Draft version July 21, 2021 Typeset using LATEX twocolumn style in AASTeX63

The far ultra-violet background

S. R. Kulkarni1

1Owens Valley Radio Observatory 249-17, Caltech Pasadena, CA 91125, USA

ABSTRACT The diffuse far-ultraviolet (FUV) background has received considerable attention from astronomers since the seventies. The initial impetus came from the hope of detecting UV radiation from the hot intergalactic medium. The central importance of the FUV background to the physics (heating and ionization) of the diffuse atomic phases motivated the next generation of experiments. The consensus view is that the diffuse FUV emission at high latitudes has three components: stellar FUV reflected by dust grains (diffuse galactic light or DGL), FUV from other galaxies (extra-galactic background light, EBL) and a component of unknown origin. During the eighties, there was some discussion that decaying dark matter particles produced FUV radiation. In this paper I investigate production of FUV photons by conventional sources: the Galactic Hot Ionized Medium (line emission), two photon emission from the Galactic Warm Ionized Medium and low-velocity shocks, and Lyman-β excitation of hydrogen at several locales in the Solar System (the interplanetary medium, the exosphere and thermosphere of Earth). I conclude that two thirds of the third component can be explained by the sum of the processes listed above. 1. BACKGROUND et al. 2005): FUV (1350–1750 A)˚ and NUV (1750– ˚ The diffuse background in the range 912–2000 A˚ is of 2800 A). The GALEX FUV band was specifically de- central importance to the physics of the diffuse atomic signed to avoid the two strongest foreground lines in the ˚ phases – the Cold Neutral Medium (CNM) and the FUV band: the geo-coronal Lyα and the 1302–1306 A Warm Neutral Medium (WNM). Historically, the var- O I airglow triplet, both of which are seen even during ious bands of ultra-violet (UV) were defined by wave- the dead of the satellite night. length properties of detectors and mirror coatings. For The diffuse FUV radiation, via photo-electric ioniza- tion of C I and photo-electric heating of dust particles, instance, mirrors coated with MgF2 lose reflectivity be- low 1150 A.˚ With advances in coating technology, along heats the two atomic phases mentioned above. The same with the rapid progress in detector technology, the radiation, via ionization of elements with ionization po- boundaries between the bands started to blur. The In- tential less than that of hydrogen, also provides residual ternational Ultraviolet Observatory (IUE) had two spec- ionization to the two atomic phases. The FUV back- trographs: the short wavelength spectrograph, covering ground is of importance for the molecular medium also. the range 1150–2000 A˚ and the long wavelength spectro- FUV photons excite the Lyman and Werner bands of H2 graph, covering 1850–3300 A.˚ The Far Ultraviolet Spec- (Duley & Williams 1980) which result in UV fluorescent troscopic Explorer (FUSE) focused on the spectral re- line emission and once in ten times leads to dissociations gion between Lyman cutoff and Lyman-α, 905–1195A.˚ (Jura 1974). In the process, the strength of FUV radia- arXiv:2107.09585v1 [astro-ph.GA] 20 Jul 2021 FIMS aboard S. Korea’s STSat-1 mission was designed tion determines the transition in diffuse clouds between to explore the far ultraviolet sky, defined by the wave- atomic and molecular phases. length range, 900–1750 A.˚ Separately, scattering of solar photons by zodiacal For this paper we use the Far Ultraviolet (FUV) and dust particles constitute an irreducible background for the Near UV (NUV) bands as defined by the passbands space missions located in the inner Solar system, e.g., of the GALaxy Evolution eXplorer (GALEX; Martin the Hubble Space Telescope (HST) in low earth orbit (LEO) or the planned Russian Spektr-UF mission (aka “World Space Observatory” or WSO) which is expected Corresponding author: S. R. Kulkarni to be in a geo-synchronous orbit (GEO; Boyarchuk et al. [email protected] 2016). However, the Sun is faint in the FUV. Ergo, the sky is dark in the FUV band. As a result, the FUV 2 Kulkarni band is most attractive for low surface brightness imag- ing Spectrometer (FIMS1; Edelstein et al. 2006 provided ing of galaxies (O’Connell 1987). Indeed, it is precisely spectral imaging of the Galactic Hot Ionized Medium this advantage of the FUV band that allowed GALEX to (HIM). discover very faint star-forming complexes well beyond The consensus from all these studies is that much of the optical disk and with sensitivity better than that the diffuse FUV emission is due to reflection of stellar provided by ground-based Hα imaging (Barnes et al. FUV photons by diffuse (“cirrus”) clouds and conve- 2011). niently traced by IRAS 100 µm band or fluorescence of In the early seventies it was speculated that the hot stellar FUV photons by molecular hydrogen. Together intergalactic medium (IGM) would be revealed by dif- this emission is called as the Diffuse Galactic Light fuse FUV emission. Searches for diffuse UV emission (DGL). However, some diffuse background emission is were undertaken with great gusto. Separately, this was not correlated with cirrus clouds, a fraction of which also the period when the first pulsar surveys showed that can be reasonably attributed to collective emission from pulsar signals are invariably dispersed (see, for example, other galaxies – the so-called Extragalactic Background Manchester & Taylor 1977; Yao et al. 2017). As a re- Light (EBL). There remains some 120–180 CU emission sult, astronomers became aware of a pervasive Galactic in the FUV band which cannot be clearly attributed to ionized medium. During the eighties, thanks primarily a single source. Here, CU (“continuum unit”) stands −1 to the work of Ronald J. Reynolds, this ionized medium for photon cm−2 s−1 A˚ sr−1 with “sr” as a short for was complementarily sensed via Hα recombination emis- steradian. In the literature, this residual component is sion (see Haffner et al. 2009). These two methods – dis- called the “isotropic [offset]” component. The purpose persion of radio signals and diffuse Hα emission – led to of this paper is to investigate possible origin(s) for this the recognition of the Warm Ionized Medium (WIM) as component. a distinct phase of the ISM. The paper is organized as follows. In §2 we review The filling factor of the WIM, by volume, is estimated the measurements of diffuse high-latitude FUV emis- to be between 20% and 40% of the Galactic disk (Haffner sion. This is followed by a summary of the essential et al. 2009). The ionizing power requirement for the physics of hydrogen two-photon decay (§3). We inves- WIM is tremendous. The conventional explanation re- tigate possible two-photon emission from the Galactic quired a significant fraction of Lyman continuum pho- WIM and low-velocity shocks and line emission from the tons from OB stars (Reynolds 1990) and also called for HIM (§4). We then investigate two-photon contribution a porous ISM so as to allow for Lyman continuum pho- from the Solar System (§5), specifically the Interplane- tons to travel significantly away from their parent stars. tary Medium2 (§6), the Earth’s atmosphere (“thermo- The observed large vertical scale height, ≈ 1 kpc, of the sphere”; §7) and the exosphere (§8). In §9 we tally the WIM was initially a mystery. This “crisis” led to a res- contributions to the diffuse FUV background and con- urrection of a hypothesis of decaying dark matter as clude that about two thirds of the diffuse FUV back- a major source of ionizing photons (e.g., Sciama 1990 ground can be accounted for by contributions discussed and earlier references therein; more recent references in- in §4-8. We conclude in §10 by discussing experimental clude Kollmeier et al. 2014; Henry et al. 2015). Miller verification of some of the proposed channels of two- & Cox(1993) and Dove & Shull(1994) provided a con- photon production and strategies to take advantage of ventional explanation based on O stars as the principal the dark FUV sky. source of ionizing radiation, radiative transfer modeling and clustering of star-forming regions, “chimneys” and 2. THE DIFFUSE FUV EMISSION “channels”. The study of diffuse FUV continued through the The “Interstellar Radiation Field” (ISRF), being a nineties with rocket-borne experiments, UV spectrom- central quantity, is almost assured of a chapter in any eters on Voyager missions, Shuttle-based experiments, serious book on the ISM (e.g., Chapter 12 in Draine FAUST and FUSE. Murthy et al.(2019) provide a com- 2011). The ISRF is composed of both resolved (bright) prehensive list of FUV experiments. Two missions, both stars and diffuse emission. Only the latter component is launched in 2003, greatly advanced the field of diffuse FUV: GALEX (Martin et al. 2005) carrying both an 1 also sometimes referred to as SPEAR FUV and an NUV wide-field imager, each with a field- 2 The term IPM has two different connotations in astronomy. In of-view (FoV) of over a square degree but with pixels radio astronomy, IPM refers to the solar wind that pervades the of few arc-seconds. STSat-1 carrying the Far-UV Imag- solar system out to the heliopause. In planetary studies, the IPM stands for the very local interstellar cloud into which the Solar system is moving. The FUV background 3 of interest to this paper. There have been extensive re- not traced by other indicators”. Akshaya et al. views of the diffuse FUV emission (e.g. see Paresce et al. (2018) state “There is an excess emission (over 1980; Henry 1991; Henry et al. 2015). Here we focus on the DGL and the EBL) of 120–180 photon units3 the measurements of the diffuse FUV by GALEX. in the FUV and 300–400 photon units the NUV. GALEX was well suited to measuring the diffuse UV ...Although we do not know its origin, we can af- emission because its FUV and NUV imagers not only firm that the excess emission cannot be accounted had good angular resolution (few arc-seconds) but also for by current models of the DGL and EBL.” a large FoV. This resolution allowed for masking out The purpose of this paper is to investigate the origin of point sources and galaxies. The resulting ten-year trove this excess emission. To this end we explore conventional of data was well suited to discerning the diffuse back- sources of FUV emission: radiative recombination, line ground. GALEX was in LEO with an orbital period and continuum emission from low velocity shocks and of 99 minutes. To minimize airglow, observations were the hot ISM, and solar Lyman-β excitation of gas within restricted to narrow periods lasting only 25–30 minutes the Solar system. Should such an exercise fail to account during Earth eclipse (Morrissey et al. 2007). for the offset emission then it would be logical to pursue We refer to two major studies of diffuse FUV carried increasingly exotic explanations. out with GALEX: one led by Jayant Murthy and as- In aeronomy, it is traditional to use “Rayleigh” (R) as sociates (e.g., Murthy et al. 2010; Akshaya et al. 2019) a unit for surface brightness of line emission. Numer- and the other, the PhD thesis of Erika Hamden (Hamden ically, one Rayleigh is 106/(4π) photon cm−2 s−1 sr−1. et al. 2013). These two papers provide an excellent sum- A related unit is “line unit” (LU) which stands for mary of past efforts to measure the diffuse FUV emission photon cm−2 s−1 sr−1). Thus 1 R ≈ 79, 577 LU. The So- at high Galactic latitudes. The conclusions from these lar system is awash with solar Lyα radiation. The min- studies are largely concordant and briefly are as follows: imum geo-coronal Lyα emission seen by HST, which is 1. At high Galactic latitudes, the FUV intensity located in LEO, is 2 kR (Ake 2012). Even for a mission scales linearly with the IRAS 100 µm intensity up in a High Earth Orbit (HEO)4, as will become clear later to an intensity of 8 MJy sr−1. This is attributed in the article, the minimum Lyα surface brightness is to reflection of stellar FUV photons from hot stars ∼ 300 R. If the FUV band of GALEX was stretched to (primarily located in the Galactic plane) by inter- 1200 A˚ then Lyα alone would contribute a background stellar dust which, heated by stellar light (includ- of about 4 × 105 CU (HST). For missions in LEO, the ing the FUV), radiates at long wavelengths (in- next brightest contributor is the airglow5 OI λ1304 A˚ cluding the IRAS 100 µm band). The FUV emis- triplet. In HST orbit, the surface brightness in this line sion saturates at higher values of 100 µm emission is 13 R, even when observed6 in the anti-Sun direction presumably because the absorption of the FUV and during deep shadow of Earth. This translates to an radiation by the clouds outweighs over reflection. increase in the background by 2.6 × 103 CU. For these two reasons the blue edge of the GALEX FUV band was 2. The FUV intensity increases in the direction of in- set to 1350 A.˚ terstellar clouds which harbor molecular hydrogen. This is attributed to fluorescence by Extreme UV 3. TWO-PHOTON DECAY (EUV) and FUV photons (see Martin et al. 1990; As will become clear from discussions below, the two- Jo et al. 2017). The reflected and the fluorescent photon process is a contributor to the FUV background. components are the primary contributors to DGL. For instance, a low density astrophysical plasma which is in collisional ionization equilibrium (CIE) and with 3. At high Galactic latitudes, diffuse FUV back- sufficient column density to be in “case B”7 and at a ground is present, even in directions towards dust- temperature . 50, 000 K, cool predominantly via Lyα, free regions. Only about half of the emission can two-photon continuum and Hα. A Lyα photon under- be reasonably attributed to emission from other galaxies, the EBL. 3 Hamden et al.(2013) conclude “ There is a ∼ The photo unit discussed here is the same as CU. 4 Geo-synchronous (GEO) and higher 300 CU FUV isotropic offset which is likely due 5 We use the term airglow for any emission that occurs up until to a combination of air glow (probably the dom- the exobase; see §E for definition of exobase. inant contributor), a small extragalactic back- 6 STIS Handbook, https://hst-docs.stsci.edu/stisihb ground component including continuum light from 7 Regions which are thick to Lyman continuum photons are in case unresolved galaxies, and/or a Galactic component B. 4 Kulkarni

