Two-photon Production in Low-velocity Shocks

The Galactic interstellar medium abounds in shocks with low velocities v s ≲ 70 km s−1. 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 Lyα and two-photon continuum, augmented by optical recombination lines (e.g., Hα), forbidden lines of metals and free-bound emission, free–free emission. The dark far-ultraviolet (FUV) sky, aided by the fact that the two-photon continuum peaks at 1400 Å, makes the FUV band an ideal tracer of low-velocity shocks. 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 bests atomic data available to date, we present convenient fitting formulae for yields of Lyα, two-photon continuum, and Hα for pure hydrogen plasma in the temperature range of 104–105 K. The formulae presented here can be readily incorporated into time-dependent cooling models as well as collisional ionization equilibrium models.

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 km s −1 .The post-shock temperature depends on the mean molecular mass, but we adopt a fiducial value of T s ≤ 10 5 K and investigate the cooling of such shock-heated hydrogen gas.These shocks cool primarily via Lyα (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: intermediatevelocity 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.
From studies with SDSS-Apogee + Gaia-DR2 (Anguiano et al. 2020), the 3D velocity dispersion of the typical (α-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 thin /n thick = 2.1 ± 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 arXiv:2302.13963v1[astro-ph.GA] 27 Feb 2023 and WIM. 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 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 10 5 K, the temperature range of interest to low velocity shocks.To this end, we first derive the probability of Lyα, two-photon continuum, and Hα resulting from excitation of the ground state of hydrogen to all n levels for n ≤ 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 Database 2 and basic formulae are from Draine (2011).

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 Lyα emission.Atoms that find themselves in the metastable 2s 2 S 1/2 level, if undisturbed over a timescale of A −1 2s→1s ≈ 0.12 s, return to the ground state by emitting a two-photon continuum.
Here, A 2s→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 × 10 8 s −1 for 1 Only a few stars are likely transiting the Cold Neutral Medium (CNM; 100 K), given its small volume filling factor, ∼ 1%. 2 https://physics.nist.gov/PhysRefData/ASD/linesform.html" 14 Note-In constructing this table we follow the notation and term values of Anderson et al. (2000), where i is the index assigned to levels, and k is the index for upper levels excited in transitions from the ground state (1s).The energy for transition k is hcL k where L k is the wavenumber (in cm −1 ).The symbol '"' is equivalent to "ditto".As can be gathered from the entries for L k , the small differences in energy due to fine structure effects are ignored.
Lyα and(1−5)×10 7 s −1 for Hα, depending on the upper levels, 3s, 3p, 3d, involved.)The goal of this section is to compute the production of Lyα photons, two-photon continuum and Hα resulting from electronic excitation of H atoms.We consider excitations to 15 n 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 × 10 4 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 −1 2s→1s and thus relax by emitting a twophoton 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.

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 Hα followed by Lyα.(We ignore forbidden transitions such as ns-1s two-photon decays; see Chluba & Sunyaev 2008.)An atom excited to 3p can decay by emitting Lyβ or decay to 2s by emitting Hα followed by two-photon decay.For the latter, the branching fraction B β for Lyβ emission is A 3p→1s /(A 3p→1s + A 3p→2s ) ≈ 88%.However, under case B, the Lyβ photon will be absorbed elsewhere in the nebula, and the situation will be repeated until de-excitation ends with emission of Hα+Lyα.For a fiducial value of optical depth (τ 0,α = 1000) of Lyα, Table 2 lists the corresponding optical depths for the Lyman series.The branching ratio B γ to emit a Lyγ line is slightly smaller that that for Lyβ.As with Lyβ under case-B conditions, Lyγ will also be converted to some combination of Lyα, optical/IR recombination lines, and a two-photon continuum.The oscillator strength, f ∝ 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 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→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 Lyα, 2γ continuum, and Hα are given in Table 3.For instance, an H atom excited to 3s (k = 3) relaxes by emitting one Hα photon and one Lyα photon.

