Two-photon production in low-velocity shocks

TL;DR

This paper investigates the emission processes of low-velocity shocks in the interstellar medium, emphasizing the role of the two-photon continuum as a key FUV tracer, and provides practical formulas for modeling their radiative yields.

Contribution

It introduces new fitting formulas for the emission yields of Ly-alpha, two-photon continuum, and H-alpha in hydrogen plasma, aiding modeling of low-velocity shock cooling.

Findings

FUV band is an effective tracer of low-velocity shocks.

GALEX FUV images reveal large interstellar structures and bow shocks.

Provided formulas facilitate incorporation into cooling and ionization models.

Abstract

The Galactic interstellar medium abounds in low-velocity shocks with velocities less than, say, about 70 km/s. Some are descendants of higher velocity shocks, while others start off at low velocity (e.g., stellar bow shocks, intermediate velocity clouds, spiral density waves). Low-velocity shocks cool primarily via Ly-alpha, two-photon continuum, optical recombination lines (e.g., H-alpha), free-bound emission, free-free emission and forbidden lines of metals. The dark far-ultraviolet (FUV) sky, aided by the fact that the two-photon continuum peaks at 1400 angstroms, makes the FUV band an ideal tracer of low-velocity shocks. Recent GALEX FUV images reaffirm this expectation, discovering faint and large interstellar structure in old supernova remnants and thin arcs stretching across the sky. Interstellar bow shocks are expected from fast stars from the Galactic disk passing through the numerous gas clouds in the local interstellar medium within 15 pc of the Sun. Using the best atomic data available to date, we present convenient fitting formulae for yields of Ly$\alpha$, two-photon continuum and H$\alpha$ for pure hydrogen plasma in the temperature range of 10^4 K to 10^5 K. The formulae presented here can be readily incorporated into time-dependent cooling models as well as collisional ionization equilibrium models.

Tables

Table 1. Table 1: The indexing scheme and spectroscopic terms

$i$	level	$L_k$ (cm-1)	$k$
1	1s	0	-
2	2s	82303	1
3	2p	”	2
4	3s	97544	3
5	3p	”	4
6	3d	”	5
7	4s	102879	6
8	4p	”	7
9	4d	”	8
10	4f	”	9
11	5s	105348	10
12	5p	”	11
13	5d	”	12
14	5f	”	13
15	5g	”	14

Table 2. Table 2: Lyman lines: Optical depths and scatterings

line	$\lambda$ (Å)	$f$	${\tau }_0$	$n\ell$	$ℬ$
Ly$\alpha$	1215.67	0.4164	1000	2p	1
Ly$\beta$	1025.73	0.07912	160	3p	0.881
Ly$\gamma$	972.54	0.02901	56	4p	0.839
Ly$\delta$	949.74	0.01394	26	5p	0.819

Table 3. Table 3: Photon yields for Ly α 𝛼 \alpha , H α 𝛼 \alpha , and 2 ​ γ 2 𝛾 2\gamma continuum

$k$	$p_k\left(\mathrm{Ly}\alpha \right)$	$p_k\left(H\alpha \right)$	$p_k\left(2\gamma \right)$
1	0	0	1
2	1	0	0
3	1	1	0
4	0	1	1
5	1	1	0
6	0.585	0.415	0.415
7	0.261	0.261	0.739
8	0.813	0.187	0.187
9	1	1	0
10	0.513	0.378	0.487
11	0.305	0.265	0.695
12	0.687	0.267	0.313
13	0.936	0.702	0.064
14	1	1

Table 4. Table 4: Polynomial fits to collision strengths a a Coefficients ( a i subscript 𝑎 𝑖 a_{i} ) for Maxwellian-averaged collision strengths, fitted by Ω ¯ k = ∑ i = 0 2 a i ​ x i subscript ¯ Ω 𝑘 superscript subscript 𝑖 0 2 subscript 𝑎 𝑖 superscript 𝑥 𝑖 \overline{\Omega}_{k}=\sum_{i=0}^{2}a_{i}x^{i} , where x = ln ​ ( T / 10 6 ​ K ) 𝑥 ln 𝑇 superscript 10 6 K x={\rm ln}(T/10^{6}\,{\rm K}) . Here, k 𝑘 k is the index of the transition (see Table  1 ).