The reader is advised to consult Draine(2011) for a description of the three primary atomic phases; the WNM, WIM and HIM. Locally, the WNM, the WIM and the HIM occupy most of the interstellar space (Cox 2005). While the WIM contributes to the FUV band by two-photon emission, the HIM primarily contributes to the FUV background via line emission. Separately, it turns out that low-velocity shocks are also emitters of two-photon decays. Below we discuss the various con- tribution to the FUV background from these Galactic sources.

4.1. The Warm Ionized Medium Figure 1. (Left y axis): The photon spectrum of a single Deharveng et al.(1982) investigated two-photon emis- two-photon decay, n (solid line) as a function of the wave- λ sion from the WIM and concluded that it could not length, λ. Since each decay leads to emission of two photons, R account for the FUV background. Martin et al. 1991, nλdλ = 2. (Right y axis): Effective area of GALEX FUV and NUV bands as a function of λ (dot-dash and dashed Reynolds 1992 and Seon et al. 2011 came to similar con- lines). The details of the fitting formula used to generate clusions. Rather than compute the expected two-photon the two-photon decay spectrum and effective area of GALEX emission from the physical parameters of the WIM (as can be found in §A. was done in the past papers) we prefer the simpler ap- proach of estimating the two-photon brightness directly goes many scatterings, ending its life upon encountering from observations of Hα. At the temperature of the a dust particle within or close to its birth site. Two- WIM (8,000 K), assuming case B, we expect that for photon continuum and Hα freely escape the plasma. In every two-photon decay there are 1.47 Hα photons. this section we first summarize the two-photon process In Figure2 we display the histogram of Galactic H α and compute the factor to convert the two-photon decay surface brightness as recorded in the Wisconsin Hα spectrum to counting rates of the GALEX bands. Mapper Sky Survey (WHAM-SS or WHAM for short; An H atom which finds itself in a 2s level will, if undis- Haffner et al. 2003, 2010). From these figures we con- turbed by a collision, radiatively decay by emitting two clude that the Galactic Hα emission towards the Galac- −1 −1 −1 photons over timescale of A2γ ≈ (8.2 s ) ; see Fig- tic polar cap is approximately 0.5 R. We subtract 20% ure1. The sum of the energies of the two photons is to account for contribution resulting from scattering of equal to that of Lyα or 10.2 eV. The factors to con- Hα emission by cirrus clouds (Witt et al. 2010; Dong & vert the two-photon spectrum to FUV and NUV count- Draine 2011). The expected two-photon decay rate is ing rates are central to this paper and a full discussion then 0.27 R which corresponds to 22.7 CU in the FUV can be found in §A. The key result is the following: a band. columnar rate of 106 decays cm−2 s−1 results in 85.1 CU in the FUV channel and 18.4 CU in the NUV channel 4.2. The Hot Ionized Medium (Table3). The three dominant metals fortunately have strong We conclude this small section by noting two compli- resonance lines (2s→2p) in the UV and also conve- cations. First, the Sun is also a source of Hα photons. niently probe a range of temperatures (Sutherland & So, solar Hα photons could pump an H atom which is Dopita 1993): CIV (λλ 1548.2, 1550.8 A;˚ 1 × 105 K), NV already in the 2s level to a 3p level (Bishop et al. 2001). (λλ 1242.8, 1238.8 A;˚ 2 × 105 K) and OVI (λλ 1037.6, This twist is addressed in §6.1. Next, there is little en- 1031.9 A;˚ 3 × 105 K). FUSE observations of OVI estab- ergy difference between the 2s and 2p levels. So colli- lished the presence of widespread HIM in the halo of sions can shift an H atom in the 2s state to a 2p state the Galaxy (Savage et al. 2000) and in the disk of the following which the atom promptly de-excites by emit- Galaxy (Bowen et al. 2008). ting a Lyα photon. This topic is of central importance FIMS/SPEAR aboard South Korea’s STSAT-1 satel- to the (non)-production of two-photons in our own at- lite undertook all sky spectral-imaging in the wavelength mosphere and is squarely addressed in §7.2. range 900–1150 A˚ and 1350–1750 A(˚ Jo et al. 2017). The authors fit high latitude data to a collisional ioniza- tion equilibrium (CIE) model and find a plasma tem- 4. THE GALAXY perature of 2 × 105 K and a vertical emission measure, The FUV background 5