ELECTRON COLLISIONAL EXCITATION
The excitation of lines of hydrogen due to collisions with electrons is a venerable topic in ISM studies.The classic review by Dalgarno & McCray (1972) summarizes the atomic physics of the 1960s.Scholz & Walters (1991) undertook detailed calculations of the n = 1 → 2 excitations and also provided an estimate for the cooling rate coefficient, Λ HI .Anderson et al. (2002; see also Anderson et al. 2000) present close-coupling Rmatrix calculations.We adopt these rates since they offer improved accuracy over previous studies (Scholz et al. 1990).The Anderson et al. (2002) theoretical cross sections were constructed with 15 physical energy states , where x = ln(T /10 6 K).Here, k is the index of the transition (see Table 1).
up to n = 5 (1s to 5g) supplemented by 24 pseudo-states described by orbitals (n, ) with n = 6 − 9 and = 0 − 5. Anderson et al. (2002) present collision strengths, Ω ij , for excitation from levels i to j, averaged over a Maxwellian velocity distribution at electron temperatures, E T ≡ k B T , ranging from 0.5-25 eV.The collisional excitation rate coefficients (in cm3 s −1 ) are then given by: where a 0 is the Bohr radius, α is the fine structure constant, g i is the degeneracy of level i, and E ij is the energy difference between level i and j.Since we are only interested in excitations from the ground state, we assume g i = 2 for 1s ( 2 S 1/2 ).Given our focus on warm plasma, we limit the model fits to 1 eV ≤ E T ≤ 15 eV.After some experimentation, we found that a cubic polynomial provides an adequate fit 3 : where x = ln(T /10 6 K).The fit is precise to about 1% for all levels except 5 levels, for which the fitting errors  4).
approach 5% (see Figure 1).The fitting coefficients can be found in Table 4.
The line cooling rate per unit volume is given by n e n HI Λ HI where n e = n p is the electron (and proton) density and n HI is the density of H atoms.The total particle density is n t = n HI + n e + n p = n H (1 + x) with n H = n p + n HI and x = n e /n H .
Figure 2. Electron collisional excitation rate coefficients, qij, for 1s→ (n, ) transitions of hydrogen derived from collision strengths provided by Anderson et al. (2002).The curves are coded by color (n = 2, 3, 4, 5 as labeled) and by line type (dash-dash 1s-ns, continuous for 1s-np, dash-dot for 1s-nd, dotted for 1s-nf, and back to dash-dash for 1s-ng.) We used the fitting model to compute the run of collisional rate coefficients, q i→j , with temperature (Figure 2).With the cooling coefficients in hand, we computed the sum of the luminosity radiated in lines up to n = 5.We consider this sum to be an adequate representation of Λ HI (T ) for warm hydrogen.The cooling rate coefficient is where the energy of transition with index k is E k = hcL k ; see Table 1 for definition of k and the adopted values for the wavenumbers, L k (in cm −1 ).Separately, in §A, we compare this cooling coefficient to previously published coefficients (Spitzer 1978;Scholz & Walters 1991;Dere et al. 1997).
The coefficient for energy loss through line X (where, for instance, X denotes Lyα, Hα, 2γ) is given by where E X is line energy and p k (X) is given in Table 3.

