depends only on Aτ m instead of depending on both A and τ m. τ m T rot, N(NH 3 ) Although analytical expressions can be derived for the errors of T rot and N(NH 3 ) as a function of the errors of the t parameters, the expressions are cumbersome and dicult to check. Its is better to use the numerical approach (see Section 5) to derive these errors. 7

8 T k The error in the kinetic temperature can be estimated from the derivative of T k over T rot, and gives (T k /T rot ) 2 ɛ(t k ) = ɛ(t rot ). (36) e 15.7/T k Alternatively, it can be estimated from the derivative of the polynomial approximation, ɛ(t k ) = ( T rot T 2 3 rot T ) rot ɛ(t rot ). (37) 5 Error estimation (numerical approach) All the derived parameters depend on m t parameters (m = 6). Let us call p i, i = 1,..., m (38) the values of the t parameters, and ɛ + i and ɛ i the + and errors of the parameters (i.e. a value of the t parameter given by p +ɛ+ ɛ ), found from the increase in the rms t residual (see Sánches-Monge et al. 2013). Let d be any of the parameters derived from the t parameters, d = d(p 1,..., p m ), for instance T rot or N(NH 3 ). For every t parameter p k (k = 1,..., m) we evaluate the values of the derived parameter when we increase the value of the k-th t parameter by its error ɛ + k, and decrease by d + k =d(p 1,..., p k + ɛ + k,..., p m), k = 1,..., m (39) d k =d(p 1,..., p k ɛ k,..., p m), k = 1,..., m (40) Since we assume that the errors of the t parameters are statistically independent, we can estimate the error in d as [ m ( d + k ɛ(d) = ) 2 ] 1/2 d k (41) 2 k=1 References Anglada, G., Estalella, R., Mauersberger, R., Torrelles, J. M., Rodríguez, L. F., Cantó, J., Ho, P. T. P., D'Alessio, P. 1995, ApJ, 443, 682 Estalella, R., Anglada, G. 1997, Introducción a la Física del Medio Interestelar, Col lecció Textos Docents, n. 50, 2nd edition: Edicions de la Universitat de Barcelona, Spain Ho, P. T. P., Townes, C. H. 1983, ARA&A, 21, 239 Kukolich, S. G. 1967, Phys. Rev., 156, 83 Mangum, J. G., Shirley, Y. L. 2015, PASP, in press Osorio, M., Anglada, G., Lizano, S., D'Alessio, P. 2009, ApJ, 694, 29 Poynter, R. L., Kakar, R. K. 1975, ApJ, 29, 87 Rosolowsky, E. W., Pineda, J. E., Foster, J. B., Borkin, M. A., Kaumann, J., Caselli, P., Myers, P. C., Goodman, A. A. 2008, ApJSS, 175, 509 Sánchez-Monge, Á., Palau, A., Fontani, F., Busquet, G., Juárez, C., Estalella, R., Tan, J. C., Sepúlveda, I., Ho, P. T. P., Zhang, Q., Kurtz S. 2013, MNRAS, 432, 3288 Sepúlveda, I., Anglada, G., Estalella, R., López, R., Girart, J.M., Yang, J. 2011, A&A, 527, A41 8

arxiv:submit/ [astro-ph.im] 14 Aug 2016

arxiv:submit/ [astro-ph.im] 14 Aug 2016 accepted by PASP Preprint typeset using L A TEX style emulateapj v. 5/2/11 HfS, HYPERFINE STRUCTURE FITTING TOOL Robert Estalella Departament de Física Quàntica i Astrofísica (formerly Astronomia i Meteorologia),