cools. The principal cooling lines of hydrogen include not only the resonance lines but also two-photon emis- sion. In fact, one of the earliest and unambiguous de- tection of two-photon emission was from a Herbig-Haro object with a 70 km s−1 shock (Brugel et al. 1982; see also Dopita et al. 1982). Two-photon emission was invoked in a recent investi- gation of the fascinating 30◦-long Hα arc in Ursa Major (McCullough & Benjamin 2001). The arc is long enough that some segments do not have overlapping cirrus emis- sion. The ratio of the GALEX FUV to NUV flux in dust-free regions is 0.31 ± 0.05 (cf. see Figure1). which is significantly smaller than that measured in directions Figure 2. Histogram of the WHAM Hα brightness towards to dusty regions. Bracco et al.(2020) investigate a grid the Galactic caps. The y-axis refers to the number of WHAM of supernova shock models and find two solutions: low −1 beams. Data obtained from WHAM. The size of the WHAM velocity (vs = 50–100 km s ; dominated by two-photon −1 beam is about a degree. (Haffner et al. 2003, 2010). cooling) and high velocity (vs > 300 km s ; dominated by strong line emission). The Hα emission breaks the tie R 2 −6 ne(z) dz of 0.01 cm pc. For these parameters, the and favors the former. Separately, recently Fesen et al. free-free emission amounts to < 1 CU in the FUV band. (2021), using GALEX FUV observations along with Hα The most prominent line in the spectrum towards the and radio imaging uncovered three very large supernova Galactic poles is the CIV doublet with a strength of remnants. 4,600 LU. Separately, Martin et al.(1991) report a sim- The work of Bracco et al.(2020) motivated us to ilar detection towards the Lockman hole, one of the re- consider the entire class of low velocity shocks (v . −1 gions with the low H I column density. We will thus 70 km s ) as a major class of two-photon emission. The assume that 4,600 LU is not reflected light but is gen- traditional approach to search for low velocity shocks uine emission from high-latitude HIM. is via the associated Hα emission. However, such Hα In order to determine the total emission from the emission will be broad in velocity which makes detec- HIM we need to account for fainter lines and contin- tion harder. The discovery of three large SNRs (Fesen uum (free-bound, primarily). To this end, we used CHI- et al. 2021) was made possible, in part, by the distinc- ANTI (Dere et al. 1997; Del Zanna et al. 2021) and ob- tive FUV signature of low velocity shocks. So, going tained the CIE spectrum of a solar-abundance plasma at forward, fortunately we have a new tool to identify low T = [1, 2]×105 K. We integrated this model with the ef- velocity shocks – FUV and NUV imagery data. 8 fective area of the GALEX FUV detector (see Figure1) The post-shock temperature of a shock with velocity and found that the CIV emission is [40%, 80%] of the vs is emission integrated over the FUV band. We take the 2 2 average of the two corrections and adopt line emission 3 µvs 5 vs  T2 = = 0.71 × 10 −1 (1) of 1.875 × 4600 = 8625 LU. The effective bandwidth of 16 kB 70 km s the GALEX FUV band is 255 A˚ (Table3). Thus, the where we have assumed cosmic abundances and that HIM contribution to the FUV background is 34 CU. both H and He are singly ionized. In CIE, hydrogen is 50% ionized at a temperature of about 2.8 × 104 K. 4.3. Low-velocity Shocks Cooling via hydrogen lines becomes ineffective once Shocks are a central topic in the study of ISM. The the post-shock temperature reaches WNM-like temper- post-shocked gas cools via line emission, free-bound atures, < 104 K. Thus, the flux of two-photon emission and free-free emission. Libraries of spectra of cooling is directly proportional to the column density of gas in gas have been published with increasing sophistication; the temperature range, 104 K to 105 K. At the present see Raymond 1979, Shull & McKee 1979 and Suther- land & Dopita 1993. Shocks, say those with velocities, 8 v 100 km s−1, are dominated by UV line emission and Assuming that the Mach number of the shock is much greater s & than unity. This assumption can be taken for granted for blast X-ray emission. In contrast, shocks with low velocities, waves from supernovae. Caution needs to be exercised for low −1 vs . 70 km s , are dominated by cooling via H with velocity shocks such as those from runaway stars, expanding HII decreasing contributions from C and Si as the plasma regions etc. 6 Kulkarni time we are not able to readily derive this fraction from In order to convert FSN to two-photon decays we make observations. So, we resort to simple modeling. the following conservative assumptions: (1) every H ion All high velocity shocks eventually cascade to low ve- undergoes only one recombination, (2) a third of case B locity shocks. In fact, the classical explanation for the recombinations lead to two-photon emissions and (3) fol- observed ≈ 10 km s−1 velocity dispersion of the CNM is lowing recombination, no H atom will be collisionally ex- the stirring of the ISM by supernovae (§12.2c, Spitzer cited to the 2s state or 3p state. With these assumption, 1 1978; Kim & Ostriker 2015). the columnar rate of two-photon decay is R = /3FSN. We draw attention of the reader to observational sup- McKee & Ostriker(1977), in the framework of the port for this “shock→stir cascade” model which comes 3-phase model for the ISM, found η = 0.05. Kim from a study of the integrated H I spectrum towards & Ostriker(2015) carry out a detailed simulation and 51 the Galactic poles (Kulkarni & Fich 1985). These au- find that each SN (with E0 = 10 erg of explosive en- 2 thors found a rather curious result: NH (v)v is flat ergy) has, at the onset of the snowplow phase, a mo- −1 5 −0.15 −1 out to 80 km s ; here, v is the velocity and NH (v) is mentum of psf = 2 × 10 n0 M km s where n0 is the column density of H atoms with velocity v. The the particle density of the medium into which the SN H I spectrum towards the South Galactic pole shows exploded. If we assume that all this momentum re- no mean motion whereas strong inflow can be inferred sults in stirring of clouds (velocity dispersion, σ) then 2 from the spectrum towards the North Galactic pole. We η = 1/2(psf /σ)σ /E0 ≈ 2σ6%. I note that in the Kim first consider the zero velocity component and postpone & Ostriker(2015) model, the expansion velocity at the −1 the discussion of in-falling (negative velocity) gas. We time of shell expansion is vsf = 200 km s . If we ac- interpret the velocity width of this component as aris- cept the inference discussed above (cf., Kulkarni & Fich ing from thermal and turbulent motions of the H I gas 1985) then the η = σ/vsf = 0.05σ6. So we elect to set 1 6 −2 −2 −1 that has recombined and is cooling. In the cascade η = 0.05 and then find R = /3×10 σ6 decay cm s . 2 model, NH (v)v can be interpreted as a product of the which corresponds to FUV brightness of 29 CU. particle flux, NH (v)v, and the momentum per particle The next major shock systems, on a Galactic scale, (∝ v). Thus, the Galactic pole H I observations in- are spiral density waves (or phenomena related to non- form us that the momentum flux is constant. This in- axis symmetric distribution of stellar matter) and in- ference is in accord with our expectation that already flow. We expect hot gas to rise into the halo, cool and by 80 km s−1 the shocks are “momentum conserving” then rain down. In addition, there could be inflow from −1 (aka “snowplow” phase of SNRs). In contrast, the en- CGM/IGM. Assuming M˙ infall ≈ 1 M yr (Shull et al. 3 ergy flux ∝ NH (v)v ∝ v. Thus, radiative shocks are 2009; Fox et al. 2019) and a radius of RG = 10 kpc for losing energy proportional to their velocity. This is per- the Galactic disk the flux towards either Galactic pole haps a simplistic analysis in that we have not tracked is ˙ the ionization state of cooling hydrogen. Nonetheless, Minfall 4 −2 −1 F = 2 = 1.3 × 10 atom cm s . (4) this analysis prepares us to accept a low efficiency for πRG stirring. Clearly, this contribution is much smaller than that due The mass of the Galactic H I gas is M = 3×109 M HI to shocking of the atomic ISM by supernovae (cf. Equa- (Chapter 1 of Draine 2011). The Galactic supernova tion3). (SN) rate is conservatively estimated to be one per cen- However, it is almost certainly the case that the infall tury or E˙ = 3 × 1041 erg s−1, assuming 1051 erg per SN is not smoothly distributed over the Galactic disk (cf., SN. Let η be the conversion efficiency of SN energy that “High velocity” and “Intermediate velocity” clouds). In goes into the stirring of the atomic phases and let σ HI fact, as noted earlier, there is clear evidence for in-fall be the effective rms velocity of CNM+WNM. Then the in the Northern Galactic polar region. The mean veloc- mean time between successive stirrings is ity of this flow is −40 km s−1, with a significant tail to −1 1 2 −70 km s . /2MHIσHI 13 −1 2 τHI = ≈ 10 η σ s, (2) ηE˙ 6 We now summarize this subsection. In low-velocity SN −1 shocks, vs . 70 km s , two-photon cooling dominates 6 −1 where σHI = 1 × 10 σ6 cm s . Locally, the vertical over line emission, excluding Lyα emission, of course. column density towards the Galactic poles is NH ≈ 2 × There are multiple origins of low velocity shocks: the 1020 atom cm−2 (Lockman & Gehman 1991). Thus, the cascade in velocity space initiated by supernova shocks, rate of successive SN impacts in a vertical column is shocks generated by fast moving stars, stellar winds and finally the infall of gas from the Galactic cooling foun- 7 −2 −2 −1 FSN = NH /τHI = 2 × 10 ησ6 atom cm s . (3) tain. We made a case for a contribution of 21 CU from The FUV background 7 the shocking and stirring of the WNM & CNM by su- 12 km s−1. The H atoms in the IPM have a mean veloc- pernovae. ity of −24 km s−1 with respect to the Sun. The thermal velocity spread of those H atoms is 8 km s−1. 5. THE SOLAR SYSTEM: SCATTERING OF The solar Lyβ line has a velocity profile with two horns SOLAR LYβ PHOTONS separated by about 0.33 A˚ and a full-width-at-zero of 1 A˚ In the Solar system, in contrast to the WIM, the pri- (Figure4). In other words the velocity width of Ly β is mary excitation of H atoms is not by photo-ionization quite narrow, ±50 km s−1. It requires a very high res- but by excitation of H atoms by solar Lyman series pho- olution spectrometer to see the structure in this line. tons (Lyα, Lyβ, Lyγ, etc.). Not surprisingly, the in- Fortunately, a linear relation exists between the inten- tensity of the Lyman photons decreases rapidly as one sity at the valley center and the integrated Lyβ emission. proceeds up the Lyman series. There is a rich literature This relation allows solar astronomers to infer the zero of both theory and observations related to Lyα in the velocity Lyβ intensity from integrated Lyβ observations. Solar system. Over the period 1996–2009 (solar cycle 23), the central photon flux density at 1 AU var- ied from 4 × 1010 phot cm−2 s−1 nm−1 to 10 × 1010 phot cm−2 s−1 nm−1 (Lemaire et al. 2015). We adopt the minimum flux for our fiducial value. We also switch to the non-CGS but traditional A˚ unit. Noting 8 8 νFν = (10 Fλ)λ where the factor of 10 takes into ac- count of our use of A˚ instead of cm we find:

9 −2 −1 −1 Fλ(β, v = 0) = 4 × 10 phot cm s A˚ , −3 −2 −1 −1 Fν (β, v = 0) = 1.4 × 10 phot cm s Hz . (5)

Figure 3. Partial Grotrian diagram for H I. Here, we fo- Here, v is the radial velocity between the absorbing H cus solely on excitation of H atoms by Lyβ photons. An H atoms and the mean velocity of the Sun. Going forward, atom thus excited has two choices: de-excite to ground state we use the short hand F (β, v = 0) = F (0) and so on. by emitting Lyβ (probability of ≈ 7/8) or de-excite to the ν ν 2 Next, the ratio of the solar Lyα to Lyβ intensity (with 2s S1/2 level by emitting Hα. For the latter case, in the −1 intensity expressed in energy units and not photons) absence of collisions, over a timescale of A2γ ≈ 0.12 s, the atom de-excites by emitting two-photons. The values of the ranges from 50 (solar maximum) to 80 (solar minimum); A-coefficients for Lyβ and Hα can be found in Table4. see (Lemaire et al. 2012). Thus the ratio in photon flux of Lyβ to that of Lyα, hereafter η , ranges from 1/95

In this paper we focus on Lyβ photons (λβ = to 1/60. We adopt η = 1/70. 1025.7220 A).˚ These photons excite H atoms to either Equation B5 gives the rate which H atoms in ground 2 2 state (“l”) at 1 AU are excited to an upper state (u) 3p P1/2 or 3p P3/2 (a partial Grotrian diagram for H I is provided in Figure3). The excited atom has two op- πe2 λ  πe2 tions: de-excite back to ground state by emitting a Lyβ R = F (0) f = 108F (0) ul f . (6) ν m c lu λ ν m c lu photon (we denote the corresponding A-coefficient by e ul e 2 A31) or de-excite to the 2s S1/2 state by emitting an Hα where flu is the oscillator strength and the odd factor 8 photon (A32). We let ηHα ≡ A32/A31 ≈ 1/7.4 (see Ta- of 10 is a result of using two different units: A˚ for ble4 for A-coefficient data). For higher order Lyman wavelength in Fλ and cm elsewhere (see §B). The A- 2 series photons the branching ratio to reach the 2s S1/2 coefficients for H lines are given in Table4 and the os- level is smaller. Furthermore, an inspection of the solar cillator strengths were computed using Equation B4. spectrum shows that the intensity of the higher lines is We first consider the simple case of a medium opti- smaller than that of the Lyβ line. As a result, we sim- cally thin to Lyα and also with low particle density (so plify by restricting our analysis to only Lyβ excitations. that the probability of an H atom colliding with another −1 Our goal in this section is to compute the rate of ex- particle, particularly a proton, over a duration of A2γ , citation of H atoms by Lyβ photons. For this exercise is negligible). Let Lα (Lβ) be the photon surface bright- we need to understand the frequency (velocity) profile of ness of Lyα (Lyβ). These quantities are ∝ R. Using both solar Lyβ and also the velocity distribution of the the atomic physics constants given in Table4 and using H atoms. The geo-coronal H atoms, with respect to the Equation B4 and noting the value of η we obtain Sun, have essentially zero radial velocity and a veloc- L ity width of no more than escape velocity of the Earth, L ≈ α . (7) β 520 8 Kulkarni

Figure 4. The profile of solar Lyβ. The continuum level in the vicinity of this line is very small, 6 × 107 phot cm−2 s−1 A˚−1 (see Figure 2 of Dudok de Wit et al. 2005 for an annotated EUV/FUV solar spectrum). The small dark vertical stub marks the rest wavelength of Lyβ. The thin vertical line marks three resonance lines of O I which can also absorb Lyβ photons (see §D.2 for further details). The top scale is the Doppler velocity in km s−1. The three OI lines are +12 km s−1 with respect to the centroid of the Lyβ line. The data used here is from the SOHO mission (Lemaire et al. 2015).

The surface brightness in Hα, LHα = ηHαLβ ≈ The entire Solar system is moving into a local inter- −1 Lα/3850. In absence of collisions over a timescale of stellar cloud at a velocity of about 25 km s . Incoming −1 A2γ , each Hα emission will lead to a two-photon de- neutral particles scatter solar Lyα photons and create a cay. If, on the other hand, the medium is optically spatially structured haze (Blum & Fahr 1970). Plane- thick to Lyα then the surface brightness of Lyα in- tary astronomers call this medium as the interplanetary creases, potentially by large factors (determined by a medium (IPM). The present picture of an incoming in- competition between recirculation, destruction by dust terstellar wind was deduced by detection of anisotropic particle or frequency-spatial random-walk escape from Lyα by OGO-5 (Bertaux & Blamont 1971; Thomas & the medium). In this case, the above calculation yields Krassa 1971) and refined by IBEX using in situ mea- an upper limit to Lyβ. If in addition, the medium is surements of He I (Frisch et al. 2013). The “downwind” optically thick to Lyβ, then all of Lyβ photons will be direction of the wind in ecliptic longitude (λ) and lati- converted to Hα and two-photon continuum. We are tude (β) coordinates is 79◦ and −5◦, corresponding to now in a position to estimate the two-photon contribu- Galactic coordinates l = 185◦ and b = −12◦ (Frisch tion from the Solar system: the interplanetary medium et al. 2013). The upwind direction is then l = 5◦ and (§6), the thermosphere of Earth (§7) and the exosphere b = 12◦ which corresponds to a direction just “above” of Earth (§8). Scorpius constellation but “below” Ophiuchus. From high-resolution spectroscopic UV studies the following 6. INTERPLANETARY MEDIUM properties of the interstellar cloud have been deduced: −14 −1 −3 The Sun has a weak wind, about 10 M yr . neutral H density of 0.1 cm , temperature of 7000 K The wind is correlated with solar activity and also has and 50% ionization (Frisch et al. 2011). These prop- dependence on solar latitude and longitude (Pr¨olss& erties seem to have physical attributes similar to the Bird 2004). At 1 AU, the typical properties are as fol- WNM, albeit with a very low column density. −3 lows: electron density, ne of 3–10 cm , magnetic field The fast solar wind comes to equilibrium with the slow strength of 10–370 µG and temperature of 105 K. By the moving interstellar medium by undergoing a shock – time the wind reaches Earth it is supersonic, vwind ∼ the so-called solar wind termination shock (SWTS). The 500 km s−1. For a spherical and steady wind, conserva- shocked solar wind is in pressure equilibrium with the to- tion of mass leads to a decrease in density as r−2. The tal pressure of the interstellar wind (which includes ram solar wind is neither steady nor spherically symmetric. pressure). In the parlance of planetary astronomers, the −2 However, averaged over a solar cycle, ne ∝ r where r surface separating the shocked solar wind and the inter- is the heliocentric radius. The FUV background 9

also can no longer serve as sources for two-photon pro- duction).