Simple Fits to Line cooling and Collision rates
The collisional excitation rate coefficient is the sum over all hydrogen levels, Both Q(T ) and the hydrogen cooling rate (from excitation to n = 2) fall off with temperature as exp(−T /T 12 ), where k B T 12 = 3I H /4 is the energy difference between n = 1 and n = 2 levels.We fit the collision rate and Λ HI over two temperature ranges: "hot" (10 4 K < T < 1.5 × 10 5 K) and "warm" (10 4 K < T < 1.5 × 10 4 K), where z = log T 4 with T 4 = (T /10 4 K).A similar expression was derived for Λ HI (T ).The fitting parameters for Q HI and Λ HI are given in Table 5, and the quality of the fit is displayed in Figure 3. Note-"Quantity" refers to the cooling coefficient (ΛHI in erg cm 3 s −1 ) or collisional excitation rate coefficient (QHI in cm 3 s −1 ).These quantities are fitted to a model displayed in Equation 4 over two temperature ranges: "hot" (10 4 K < T < 1.5 × 10 5 K) and "warm" (10 4 K < T < 1.5 × 10 4 K.) In Figure 4 we plot the line production efficiency4 .Consistent with the collisional coefficients displayed in Figure 2 we see that Lyα has the highest efficiency, approximately 2/3, followed by 2-photon emission, 1/3.Hα is quite weak, even when measured by photons emitted.Hα emission requires excitation to the n = 3 level, whereas two-photon and Lyα emission are obtained by excitation to n = 2 (and cascade from higher states).However, Hα has a major advantage -it can be observed with existing ground-based observatories.For this reason, we provide a fitting formula for y Hα , the fraction of Hα photons per ionization, where z = log T 4 , as before.The model fit for Hα is shown in Figure 5, and the values for the model coefficients are given in Table 6, as well as those for Lyα and two-photon emission.Note-Electron collisions with hydrogen atoms produce Lyα, twophoton continuum (2γ), Hα, and other lines.For each of these categories, the efficiency of photon production, f , depends on temperature and is fitted to a model given by Equation 5.The model fits are accurate to 2%.

ELECTRON COLLISIONAL IONIZATION
The collisional ionization rate coefficient is derived as the integral, where f (E) is the Maxwellian energy distribution, I H = 13.598eV is the ionization energy of hydrogen, and σ ci (E) is the collisional ionization cross section as a function of electron energy in the center-of-mass frame, E = 1 /2µv 2 with µ = m e m H /(m e + m H ) ≈ m e .Here, m e and m H are the mass of the electron and hydrogen atom, respectively.The collisional ionization rate coefficient can be sensibly written as (6) Black (1981) provided an approximate form for k ci T ), based on ionization cross sections tabulated by Lotz (1967), This expression is consistent with the approximation, σ ci ∝ (1−I H /E), quoted in Draine ( 2011), but only valid at low collision energies (I H ≤ E ≤ 3I H ). As shown by Lotz (1967), the high-energy behavior is σ ci ∝ ln E/E.Scholz & Walters (1991) provided a better approximation (for 10 4 K to 2 × 10 5 K) using a sixth-order polynomial, where y = ln T .A comparison (Figure 6) between Equation 7 (Black 1981 fit) and the more accurate Scholz &  Walters fit (Equation 8) shows that the former breaks down at high temperatures.We offer a modified formula with the correct asymptotic behavior (k B T > I H ) as used in the shock models of Shull & McKee (1979), As can be seen from Figure 6, this modified formula provides a good fit at both low and high temperatures.For quick estimates we use Equation 9, but the polynomial formulation of Scholz & Walters (1991) is preferred when precision is needed (e.g., in numerical integration of differential equations).The resulting ionization power loss per unit volume is n e n HI Λ ci where Λ ci = k ci I H is the collisional ionization energy loss coefficient.

THE HYDROGEN COOLING CURVE
The goal in this section is to construct a cooling curve for "warm" (T 10 5 K) hydrogen plasma.In §3 we formulated the cooling coefficient due to line cooling, while in §4 we presented the same for ionization losses.
Here, we summarize the cooling coefficients for radiative recombination (free-bound) and free-free losses.Armed thus, we formulate the cooling curve for hydrogen in the temperature range 10 4 K to 10 5 K.
The kinetic energy of recombining electrons is a loss to the thermal pool.The model fits for α k where k = 1, A, B (corresponding to recombinations to n = 1s level, case A and case B) can be found in Table 7 ( §B).The radiative recombination power loss per unit volume is n e n p αΛ rr where α is either case A or case B, as appropriate.Here, the radiative recombination energy rate coefficient Λ rr = α E rr where E rr is the mean thermal energy lost by electron upon recombination.Following Draine (2011) we let E rr = f rr k B T .
The free-free emission rate per unit volume is n e n p Λ ff , where Λ ff is the free-free emissivity.The free-free power per electron is n p Λ ff .The mean time for an electron to recombine is p α) −1 .Thus, the free-free energy lost up to the point of recombination is Λ ff /α, which we equate to f ff k B T .The combined recombination and free-free cooling rate coefficient is then where f rf = f rr +f ff .The run of f rf (T ) with temperature is displayed in Figure 15 ( §B), and the model fits are presented in Table 7 ( §B).
We now have all the elements to formulate the cooling rate per unit volume, C(T ), expressed as a negative value (for energy losses): The three RHS terms are given by Equations 3, 6, and 10, respectively.