$k$	trans	$a_0$	$a_1$	$a_2$
1	1s-2s	$0.5532$	$0.1044$	$0.0105$
2	1s-2p	$5.4261$	$2.2029$	$0.2481$
3	1s-3s	$0.1121$	$0.0131$	$0.0008$
4	1s-3p	$0.9355$	$0.3518$	$0.0382$
5	1s-3d	$0.1957$	$0.0517$	$0.0050$
6	1s-4s	$0.0390$	$-0.0005$	$-0.0008$
7	1s-4p	$0.3224$	$0.1124$	$0.0114$
8	1s-4d	$0.0944$	$0.0213$	$0.0016$
9	1s-4f	$0.0117$	$0.0011$	$0.0002$
10	1s-5s	$0.0175$	$-0.0019$	$-0.0004$
11	1s-5p	$0.1464$	$0.0501$	$0.0055$
12	1s-5d	$0.0471$	$0.0094$	$0.0008$
13	1s-5f	$0.0108$	$0.0003$	$-0.0000$
14	1s-5g	$0.0005$	$-0.0004$	$0.0001$

Table 5. Table 5: Fits to Cooling and Collisional Coefficients

Quantity	$A$	$a_0$	$a_2$	$a_3$	$a_4$
${\Lambda }_{\mathrm{HI}}$:hot	$6.0\times {10}^{-19}$	$1.018$	$-0.771$	$1.537$	$-0.716$
${\Lambda }_{\mathrm{HI}}$:warm	$6.0\times {10}^{-19}$	$1.032$	$-1.138$	$3.376$
$Q_{\mathrm{HI}}$:hot	$1.0\times {10}^{-7}$	$0.371$	$-0.304$	$0.560$	$-0.255$
$Q_{\mathrm{HI}}$:warm	$1.0\times {10}^{-7}$	$0.376$	$-0.433$	$1.220$

Table 6. Table 6: Efficiency of photon per collision

phot	$a_0$	$a_1$	$a_2$
H$\alpha$	$0.031$	$0.302$	$-0.149$
2$\gamma$	$0.377$	$-0.095$
Ly$\alpha$	$0.623$	$0.095$

Table 7. Table 7: Fits to α 𝛼 \alpha and f 𝑓 f

qty	$A$	$n$	$b$
${\alpha }_1$	$1.58\times {10}^{-13}$	$-0.518$	$-0.039$
${\alpha }_A$	$4.16\times {10}^{-13}$	$-0.708$	$-0.030$
${\alpha }_B$	$2.58\times {10}^{-13}$	$-0.822$	$-0.045$
$f_{\mathrm{rr}}^A$	0.784	$-0.042$	$-0.020$
$f_{\mathrm{rr}}^B$	0.672	$-0.109$	$-0.021$
$f_{\mathrm{rf}}^A$	1.09	$0.035$	$0.019$
$f_{\mathrm{rf}}^B$	1.17	$0.087$	$0.061$

Equations

$$q_{i\to j}$$

$$=$$

$${\overline{\Omega }}_{ij}(T)=a_1+a_{2}x+a_3{x}^2,$$

$${\Lambda }_{\mathrm{HI}}(T)=\sum _{k=1}^{14}{q}_k(T)E_k,$$

$${\Lambda }_X(T)=\sum _{k=1}^{14}{q}_k(T)p_k(X)E_X,$$

$$Q(T)=\sum _{i=1}^{14}{q}_k(T).$$

$$Q_{\mathrm{HI}}(T)=A\exp (-T/T_{12})\sum _{i=0}^n{a}_i{z}^i,$$

$$f_{\mathrm{H}\alpha }=\sum _{k=0}^2{a}_k{z}^k,$$

$$k_{ci}(T)={\int }_{I_{\mathrm{H}}}^{\infty }{\sigma }_{ci}(E)vf(E)dE,$$

$$k_{ci}(T)=A(T)\exp (-I_{\mathrm{H}}/k_{B}T).$$

$$k_{ci}(T)=5.85\times 10^{-11}{T}^{\text{nicefrac}12}\exp (-I_{\mathrm{H}}/k_{B}T)\mathrm{c}{\mathrm{m}}^3{\mathrm{s}}^{-1}.$$

$$A(T)=\exp\Big{(}\sum_{i=0}^{6}a_{i}y^{i}\Big{)}\ {\rm cm^{3}\,s^{-1}}$$

$$k_{ci}(T)=\frac{5.85\times 10^{-11}T^{\nicefrac{{1}}{{2}}}}{\big{[}1+0.1k_{B}T/I_{\rm H}\big{]}}\exp\Big{(}-\frac{I_{\rm H}}{k_{B}T}\Big{)}\,{\rm cm^{3}\,s^{-1}}\;.$$