Figure 5. The Lyα glow of the IPM. URL: https:// sohowww.nascom.nasa.gov/home.html stellar gas is called the heliopause (aka “contact discon- tinuity” for astronomers). Thanks to the Voyager mis- Figure 6. High-resolution spectrum of the Sun in the vicin- sions we know that the SWTS is located at 90 AU while ity of Hα (data from Chance & Kurucz 2010). The abso- the heliopause is at 120 AU. Neutral particles from the lute level at 6500 A˚ agrees to within a percent of the spec- interstellar wind stream undeflected into the Solar sys- trum from SOLSPEC, aboard the International Space Sta- tem. The interstellar H atoms advance into the Solar tion (Meftah et al. 2021). The dotted curve is a “chi-by- eye” fit to the narrow Hα core. The model is given by system at the rate of 6 AU per year. The neutral par- y(λ) = 1.6 × [1 − α exp(−βx2)] where x = λ − λ(Hα), ticles in the IPM are subject to gravitational attraction α = 0.79 and β = 1.5 A˚−2. The thick vertical stub is the by the Sun, radiative repulsion, photoionization by solar vacuum wavelength, λ(Hα) = 6564.614 A.˚ The thin stub is Lyman continuum and charge exchange with protons in the rest air wavelength of Hα (6562.801 A),˚ shown merely for the solar wind. As it so happens, for H atoms, the ra- reference. diative repulsion is almost canceled by the gravitational acceleration (see see Kurt et al. 2009 for a review). In contrast, for He atoms the radiative pressure is small 6.1. Solar Hα pumping of excited H atoms and so they are gravitationally focused by the time they In this subsection we investigate the pumping of H arrive in the the inner Solar system. atoms by solar Hα. Recall that an H atom in the 2s state The rate of photo-ionization by the solar Lyman con- has a lifetime of about 0.12 s (Figure3) and so if there is −8 −2 −1 tinuum is Rpi = 6 × 10 dAU s where dAU is the a sufficiently large Hα flux from the Sun then such an H heliocentric distance in AU. The rate of charge ex- atom could get pumped back up to a 3p level. In effect, change, assuming mean solar wind properties, is Rce = Hα pumping effectively reduces the value of A32. For an −7 −2 −1 2 6 × 10 dAU s . The recombination timescale for a H atom in the 2s S1/2 state (level degeneracy, gl = 2) + 2 newly ionized H particle is considerably longer than the Hα absorption can be to either 3p P1/2 (gu = 2) 2 the crossing time across the Solar system. So the ion- or 3p P3/2 level (gu = 4) The A-coefficients for the two ized cavity is highly anisotropic with strong Lyα haze in transitions are given in Table4. The oscillator strength, the upwind direction and a weaker haze in the downwind flu ∝ Aulgu/gl (Equation B4) and so the sum of the direction. The Lyα production peaks in the heliocentric oscillator strengths for the two transitions is f2s→3p = annulus 2–5 AU in the upwind direction. A visual map 0.4360. of the cloud, as traced by solar Lyα, can be found in The thermal velocity dispersion of H atoms in the p −1 Figure5. IPM is σv = kT/mH or 7.6 km s ; here, mH is Upon charge exchange, the newly minted H+ is sub- the mass of an H atom. The corresponding full- 1/2 ject to electrical and magnetic forces of the solar wind width at half-maximum (FWHM) is (ln(256)) σv = and acquires the velocity of the solar wind. However, 18 km s−1. We fit the Hα absorption feature to a con- the narrow velocity width of the solar Lyβ emission (Fig- tinuum+Gaussian absorption model (Figure6) and de- ure4) means that the fast H atoms can no longer absorb rive an FWHM of 2pln(2α)/β = 46 km s−1. Thus, solar Lyα photons. Thus, H atoms which have under- if the H atom has no or little Doppler shift with re- gone charge exchange become effectively invisible (and spect to the Sun (which is the case for geo-coronal H atoms) then we can assume that the flux of the solar 10 Kulkarni

Hα line is constant over the thermal frequency spread We now summarize this section. From our vantage of an H atom. The Hα flux at the bottom of the point, in the “upwind” direction of the IPM, we see −2 −1 absorption line is 0.34 W m nm (Figure6). This Lβ = 2.4 R (Figure5). H α pumping of H atoms in 13 −2 −1 −1 corresponds to Fλ(0) = 1.1 × 10 phot cm s A˚ the 2s state is not powerful enough to prevent or even and the corresponding photon spectral flux density is significantly modify the rate of two-photon decay. The −2 −1 −1 Fν (0) = 158 phot cm s Hz . Applying f2s→3p and expected two-photon decay rate is ηHαLβ = 0.33 R. The Fν (0) to Equation B5 we compute the rate of Hα pump- corresponding brightness in the FUV band is 27.6 CU. ing, per atom, to be 7. THE EARTH: THERMOSPHERE 2 πe −2 −1 −1 RHα = Fν (0) f2s→3p ≈ 1.8d atom s . (8) The same solar Lyman photons that excite the IPM m c AU e also excite the upper atmosphere and the exosphere. In Here, d is the distance in AU. RHα should be compared fact, a by-product of Lyβ absorption is “geo-coronal” −1 with A2γ ≈ 8.2 s . Parenthetically, we note that the Hα which is routinely detected by the ground-based pumping by solar Hβ photons is smaller, relative to that WHAM. Typically the Hα surface brightness measured 9 for Hα, by a factor of 6.6 and so can be ignored. at zenith in the middle of the night is about 4 R (e.g., However, there is a complication: the IPM has a ve- Nossal et al. 2008, 2019). If we assume, as was the case −1 locity of about 24 km s with respect to the Sun which for the IPM, that each Hα emission led to a two-photon ˚ amounts to a wavelength shift of δλ = 0.5 A. As can decay then the FUV background from geo-coronal emis- be seen from Figure6 the H α flux increases on either sion, even in the dead of the night, would be 304 CU ˚ side of rest wavelength. For δλ = 0.5 A the increase which exceeds the level of the isotropic [offset] compo- in flux is a factor of two. However, most of the Ly- nent! Obviously, our reasoning is incorrect. The flaw man scattering happens in a heliocentric 2–5 AU annu- is that, unlike the situation with the WIM, the HIM lus in the upwind direction. Taken together, RHα . and the IPM, the densities in the thermosphere are high −2 −1 −1 0.5(d/4 AU) atom s which should be compared enough that collisions suppress two-photon decay. Since −1 with A2γ = 8.2 s . Thus, for bulk of the IPM, we the author was quite ignorant of the gross physical fea- can afford to ignore the effect of Hα pumping of excited tures of Earths’s atmosphere a brief summary of the H atoms. densities in the thermosphere and exosphere is provided 6.2. Measurements in §D and §E, respectively. Voyager-2 was launched on 1977 August 20. One month later, when the spacecraft was at a heliocen- Table 1. STP 78-1 EUV observations of night glow tric distance10 of 1.02 AU, the UV spectrometer was Line “Up” (R) “Down” (R) ◦ ◦ pointed to α = 324 and δ = −23 . It detected Lyα Lyβ, OI 8.76 ± 0.3 2.3 ± 0.23 at 722 ± 0.5 R, Lyβ at 2 ± 0.16 R and Helium 584 A˚ at Lyα 3533 ± 5.8 1712 ± 4.7 3.8±0.04 R (Sandel et al. 1978). Next, long observations OI 1304 7.1 ± 0.4 53.8 ± 1.3 of several high Galactic latitude fields were undertaken OI 1356 < 2 52.2 ± 0.8 in February and March, 1981 (heliocentric distance of 8.4 AU). These observations detected not only Lyβ but also higher order Lyman lines (Holberg 1986). Of in- 7.1. Observations terest to us were Lα = 1080 R and Lβ = 2.4 R. The ratio between Lyα and Lyβ seems to be consistent with The Space Test Program 78-1 (STP 78-1; aka “Sol- that expected for optically thin medium (Equation7). wind”)11 was launched in 1979 into a sun-synchronous The New Horizons mission also carried a UV spectrom- orbit (600 km height). The satellite spin-orbital axis eter and detected Lyα surface brightness of 550 R at a was perpendicular to the Earth-Sun axis (i.e., a “noon- heliocentric distance of 7.6 AU decreasing to 100 R at midnight” orbit). It carried a number of instruments 38 AU (Gladstone et al. 2018) – confirming that peak for aeronomy including an extreme-ultraviolet spectro- Lyα production takes place in the inner Solar system. graph. Chakrabarti et al.(1984) reported satellite night- time EUV spectrum of airglow, both looking “down”