LOW-VELOCITY SHOCKS: A SIMPLE COOLING MODEL
The investigation of time-dependent cooling of gas heated to T ≈ 10 5 K is a classic endeavor, constituting the Ph.D. thesis topics of Michael Jura (Jura & Dalgarno 1972) and Minas Kafatos (Kafatos 1973).The motivation in the 1970s seems to have been ambient gas heated by an FUV shock-breakout pulse.Separately, Draine & Salpeter (1978) investigated the production of Lyα from SN shocks, and Shull & Silk (1979) computed UV emission from SNRs in primeval galaxies.
In unmagnetized plasma, the post-shock temperature of an adiabatic shock is given by Here, µ is the mean mass per particle and γ is the ratio of specific heats at constant pressure and constant volume; γ = 5/3 for mono-atomic gas.If y is the number density of helium relative to that of hydrogen, the mean molecular mass for H 0 and He 0 is µ = [(1 + 4y)/(1 + y)]m H = 1.23mH for y = 0.0819 (Planck Collaboration et al. 2020).For H + and He 0 , µ = 0.64m H .For H + and He + , µ = 0.61m H , and for H + and He +2 , µ = 0.59m H .
Three timescales come into play for post-shocked gas: τ r , the recombination timescale; τ ci , the collisional ionization timescale; and τ c , the cooling timescale.For gas around 10 5 K, we have τ ci τ r .With this inequality, the cooling gas does not obey the conditions for collisional ionization equilibrium.Thus, it is often essential to undertake a full time-dependent calculation.

Electron-Proton Equilibration
At the collisionless shock front, the electrons and protons receive similar amounts of random motion.Being more massive, the protons acquire more energy and are initially much hotter than the electrons.The equilibration timescale for electrons to be heated up to the temperature of the protons via electron-proton encounters is approximately where lnΛ is the Coulomb logarithmic factor accounting for distant encounters (Spitzer 1978; Chapter 2).The current view (Laming et al. 1996;Ghavamian et al. 2007) is that plasma instabilities and electromagnetic waves drive electron-proton equilibration faster than two-body interactions.We assume that equipartition occurs before collisional ionization sets in ( §6.2).

Collisional Ionization
The rate equation for the number density of electrons is where α = α A , α B as needed.Given our assumpton of hydrogen plasma, the number density of protons, n p = n e .As noted in §C a solution to this equation at cosnatnt T is Here, x 0 = x(t = 0), τ r ≡ (αn H ) −1 and τ ci ≡ (k ci n H ) −1 are the characteristic time scales for recombination and collisional ionization, respectively, and The lowest probable value for the ionization fraction in a realistic diffuse atomic medium, before heating commences, is x 0 ≈ 2 × 10 −4 .In this case the electrons come from stellar FUV photoionization of C, S, Mg, Si, Fe and other trace metals.The timescale for electron ionization to reach fraction x is τ ci ln(x/x 0 ).

Recombination
As can be seen from Figure 7, collisional ionization is a strong function of temperature.At late times, when the plasma has cooled, collisional ionization can be ignored and Equation 13simplifies to dx/dt = −x 2 /τ r , with the solution 1 x − 1 x 0 = t τ r .9) and recombination time (τr; case B) as a function of temperature.
The two timescales cross at about 15,000 K, at which point the ionization fraction would be 50%, if collisional ionization equilibrium were to hold (see Equation 14).
The ionization fraction decreases from x 0 to x 0 /m on a timescale of t = (m − 1)τ r /x 0 .The run of recombination time scale as a function of temperature is shown in Figure 7.