$${\Lambda }_{\mathrm{rf}}={\Lambda }_{\mathrm{rr}}+{\Lambda }_{\mathrm{ff}}=\alpha f_{\mathrm{rf}}(T)k_{B}T.$$

$$\mathcal{C}(T)=-n_{e}n_{\rm HI}\big{[}\Lambda_{\rm HI}(T)+k_{ci}(T)I_{\rm H}\big{]}-n_{e}n_{p}\Lambda_{\rm rf}\;.$$

$$T_{s}=\frac{2(\gamma-1)}{(\gamma+1)^{2}}\frac{\mu v_{s}^{2}}{k_{B}}=(1.12\times 10^{5}~{}{\rm K})\frac{\mu}{m_{\rm H}}\Big{(}\frac{v_{s}}{70\,{\rm km\,s^{-1}}}\Big{)}^{2}\ \;.$$

$$t_{\rm loss}=14\ \Big{(}\frac{T}{10^{5}\,{\rm K}}\Big{)}^{3/2}\Big{(}\frac{\rm cm^{-3}}{n_{e}}\Big{)}\Big{(}\frac{25}{{\rm ln}\Lambda}\Big{)}\ {\rm yr}\;,$$

$$\frac{d{n}_e}{dt}=n_e{n}_{\mathrm{HI}}{k}_{ci}(T)-n_e{n}_p\alpha (T),$$

$$x(t)^{-1}=x_{0}^{-1}\exp(-t/\tau_{ci})+x_{eq}^{-1}\Big{[}1-\exp(-t/\tau_{ci})\Big{]}\;.$$

$$x_{eq}(T)=\frac{k_{ci}(T)}{\alpha +k_{ci}(T)}=\frac{{\tau }_r}{{\tau }_r+{\tau }_{ci}}.$$

$$\frac{1}{x}-\frac{1}{x_0}=\frac{t}{{\tau }_r}.$$

$$q\frac{d}{dt}(nk_{B}T)=-n_{e}n_{\rm HI}\big{[}\Lambda_{\rm HI}+k_{ci}I_{\rm H}\big{]}-n_{e}^{2}\alpha_{B}f_{\rm rf}k_{B}T\;,$$

$$\frac{dn}{dt}=n_e{n}_{\mathrm{HI}}{k}_{ci}-n_e^2{\alpha }_B.$$

$$\begin{split}qnk_{B}\frac{dT}{dt}=-n_{e}n_{\rm HI}\big{[}\Lambda_{\rm HI}+k_{ci}I_{\rm H}+k_{ci}qk_{B}T\big{]}\\ -n_{e}^{2}\alpha_{B}\big{(}f_{\rm rf}-q\big{)}k_{B}T\;,\end{split}$$

$$\begin{split}qnk_{B}\frac{dT}{dt}=-n_{e}n_{\rm HI}\big{[}\Lambda_{\rm HI}+k_{ci}I_{\rm H}\big{]}-n_{e}^{2}\alpha_{B}f_{\rm rf}k_{B}T\\ +qk_{B}T\big{[}n_{e}^{2}\alpha_{B}-n_{e}n_{\rm HI}k_{ci}\big{]}\;.\end{split}$$

$$N_r=\int \frac{1}{n_{\mathrm{H}}}{n}_{\mathrm{H}}^2{x}^2{\alpha }_B(T)dt,$$

$$N_c=\int n_{\mathrm{H}}x(1-x)\sum _{k=1}^{14}{q}_k(T)dt.$$

$$\frac{{\alpha }_{2s}}{{\alpha }_B}\approx 0.328T_4^{0.115},\frac{{\alpha }_{\mathrm{H}\alpha }}{{\alpha }_B}\approx 0.450T_4^{-0.11}.$$

$$\Lambda_{\rm H}^{\rm SW}(T)=10^{-20}\times\exp\Big{(}\sum_{i=0}^{5}d_{i}y^{i}-\frac{T_{12}}{T}\Big{)}\ {\rm erg\,cm^{3}\,s^{-1}}$$

$$q_{1s\to f}(T)={\Gamma }_{1s\to f}(T)\exp (-T_{12}/T)$$

$$\Gamma_{1s\rightarrow f}(T)=\exp\Big{(}\sum_{i=0}^{5}b_{i}y^{i}\Big{)}\,{\rm cm^{3}\,s^{-1}}\;.$$