9 The reduction is due to a smaller oscillator strength, f2s→4p = 0.1028 and a slightly smaller spectral flux, F (0) = 11 Unfortunately, towards the end of the mission, Solwind was as- −2 −1 −1 104 phot cm s Hz . signed as target for a pilot demonstration of anti-satellite mission 10 The distance to Voyager-2 was computed using a tool provided (ASM) technology. On September 13, 1985 the satellite was de- at https://omniweb.gsfc.nasa.gov/coho/helios/heli.html stroyed by a US Air Force missile. The FUV background 11

(zenith angle between 120◦ and 150◦) and looking “up” Normally, electrons are effective in collisional interac- (zenith angle between 30◦ and 80◦); here zenith angle of tion with neutral particles. However, given the small 0◦ corresponds to the anti-sun direction. The relevant energy difference between 2s and 2p levels, the slower key results are summarized in Table1. moving protons are more effective than electrons in ef- The Spanish Minsat-01 spacecraft (Morales et al. fecting these (primarily) angular-momentum changing 1998) carried a high spectral resolution EUV spectrom- 2s→2p transitions (Purcell 1952; see §C). A reasonable eter, EURD (“Espectr´ografoUltravioleta extremo para night-time temperature for the thermosphere is 1,000 K la Radiaci´onDifusa”; Edelstein et al. 2006). The or- (§E). Let qe+p be the sum of the collisional coefficients bit was a circle with height of 580 km and inclined 151◦ for electrons and protons.12 Matching the inverse mean with respect to the equator. Observations were obtained time between collisions neqe+p to A2γ yields the criti- 4 −3 only during satellite midnight, specifically restricted to cal density ncrit = A2γ /qe+p ≈ 10 cm . The produc- ◦ zenith angle of −85 (just before ground dawn) and tion of two-photons becomes inefficient by (1+ne/ncrit). +80◦ (just after dusk). Restricting to absolute zenith From Figure 13 we see that the proton density is below angle of < 70◦, Lyβ+OI was detected at a level of 6.4 R this critical density for heights above 1,000 km. (L´opez-Moreno et al. 2001). We now summarize the situation. The main consid- The STP 78-1 and EURD measurements are listed as erations for determining the brightness of the thermo- Lyβ+OI because there happens to be a near coincidence sphere are as follows: (1) Ground-based observatories between the wavelength of Lyβ and some resonance lines find 4 R of Hα emission for observations taken towards of O I (see Figure 15 and §D.2 for details). The tran- the zenith at midnight. (2) In LEO the night-time sition is composed of six lines of which three (here- zenith surface brightness values are as follows: Lα ≈ after, the “trio”) lie only +8 km s−1 of the rest wave- 2 kR (HST) and Lyβ+OI ≈ 6–9 R (STP 78-1, Minisat- length of Lyβ and the other three are several hundreds 01). (3) Above 700 km, the contribution to O I 1025 A˚ of km s−1 away. The trio, thus, are readily excited by decreases due to the reduction of the OI column density. solar Lyβ photons. The sum of the oscillator strengths (4) Finally, two-photon decay is suppressed up to an al- for the trio is ftrio = 0.0199 which can be compared titude of 1,000 km due to collisions between H atoms with fβ = 0.0791, the oscillator strength for Lyβ. Above with protons. 700 km the density of H atoms is increasingly larger than We compute an upper limit to the two-photon bright- that of O I atoms (Figure 12). Thus, above 1000 km we ness for LEO missions. We do so by (1) attributing all can ignore excitation of OI λ1025 A.˚ of the observed Lyβ+OI brightness in Table1 to Ly β In order to minimize airglow, HST FUV observations and (2) further assuming that there is no collisional sup- are only taken around orbital midnight. Ake(2012) re- pression of two-photon production above the height of port night time Lyα surface brightness of 2 kR. The dif- STP 78-1 or Minisat-01. In this case, the two-photon ference between this value and the one given in Table1 decay will be ηHαLβ which translates to an upper limit is probably because the latter were taken over larger of 90 CU in the FUV band. zenith angle range and the larger angles probe a larger volume of sun-lit H atoms outside the umbra cast by 8. THE EXOSPHERE Earth (see Figure9). The exobase is the bottom of exosphere (and the top of the thermosphere; see §D). At the high temperature 7.2. Collisions: Critical Density of the exobase, H and He achieve sufficient velocity to es- An H atom will undergo collision on a timescale of cape from Earth (§E). As can be seen from Figure7 the (nq)−1 where q = hσvi, n is the density of the collider, exosphere appears to extend to a distance well beyond σ is the cross-section for the interaction, v is the rel- that of the orbit of the moon. For the purpose of this ative velocity between the collider and the excited H section, we place the exobase conveniently at 1,000 km. atom, and the angular brackets indicate averaging over In Figure8 we display the column density integrated a Maxwellian velocity distribution. The colliders can be along the radius. neutral particles, protons and electrons. For the neutral We now compute Lyβ scattering during Earth eclipse particles we adopt a “hard sphere” model and set the as observed by a satellite in a circular orbit of height cross-section to σ = 10−16 cm2. We compute the sum h. To this end, we integrate the H atom density along of the rates for collisions with all neutral species as a function of height. We find that the time scale for col- 12 Ions such as O+ are even more effective because of their slower −1 lisions is longer than A2γ throughout the thermosphere velocities. However, by 1,000 km heigh the dominant species is (see right panel of Figure 12). H and not O; see Figure 12. 12 Kulkarni

given by Equation6. Adopting F0(ν) as given by Equa- −6 −1 −1 tion5 results in Rβ = 2.94 × 10 atom s . Thus,

N L = 2.94 H R. (11) β 1012 cm−2 3 We set h = 10 km and display Lβ as a function of the satellite zenith angle (Figure9). We see that our simple single-scattering model there is no Lyβ at zenith. However, there is evidence of Lyβ photons in the shadow Figure 7. Top: The density profile of H atoms as derived cylinder. The evidence comes in the form of O I λ 1302- from SWAN-SOHO measurements (Baliukin et al. 2019). 1306 A˚ triplet seen by HST. This triplet is a result of Here, RE is the radius of Earth. The dashed vertical line Bowen fluorescence of O I powered by Lyβ photons (§D). marks orbit of a satellite at height h. Owing to the angular We conclude that there is some two-photon decay in resolution of SWAN the model is not reliable for radius less the thermosphere above HST altitude. We make the than 1.5 RE (shaded region). φ is the angle between line- simplistic assumption that the scattered photons “fill of-sight and the Sun. Thus the night-time radial profile is described by φ = 180◦ model whereas φ = 0◦ applies to the up the trough” in Figure9. If so, the mean value for model at noon time. Note the exosphere has a larger radial Lyβ brightness in the night sky above 1,000 km is ≈ 2 R extent on the night side, compared to the day side (“geo- – consistent with the upper limits from STP 78-1 and tail”). On a timescale of about six months the H atoms EURD observations (§D). which escape Earth are ionized by the solar Lyman contin- The excited H atoms in the 2s state which are exposed uum (see §6). The model data were supplied by I. Baliukin. to sunlight will suffer from solar Hα pumping (Equa- tion8). However, the two-photon decays that matter for observations at satellite midnight arise primarily by H atoms in the shadow cylinder. If so, solar Hα pump- ing is not relevant. In summary, we suggest that, at zenith (anti-sun) and in Earth’s shadow, Lβ ≈ 2 R. The two-photon decay rate is then ηHαLβ ≈ 0.27 R. The cor- responding two-photon emission in the GALEX band is 23 CU.