Basic Cooling and Recombining Framework
The path in the phase diagram of density, ionization fraction, and temperature along which the gas cools depends on the cirucmstances.For planar radiative shocks, the pressure behind the shock is P 0 + (3/4)ρ 0 v 2 0 , rising to P 0 + ρ 0 v 2 0 downstream when ρ ρ 0 .Here, the pre-shock parameters have subscript 0. Thus, radiative shocks are good examples of cooling at nearly constant pressure ("isobaric").As the gas cools, its density rises to maintain the pressure.A second possibility is cooling at constant density ("isochoric").The decrease in temperature, following cooling, leads to lower pressure.Pressure changes are conveyed at the speed of sound, c s .The time scale for adiabatic sound waves to cross a nebula of length L is τ a = L/c s .Isochoric cooling will take place when the cooling time is short, τ c τ a .The first law of thermodynamics states that any gain in the internal energy (U ) of the system is due to increase in internal heat and work done: dU = dQ − P dV .For mono-atomic gas, the internal energy of the nebula per unit volume is U = (3/2)nk B T , while the pressure is given by Boyle's law P = nk B T .Here, N = nV is the total number of particles in the nebula whose volume is V .Let N H = n H V be the total number of hydrogen nuclei.Ionization can produce changes in N , whereas N H is fixed.
The three physical parameters governing the cooling hydrogen plasma are n e , T and n H .We have two differential equations, one for ionization balance (n e ; Equation 13) and one for for energy loss (T ; discussed below).A third differential equation follows from the assumed framework: dP/dt = 0 (isobaric cooling) or dV /dt = 0 (isochoric cooling).
For an isochoric system, no work is done by or on the nebula.Adopting case B framework the energy balance equations becomes where q = 3/2 and the RHS gives the total cooling rate.Note that n H remains constant, whereas n = n H + n e varies as the ionization fraction changes.The ionizationrecombination equation (Equation 13) can be restated We combine the above two equations to obtain In this formulation, the meaning of Equation 15is clear.
The LHS arises from the loss of internal energy.The first term on the RHS represents energy loss from H I collisional line excitation (Λ HI ) and collisional ionization (loss of I H per collision).The loss of kinetic energy per recombination (including the free-free radiation up until the recombination event) is given by the second term.
The final term accounts for losses/gains to the thermal pool of electrons during recombination and ionization.
For plasma in collisional ionization equilibrium this term vanishes, as expected.
For isobaric cooling, the pressure, P = (n H + n e )k B T is fixed.In this case, we compute n e and T and then deduce n H through the pressure equation.As the nebula cools, the ambient gas, in order to maintain the pressure, P a , does work on the nebula by compressing the nebula.The work done by the medium on the nebula is P dV .However, since P is constant, d(P V ) = P a dV .The internal energy of the nebula is then the enthalpy, HV where H = U + P a .Going forward, we will drop the subscript to P .It is this store of enthalpy that powers the nebular cooling, CV .Since (U + P )V = (5/2)N k B T we see that Equation 15 still applies but with q = 5/2.