Full text

1 Motivation

Supernova remnants and stellar wind bubbles are iconic examples of shocks in the interstellar medium (ISM). These shocks, with the passage of time, descend to lower velocities. Our interest here is shocks with velocities less than $70\,{\rm km\,s^{-1}}$. The post-shock temperature depends on the mean molecular mass, but we adopt a fiducial value of $T_{s}\leq 10^{5}$ K and investigate the cooling of such shock-heated hydrogen gas. These shocks cool primarily via Ly$\alpha$ (whose photons are trapped within the shocked region and eventually die on a dust particle) and two-photon continuum. The latter can be detected by Far Ultra-Violet (FUV) imagers. Low-velocity shocks can also arise on Galactic length scales: intermediate-velocity and high-velocity clouds raining down from the lower halo into the disk and gas that is shocked as it enters a spiral arm. Vallée (2017) provides a good description of the Milky Way’s spiral arms, and Kim et al. (2008) discuss Galactic interstellar shocks.

Stellar bow shocks are another major source of low-velocity shocks. For instance, consider our own Sun, a generic G5V star with a weak stellar wind ($2\times 10^{-14}\,M_{\odot}\,{\rm yr^{-1}}$) moving into a warm ($\sim 7,000\,$K) and partially ionized cloud (ionization fraction, $x\approx 1/3$) at a relative speed of 23–26 km s*-1* (Frisch et al., 2011; McComas et al., 2012; Zank et al., 2013; Gry & Jenkins, 2014). Because this velocity is not larger than the magnetosonic velocity of the interstellar cloud, there is only a “bow wake” instead of a bow shock (McComas et al., 2012). In the Galactic disk, interstellar space is occupied by the Warm Neutral Medium (WNM; $10^{3}$ K to $8\times 10^{3}$ K), the Warm Ionized Medium (WIM; $8\times 10^{3}$ K), and the Hot Ionized Medium (HIM; $10^{5}$ K to $10^{6}$ K), in roughly equal proportions.

From studies with SDSS-Apogee + Gaia-DR2 (Anguiano et al., 2020), the 3D velocity dispersion of the typical ($\alpha$-abundance tagged) thin-disk star is 48 km s*-1*, whereas those belonging to the thick disk have dispersion of 87 km s*-1*. The majority of these local stars reside in the thin disk with a density ratio $n_{\rm thin}/n_{\rm thick}=2.1\pm 0.2$. As discussed in a previous study (Shull & Kulkarni 2023), a sizeable number of stars should be moving supersonically through ambient gas in the WNM and WIM.111Only a few stars are likely transiting the Cold Neutral Medium (CNM; 100 K), given its small volume filling factor, $\sim 1\%$. The sizes of the resulting bow shocks will be determined by the stellar velocity and the magnitude of the stellar wind.

Separately, recent developments warrant a closer look at low-velocity shocks. We draw attention to the discoveries of three large-diameter supernova remnants (Fesen et al., 2021) and a 30-degree long, thin arc in Ursa Major (Bracco et al., 2020). In large part, these findings were made possible with a new diagnostic – GALEX FUV continuum imaging. The detection of such faint, extended features demonstrates simultaneously the value of the dark FUV sky (O’Connell, 1987) as well as the value of the FUV band in detecting two-photon emission, a distinct diagnostic of warm ($T\lesssim 10^{5}\,$K) shocked gas (Kulkarni, 2022).

The primary goal of this paper is to develop accurate hydrogen plasma cooling models, paying attention to the production of the two-photon continuum in warm plasma, $T\lesssim 10^{5}\,$K, the temperature range of interest to low velocity shocks. To this end, we first derive the probability of Ly$\alpha$, two-photon continuum, and H$\alpha$ resulting from excitation of the ground state of hydrogen to all $n\ell$ levels for $n\leq 5$ (§2). Next, we review rate coefficients for line excitation by collisions with electrons (§3), followed by a review of collisional ionization (§4). The results are combined to construct a cooling curve for warm hydrogen plasma (§5). We then present a comprehensive (isobaric and isochoric) cooling framework and apply it to gas shock heated to $10^{5}\,$K (§6). In §7 we summarize our results and discuss future prospects. Unless otherwise stated, the atomic line data (A-coefficients, term values) were obtained from the NIST Atomic Spectra Database222https://physics.nist.gov/PhysRefData/ASD/lines_form.html and basic formulae are from Draine (2011).