9. SUMMING UP

Figure 8. The colum density of H I atoms obtained by Table 2. Inventory of isotropic diffuse FUV emission integrating the density displayed in Figure7. Source tracer value B (CU) a line-of-sight bearing a constant zenith angle, z, for WIM Hα 0.4 R 23 such a satellite on the night side. The exosphere density HIM C IV 8625 LU 34 is decreasing reasonably rapidly with radius. So, not a much error is made in replacing the conical umbra with Low-velocity shocks model 0.34 R 16–29 b a cylindrical shadow. IPM Lyβ 2.4 R 7–28 b From vector algebra and trigonometry we find Exosphere Lyβ ∼ 2 R ∼ 23 Total 114–137 d r2 = r2 + l2 + 2r l cos(z), sin(φ) = sin(z), (9) Isotropic [offset] bckg GALEX - 120–180 s s r Note—a Does not include contribution from collisional where r = R + h is orbital radius of a satellite at s E excitations; does not include contribution local HVC height h and l is the distance measured from the satellite contribution. b Scales directly with solar activity (EUV along a given line-of-sight (see Figure9 for geometry). flux); estimates were computed for solar minimum. The resulting Lyβ surface brightness in photon units is 1 L = N R (10) β 4π H β R where, NH = nH (r)dl, the column density along the We started the paper by noting that at high latitude line-of-sight, and the per atom rate of excitation, Rβ, is 120–180 CU of the GALEX FUV could not be accounted The FUV background 13

Figure 9. (Left): Geometry of the Exosphere. The Cartesian coordinate system is centered at the Earth’s center with x towards the Sun and y towards the terrestrial North pole. For simplicity we assume that the Earth’s rotation axis is perpendicular to the Sun-Earth line. The inner-most dark circle with radius RE represents Earth. The radius of the next circle is rs = RE + h and the circle represents the orbit of a satellite at height h (marked “S”). The next two circles have radii of 2RE and 3RE . The impact parameter, a, is the segment OA. The umbra cast by Earth is a cone with a depth of 215 RE . We simplify by replacing the umbra-penumbra combination with a cylinder (dashed lines; “Shadow”). Consider a line-of-sight originating from the aforementioned satellite going towards point B (which is at radius r). Let z be the zenith angle and φ the corresponding geo-centric polar angle. l (OB) is the distance measured from the satellite to point B whereas r is geo-centric distance to point B. (Right): The model Lyβ surface brightness in Rayleigh expected at night time as a function of the fraction of the sky, as measured along angle z of the satellite. by EBL and DGL (§2). In successive sections (§4-8) modeling (radiative transfer of Lyβ in the night sky). we undertook a systematic examination of two-photon Firming up this estimate requires addressing both these emission from the WIM, low-velocity shocks, and the So- lacuanae (see §10.1) lar system as well as line emission from the WIM. The resulting estimates are summarized in Table2 above. 9.1. Temporal Variation We find that 114–129 CU of the GALEX FUV back- The two-photon emission from the IPM as well as the ground can be attributed to these sources. exosphere scales directly with solar activity (solar EUV We now address the uncertainties of the entries in Ta- radiation). The solar EUV irradiance can, over a typi- ble2. In my view, the contributions from the WIM, cal solar cycle, vary by factor of 2.5. In computing the the HIM and IPM rest on sound measurements and ro- contribution from the exosphere we adopted the solar bust theory (WIM: classical theory of recombination; minimum value for solar Lyβ irradiance (Equation5). HIM: direct observations; IPM: basic physics involving Thus, on statistical grounds the estimate given in Ta- branching ratio of Hα versus Lyβ). The mean contri- ble2 is assuredly an underestimate. bution from low-velocity shocks depends directly on the A convenient surrogate for the EUV irradiance is pro- value of η which we have set to 0.05 (McKee & Ostriker vided by the 10.7 cm solar radio flux (“F10.7” index; 1977). If, on the other hand, it is 0.02 (Kim & Ostriker see Tapping 2013). In Figure 10 we display this sur- 2015) then the contribution from low-velocity shocks will rogate over the last two decades. The Sun appears to be reduced to 16 CU. On the other hand, as explained be quite variable not just on long timescales (solar cy- in §4.3 the model used to compute the two-photon emis- cle) but also much shorter timescales. We can expect sion ignores collisional excitation and so undercounts the that solar Lyman lines to equally variable. Parentheti- two-photon decays. A careful modeling of two-photon cally, we note that evidence for variability in the FUV emission from radiatively cooling shocks (starting shock background can be seen in wide-field FUV images of −1 velocity of say, 70 km s ) would address this concern. the sky in which some FUV pointings have a discordant A second concern is that while the two-photon emission “sky” (mode) value when compared with neighboring from the mean infall is small it is possible that localized fields (e.g., Figure 2 of Fesen et al. 2021). infall as seen towards the North Galactic pole Kulkarni In conclusion, it it appears that conventional possibil- & Fich(1985) dominate. ities can explain a good fraction of the GALEX FUV In Table2 we list the IPM contribution in the upwind “isotropic offset” background. direction. In the opposite direction, the two-photon con- tribution will be a quarter of this value or only 7 CU. 9.2. Diffuse NUV emission Our estimate of the contribution from the exosphere is limited by lack of data (Lyβ above 1000 km) and also So far, we have focused on the FUV background. The NUV background, in addition to DGL and EBL, has 14 Kulkarni

photon emission), HIM (line emission) and two-photon emission from three locales in the Solar system: the in- terplanetary medium (IPM), the thermosphere and the exosphere of the Earth. It appears that these contribu- tions collectively can account for two thirds or perhaps even all of the isotropic component. In view of the results presented in this paper it would be useful to carefully review contributions from extra- galactic sources. Akshaya et al.(2019) list 60–81 CU from other galaxies, 16–30 CU from QSOs and < 20 CU from the IGM (and attributed to Martin et al. 1991). Our view of the IGM/CGM has changed substantially since 1991 and it would be useful to revisit two-photon contribution from IGM/CGM. Figure 10. (Top) Light curve of solar 10.7-cm flux density (F10.7) since year 2000. (Bottom) Same data as top but UV missions suffer less background by being in a high with a smaller vertical scale. The shaded region is the pe- orbit (cf. IUE and the planned Spektr-UF/WSO mis- riod during which GALEX was operational. The short spikes sion). This is particularly important for FUV spectro- are probably due to magnetic reconnection. The unit SFU scopic missions for which Lyα is both an important diag- 4 stands for solar flux unit and is equal to 10 Jy. The measure- nostic and also the brightest background. In Figure 11 ments were obtained at the Dominion Radio Astrophysical we compare the Lyα background for a mission in GEO Observatory, Penticton, British Columbia, Canada and the and in HEO. In either case, for the deepest studies, it data were obtained from https://lasp.colorado.edu/lisird/. is advantageous to focus on high Galactic latitude fields in the downwind direction of the IPM (especially if the an additional and dominant source: scattering of so- Sun were to become highly active). lar photons by zodiacal dust. Furthermore, it is bedev- The rest of this section is organized as a collection of ˚ iled by a bright airglow line (O II λλ 2470.2, 2470.3 A). conclusions that were obtained whilst undertaking re- These complications not withstanding, Akshaya et al. search for this paper. First, it is critical to seek observa- (2018) report “an excess emission (over the DGL and tional confirmation of the modeled two-photon emission the EBL) of 120–180 CU in the FUV and 300–400 CU from the Earth’s exosphere (§10.1). Next, two-photon in the NUV”. Applying the conversion factor given in emission from the IPM is a direct consequence of Hα Table3 the two-photon excess in the NUV is modest, emission from the IPM. Modern instrumentation is sen- 20 CU or so. It would useful to undertake a full calcula- sitive enough that direct observations of Hα emission tion to see if processes which are expected to dominate from the IPM can and should be undertaken (§10.2). in the NUV – free-bound and line emission from both Finally, one of the starting point for this paper was the WIM and HIM – are sufficient to explain the NUV offset opportunity offered by the dark FUV sky for low surface emission. brightness imaging. However, to realize this opportunity 10. CONCLUDING THOUGHTS we need detectors which have exacting demands on dark current and read noise. This requirement is reviewed in Diffuse FUV radiation plays a major role in the §10.3. physics and chemistry of the Galactic diffuse interstel- lar medium. It provides primary inputs (heating, ion- ization, dissociation) to the diffuse atomic and molecu- 10.1. Emission from the Lower Exosphere lar medium. As such, there has been considerable ob- We derived an upper limit of 90 CU for the emission servational effort in measuring the FUV background. from radial columns with height above LEO, say 600 km. The diffuse FUV radiation seen at high Galactic lati- In the column between LEO and h = 103 km, two- tudes has two components: FUV emission that is cor- photon emission suppressed. Next, the SWAN-SOHO related with cirrus clouds and FUV that is indepen- model is reliable for geocentric distances > 1.5 RE. dent of cirrus clouds. Some fraction of the latter must Nonetheless, lacking measurements of H density between be due to collective emission of extra-galactic galax- rs and 1.5 RE we an analytical extrapolation of the ies. About 120–180 CU of the diffuse emission, some- the SWAN-SOHO model and computed what is likely time called as “isotropic [offset]” emission, has no ob- a lower limit of 23 CU. Equally, this means that do not vious source(s). This paper investigated FUV emission have a robust understanding of the vertical distribution from the Galactic WIM and low-velocity shocks (two- of Hα production. The FUV background 15