A shock heated nebula
Consider a nebula composed of hydrogen which has been shocked heated to, say, T s = 10 5 K.The electronproton equilibration timescale will be shorter than 14n −1 yr (see §6.1).The collisional ionization time, t ci , is short, 10n −1 H yr at T = 10 5 K, rising to 60n −1 H yr at 50,000 K. Thus, even if the pre-shocked gas has minimal ionization (x 0 ≈ 2 × 10 −4 ) it will take a time, τ ci ln(x/x 0 ) = 7.8tci for the ionization fraction to reach x = 0.5.The initial losses are large, owing to both collisional ionization and collisional excitation by the newly liberated electrons and subsequent radiation.An exception is if the pre-shocked gas is pre-ionized.If the shock is strong, pre-ionization (H + and He + ) will be achieved, which diminishes the hydrogen Lyα emission.(There will still be cooling from He II Lyα λ304 and lines from metal ions.)As the gas cools, recombination becomes more efficient.Once the gas reaches 10 4 K, cooling by forbidden lines of metals will occur.The gas will eventually settle down at T 1 ≈ 5000-8000 K, the temperature of the stable WNM phase (see Heiles & Troland 2003;Kanekar et al. 2003;Patra et al. 2018;Murray et al. 2018).
Ignoring "metals", the mean particle mass is µ = m H (1 + 4y)/(1 + y + x 0 ).The shock velocity and µ determine the post-shock temperature, T s through Equation 12. Recall that, in our simplified model, the losses from the shocked nebula are only those associated with hydrogen (line radiation, ionization, free-bound, and free-free).In particular, while we include helium in computing the reduced mass, we do not include losses due to helium.In short, we treat helium as a silent and inactive partner.The energy per H-nucleus and associated electron is E 0 = qk B T s (1 + x 0 ).The end state is when hydrogen has largely recombined and thus the energy per H-atom is E 1 = qk B T 1 .
Example runs are shown in Figures 8 and 9.In addition to the run of physical quantities (T , x, n H ) we also plot the total number of recombinations per H nucleus, and the total number of collisions per atom q k (T )dt .

CONCLUSION & PROSPECTS
Low velocity shocks with velocities near 70 km s −1 abound in our Galaxy.Some descend from higher velocity shocks (e.g., supernova remnants) while others start at low velocity (e.g., stellar bow shocks, high velocity cloud shocks).These shocks do not have strong precursor ionization fronts, and as such the post-shocked gas is partially neutral.Such shocks cool primarily through Lyα, two-photon continuum, Hα, and metal emission lines.Lyα is the brightest line, although resonant scattering traps usually traps these photons within the plasma, resulting in absorption by dust grains.Hα is weak but has the great advantage of being observable from the ground.
Two-photon continuum emission is about 50% of Lyα emission (see Figure 4).It is several times brighter than Hα, even when one compares photon fluxes rather than energy fluxes.Fortunately, two-photon emission can be observed with space-based observatories.Furthermore, the two-photon continuum has a distinct FUV/NUV ratio.In fact, GALEX FUV and NUV imagery has led to the recent discovery of large middle-aged supernova remnants (Fesen et al. 2021) and exotic shocked stellar bow shocks with angular scales of hundreds of degrees (Bracco et al. 2020).The Ultraviolet Explorer (UVEX) is a NASA Explorer mission currently under development (Kulkarni et al. 2021).Amongst other goals, UVEX aims to undertake FUV and NUV imaging of the entire sky with higher sensitivity and finer spatial resolution, relative to GALEX.The aforementioned successes with GALEX imagery show great promise of identifying and studying low velocity shocks in a future all-sky survey with UVEX.
With this motivation, and using the best available atomic physics data and atomic calculations, we computed the collisional and cooling coefficients for warm hydrogen (T 10 5 K).The primary application of our results is in computing two-photon continuum from bow shocks and old supernova remnants.We allow for preionization by keeping the ionization fraction of the preshocked gas as a free parameter that can be set to values computed from more sophisticated shock models (ibid; Dopita & Sutherland 1996).Our expectation is that the accurate H-cooling developed here can be incorporated into time-dependent models (e.g., Gnat & Sternberg 2007).
For completeness, we discuss two-photon emission from photoionized gas (e.g., H II regions, the Warm Ionized Medium).Draine (2011;Table 14.2) provides the recombination coefficient to the 2s level, α 2s and the recombination coefficient for Hα emission.From this we find ratios Thus, at typical temperatures of photoionized gas, recombination process result in similar diffuse emission for two-photon continuum and Hα.However, while Hα emission is concentrated in a narrow line, the twophoton continuum is distributed over the FUV band.
Compensating for this effect, the FUV sky is incredibly dark relative to the optical band (see Kulkarni 2022 for detailed analysis of the FUV background).
In the Galactic plane and at low latitudes, two-photon emission will be attenuated by dust in the intervening neutral ISM and contaminated by reflected light from dust grains.In practice, this means that the use of twophoton continuum as a diagnostic will be restricted to high Galactic latitudes and will require careful modeling of reflected light.However, the early success with GALEX promises rich returns from the all-sky survey in both the FUV and NUV planned with UVEX.
small.This to Λ HI (T ) was claimed to be accurate to 3% over the range 4,000 K to 12,000 K.This formula does not include losses from collisional ionization.Spitzer's fit was actually made to H I cooling rates taken from Table 2 of Dalgarno & McCray (1972) which were based on H I excitation cross sections assembled by Gould & Thakur (1970) from theoretical calculations in the 1960s.Those atomic calculations are superseded by more recent work used in this paper (Anderson et al. 2002).In the left panel of Figure 11 we display the ratio of our cooling curves, Λ HI and Λ H = Λ HI + k ci I H to the cooling rate of Spitzer (Equation A3).Our fit is tuned to be accurate over the range 10 4 K to 10 5 K.Even bearing this in mind, it appears that Spitzer's formula over-estimates line cooling at temperatures above 10 4 K by about 25%.