2 Two-photon production

Colliding electrons excite hydrogen atoms to various levels and, if sufficiently energetic, ionize H I to H II. Excited levels are also populated by radiative recombination. Excited hydrogen atoms return to the ground state, some by emitting a Lyman-series photon and others via a cascade of optical/IR recombination lines and ending with Ly$\alpha$ emission. Atoms that find themselves in the metastable 2s ${}^{2}{\rm S}_{1/2}$ level, if undisturbed over a timescale of $A_{2s\rightarrow 1s}^{-1}\approx 0.12\,$s, return to the ground state by emitting a two-photon continuum. Here, $A_{2s\rightarrow 1s}$ is the Einstein A-coefficient for the 2s-1s transition (Drake, 1986). Its value should be compared to those for allowed transitions (e.g., $6.26\times 10^{8}\,{\rm s}^{-1}$ for Ly$\alpha$ and$(1-5)\times 10^{7}\,{\rm s^{-1}}$ for H$\alpha$, depending on the upper levels, 3s, 3p, 3d, involved.)

The goal of this section is to compute the production of Ly$\alpha$ photons, two-photon continuum and H$\alpha$ resulting from electronic excitation of H atoms. We consider excitations to 15 $n\ell$ levels; see Table 1 for term values and index scheme. We make the following assumptions. (1) The proton density in the plasma is less than the “2s critical density” of $1.5\times 10^{4}\,{\rm cm^{-3}}$ (see Chapter 14 of Draine 2011). This ensures that atoms in the 2s level are not collisionally mixed to the 2p level over a timescale of $A_{2s\rightarrow 1s}^{-1}$ and thus relax by emitting a two-photon continuum. (2) The cooling plasma is optically thick to Lyman lines (case B), so that Lyman photons are absorbed in the vicinity of where they are emitted. Thus, when computing branching ratios, all allowed Lyman series recombinations can be ignored.

2.1 Photon Yields

Consider, for example, an atom excited to one of the $n=3$ levels. An atom excited to 3s or 3d will decay to 2p by emitting H$\alpha$ followed by Ly$\alpha$. (We ignore forbidden transitions such as ns-1s two-photon decays; see Chluba & Sunyaev 2008.) An atom excited to 3p can decay by emitting Ly$\beta$ or decay to 2s by emitting H$\alpha$ followed by two-photon decay. For the latter, the branching fraction $\mathcal{B}_{\beta}$ for Ly$\beta$ emission is $A_{3p\rightarrow 1s}/(A_{3p\rightarrow 1s}+A_{3p\rightarrow 2s})\approx 88\%$. However, under case B, the Ly$\beta$ photon will be absorbed elsewhere in the nebula, and the situation will be repeated until de-excitation ends with emission of H$\alpha$+Ly$\alpha$.

For a fiducial value of optical depth ($\tau_{0,\alpha}=1000$) of Ly$\alpha$, Table 2 lists the corresponding optical depths for the Lyman series. The branching ratio $\mathcal{B}_{\gamma}$ to emit a Ly$\gamma$ line is slightly smaller that that for Ly$\beta$. As with Ly$\beta$ under case-B conditions, Ly$\gamma$ will also be converted to some combination of Ly$\alpha$, optical/IR recombination lines, and a two-photon continuum. The oscillator strength, $f\propto n^{-3}$, where $n$ is the principal quantum number of the excited state. Thus the Lyman-line optical depths decrease rapidly with increasing $n$ (up the series). In contrast, the branching factors $\mathcal{B}$ decrease slowly with $n$.

Each state other than 4s and 5s has two fine-structure levels. For example, the 4p state has two levels, $P_{1/2}$ and $P_{3/2}$, with very little energy difference between the fine structure levels. However, the electron collisional excitation rate coefficients presented below (§3) refer to the sum of transitions to the entire level, e.g., 1s$\rightarrow$4p. The excitation coefficient is divided in proportion to the number of levels of the excited state, $g_{u}=2J+1$ where $J$ is the total angular momentum of the excited state. The photon yields for Ly$\alpha$, 2$\gamma$ continuum, and H$\alpha$ are given in Table 2.1.