A.3. Collisional Ionization Equilibrium: CHIANTI
We conclude this section by briefly discussing hydrogen plasma which is in collisional ionization equilibrium (CIE; electron ionization balanced by radiative recombination).The volumetric cooling rate in CIE is given by C = x eq (1 − x eq )n 2 H Λ CIE where x eq is given by Equation 14.It is mainly dominated by line cooling.CHIANTI (Dere et al. 1997;Del Zanna et al. 2021) is a major resource for astronomers working on collisionally excited gas, especially hot plasma which are in CIE.CHIANTI returns x eq and cooling function C ≡ n e n H Γ (under assumption of case A).Thus, Γ = (1 − x eq )Λ H .The run of Γ from with temperature is displayed in Figure 12.It is evident that Γ from CHIANTI is brighter relative to the calculations reported here at higher temperatures.There is a small difference As can be seen from Figure 13 our x eq agrees with that used in CHIANTI for T > 2 × 10 4 K.Standard textbooks (e.g., Draine 2011) provide fitting formulae for hydrogen (recombination, free-bound losses) tuned for study of H II regions.Here, we present fitting formula in the temperature range interest to this paper: 10 4 K to 10 5 K.Our starting point is Hummer (1994) who, over an impressive range of 10 K to 10 7 K, present recombinations rate coefficients, α i where i = 1 (recombination to n = 1) and i = A, B for case A and case B; see Figure 14.
Separately, Hummer (1994) also tabulate the kinetic energy loss due to recombination and free-free emission.In Figure 15 we plot the mean kinetic energy, E rr versus T ).Note that E rr = f rr k B T .The resulting fits are presented in Table 7 and are accurate to one percent.
For free-free emission rate coefficient, Λ ff we use Equation 10.12 of Draine (2011) and from that derive f ff (T ) (see §5 for definition of f rf ).The values of f rf = f rr + f ff for case A and case B are displayed in Figure 15 (left panel).We assume a simple linear relation and obtain the following fits for f rf : f rf = 0.71 + 0.0154T 4 (case A), f rf = 1.10 + 0.0898T 4 (case B) .
We conclude this section with a discussion of collisional ionization.The energy of the electron, E, upon collision goes into ionizing the atom and imparting kinetic energy to the newly liberated electron.We use the low-energy approximation (Equation 7) for the collisional cross-section, σ ci ∝ [1 − (E/I H )] (valid for I H ≤ E ≤ 3I H ) to find a mean kinetic energy following ionization, This formula becomes inaccurate at high temperature (kT 3I H ) where the adopted fit to σ ci breaks down (σ ci peaks and then falls off as ln E/E).The energy E k is shared between the colliding electron and the ionized electron.This energy is not a loss since it is returned to the thermal pool.However, over time, the ionized electron will draw qk B T energy (q = 3/2 in isochoric framework and q = 5/2 in isobaric framework) from the thermal pool.This is a genuine loss (see the discussion following Equation 15).where the temperature range is 5 × 10 3 K < T < 2 × 10 5 K; here, T4 = T /(10 4 K).See text for definition of subscripts.The superscripts stand for case A or case B. The unit for α is cm 3 s −1 while that for f is dimensionless.The fits are accurate to 1% over the temperature range of interest.is the harmonic mean of the two timescales.The equilibrium value for ionization is obtained by setting the LHS to zero, x eq = τ h /τ ci .The above equation can be re-arranged to yield where x = x/x eq .This equation can be integrated using the method of partial fractions to yield ln x 1 − x = t τ ci + const.
This equation is useful to compute the time-scale to achieve a particular level of ionization.As t → ∞, x → x eq , as expected.Alternatively, we apply the transformation, u = 1/x: Multiply both sides by e t/τci to obtain e t/τci du dt + u τ ci e t/τci = 1 τ h e t/τci .

Figure 1 .
Figure 1.The electron-impact collision strengths, Ωij, for 1s to (n, ) excitations of hydrogen as a function of temperature for n = 2, 3, 4, 5. Open circles are model data from Anderson et al. (2002), and the lines are second-order polynomial fits (see Equation 2 and Table4).

Figure 3 .
Figure 3. (Top Panel.)Left axis shows the line cooling coefficient, ΛHI(T ), for hydrogen (black line), where the total cooling rate is nenHIΛHI(T ).Right axis shows the percent residuals (red-dashed lines) in the form of [1−(ΛHI/fit)].The model form is given by Equation4, and model parameters are given in Table5.(Bottom Panel.)The same, but for Q, the collisional coefficient.The rate of collisions per unit volume is nenHIQ.

Figure 4 .
Figure 4. (Left) Case B photon production per collision as a function of temperature, T for Lyα, Hα and two-photon continuum.(Right) The ratio of yields of two-photon decays to that of Hα photons.

Figure 5 .
Figure 5. (Left axis) The fraction, fHα, of electron collisions with an H I atom that result in emission of an Hα photon as a function of temperature.A polynomial model for yHα is described in Equation 5.The ratio of this model to calculations is displayed by the dashed red line.

Figure 7 .
Figure 7. Ionization time (τci; see Equation9) and recombination time (τr; case B) as a function of temperature.The two timescales cross at about 15,000 K, at which point the ionization fraction would be 50%, if collisional ionization equilibrium were to hold (see Equation14).

Figure 8 .
Figure 8. Run of temperature and density of a pure hydrogen plasma suddenly heated to 10 5 K and subsequently cooling down via an isobaric process.The hydrogen density, nH and ionization, x0 at t = 0 are given in the legend in the top-most panel.

Figure 11 .
Figure 11.(Left).The ratio of Spitzer's H-line cooling function to our computations of ΛHI (line cooling) and ΛH (including collisional ionization cooling) as a function of temperature.(Right).Comparison of the total cooling curve (line and ionization losses) between our cooling curve and that from CHIANTI.

Figure 12 .
Figure 12. (Left): The run of Γ as obtained from CHIANTI and the calculations reported here (case A, CIE) with temperature T .Here, the ΓCIE is defined as follows: C = nenHΓCIE where it is understood that C and ne reflect CIE conditions.(Right):The ratio of Γ (this work) to that reported from CHIANTI.

Figure 13 .
Figure 13.(Left) The run of xeq under CIE conditions as a function of temperature, T .(Right) The ratio of xeq (this paper) to that provided by CHIANTI.

Figure 15 .
Figure 15.(Left): The run of the average kinetic energy of the recombining electron, Err , as a function of temperature.(Right):The run of f as a function of temperature (T ).The superscripts denote case A or case B. The subscript is "rr" (free-bound radiative recombination), "ff" (free-free emission), and f rf = f ff + frr.

Table 1 .
The indexing scheme and spectroscopic terms i level L k (cm −1 ) k

Table 4 .
Polynomial fits to collision strengths a a Coefficients (ai) for Maxwellian-averaged collision strengths, fitted by Ω k = 2 i=0 aix i

Table 5 .
Fits to Cooling and Collisional Coefficients

Table 6 .
Efficiency of photon per collision