Figure 11. (Left): The geometry of a mission in HEO. The dark central circle represents the Earth. The satellite, “S”, is in a circular orbit of radius R. The zenith angle, z, is measured with respect to the satellite-Earth axis while φ is the corresponding geo-centric polar angle. l is the length along the line-of-sight and r is the geocentric radius to a point along the line-of-sight. (Right): The expected Lyα background (in R) as a function of z for a satellite located in geo-synchronous orbit (7 RE ; e.g., IUE ◦ or the planned Spektr-UF mission) and in HEO (16 RE ). The shaded region covers the zenith angle range of 180 to 180 − 4θE −1 where θE = sin (RE /R) is the angular radius of Earth as seen from the vantage point of the satellite. The Lyα contribution from the IPM can range from 1,000 R to 200 R (see §6). In this figure, the IPM contribution is fixed to 500 R. I undertook a literature search and found no reports of The ISM-Solar system interaction is a major area of Lyβ measurements between rs and 1.5ER. A cube-sat research in space sciences and as such has received con- in a highly elliptical orbit (500 km to 10,000 km) which siderable investment from NASA. For a modest funding can undertake Lyβ and Hα observations would resolve a ground-based facility can be built to detect the ex- the uncertainty of two-photon production in our own pected but faint Hα emission from the IPM. In the up- exosphere. wind direction the surface brightness will be 0.14 R (Ta- A separate concern is that the 2s to 2p collisional co- ble2) and negligible in the down wind direction. This efficients that we used were computed with asymptotic is a challenging measurement for existing ground-based formalism that applies for particles moving at velocities instruments (see Reynolds 1984) but it would be sci- −1 & 10 km s (Purcell 1952). The thermal speeds of pro- entifically rewarding by providing an entirely different tons in the exosphere is smaller, ≈ 4 km s−1. We used measurement of the IPM, particularly given the natural an empirical model (power law fit to the cross-section high velocity precision of Hα spectrometers. as a function of temperature) to extrapolate the colli- sional coefficients at the temperature of the exosphere. 10.3. Dark Current & Read Noise A definitive calculation of the coefficients at T ≈ 103 K Up until now, FUV detectors have been based on some would be reassuring. sort of photo-electric detectors which inevitably involve In closing this subsection I note that HST, operating high voltage to convert the ejected photoelectron into over three decades, provides a trove of geo-coronal Lyα a detectable current. As such, these detectors suffer measurements and indirectly Lyβ measurements (via O I from “dark current” (usually specified by electrons per λ1302-1306 A˚ triplet). The three decades cover nearly second per unit physical area of the detector). The tele- three solar cycles. The analysis of this archival data will scope and camera optics determine the “plate scale” – provide an experimental basis to account for the verti- the relation between the coordinate on detector and the cal distribution of two-photon emission from the ther- angle on the sky. The plate scale fixes the pixel solid mosphere, independent measurements of Lyα emission angle, ∆Ω which when combined with the physics of the from the exosphere and the IPM. Careful modeling of detector fixes the value of D, the rate of dark current per Lyα and the O I 1304 A˚ airglow triplet may inform us of pixel (units: electrons pixel−1 s−1). The Poisson fluctu- the production of two-photon above the orbit of HST. ations of D then translate to the “dark current surface brightness”. Let us consider a “minimum” FUV background of 10.2. Ground-based Hα Mapping of the IPM 300 CU. This translates to 29.2 AB mag arcsec−2. Mul- 16 Kulkarni tiplying this value by a hypothetical FUV bandpass of limited telescope is ηBλ2, regardless of the size of the −6 −2 −1 −2 2 400 A˚ leads to B = 2.8 × 10 phot cm s arcsec . telescope. The ideal detector would have D . ηBλ . Let Aeff = ηAg be the effective collecting area while We now review the properties of an ideal FUV imag- 2 Ag = π(D/2) is the geometric area with D being the ing detector. We start by first considering the case diameter of the circular aperture. The background re- of a photon-counting FUV detector. From the dis- sults in a photoelectron rate per pixel of Aeff B∆Ω. The cussion above, such detectors must have dark current, −5 −1 −1 variance of the counts from the natural background and D . 2.7 × 10 η count pixel s . However, at the the detector dark current increase linearly with time. present time, large format detectors are of the inte- Thus the ratio of the variances, V = BAeff ∆Ω/D  1 grating sort (e.g., CCDs). The output amplifiers result is independent of integration time. We naturally wish in a “read noise” which can be described by Gaussian 2 V  1. This inequality motivates astronomers to build statistics with zero mean and variance σr where σr is bigger telescopes. Fatter pixels are ideal from the per- the noise (electrons) per pixel. Read noise can be re- spective of minimizing the deleterious effect of dark cur- garded as an integration time dependent dark current, 2 rent. However, smaller pixels carry valuable spatial in- Dr = σr /τ where τ is the integration time. If, we as- formation. So, there is inevitable trade between limiting sume τ = 1τhr hour as a typical long exposure, then −4 2 −1 −1 −1 surface brightness and quest for spatial resolution. Dr = 2.8 × 10 σr τhr count pixel s . So, an ideal The FUV camera on the Space Telescope Imaging CCD/CMOS detector for FUV imaging, should have (1) Spectrograph (STIS; Kimble et al. 1998; Woodgate & dark current below 2.7×10−5η count pixel−1 s−1 and (2) Kimble 1999) has exquisitely fine pixels (25 mas). The read noise σr . 0.3 electrons. dark current of the STIS FUV MAMA detectors is a complex subject. Upon turning on the high voltage, D = 6.4 × 10−6 count pixel−1 s−1. Given η ≈ 0.1, the ACKNOWLEDGMENTS nominal value of D is well matched to the demands of Many of the topics discussed here new to me. I am the dark FUV sky. However, other than a small region grateful to my colleagues for help received, education of the detector, the dark current increases at the rate of and feedback. What follows is an incomplete list of −6 −1 −1 1.6 × 10 count pixel s per HST orbit. The value these who helped me. For help with ISM & Atomic adopted for D by the STIS ETC (Exposure Time Cal- Physics: Ronald Reynolds; Michael Shull, University of −4 −1 −1 culator) is about 2 × 10 count pixel s . Thus, in Colorado at Boulder; E. Sterl Phinney, California In- practice, the noise from the dark current of the MAMA stitute of Technology. For help with FUV background: detectors completely dominates the noise budget in the Jayant Murthy, Indian Institute of Astrophysics. For 13 FUV band. help with Aeronomy: Edwin J. Mierkiewicz & Matthew In contrast, GALEX chose the fat pixel approach. D. Zettergren, Embry-Riddle Aeronautical University; I. The total detector dark current of the GALEX FUV I. Baliukin, Space Research Institute (IKI), Moscow. E. −1 detector was 78 s (of which more than half was from C. Stone, Project Scientist for Voyager Mission, for help “hotspots”) while the total detector rate from the dark- with location of the Voyager missions. For help with −1 est sky was 1000 s (Morrissey et al. 2007). The 5- CHIANTI: Nik Zen Prucinski, California Institute of arcsecond PSF of GALEX was well suited to the prin- Technology. For detector physics: Roger Smith, Caltech cipal objective of the mission: matching GALEX sky Optical Observatories. For feedback: Kevian Stassun, survey to the Palomar Observatory Sky optical Survey Vanderbilt University and Jerry Edelstein, University at the arcsecond level. of California at Berkeley. Finally, I am most grateful to Now let us consider the opposite regime: a diffraction- Robert Benjamin,University of Wisconsin at Madison; limited telescope of diameter D. In this case, diffraction Hannah Earnshaw, California Institute of Technology; 2 theory informs us that AgΩ ≈ λ where, for convenience, Eran Ofek, Weizmann, Institute of Science; and Michael we set the pixel length equal to the diffraction limit, Shull, University of Colorado at Boulder for their careful ≈ λ/D. Thus, the background rate for a diffraction- reading and constructive feedback.

REFERENCES

13 The STIS ETC does not mention the isotropic FUV background – the subject of this paper. However, this omission is of little practical consequence given the high dark noise of the MAMA detector. The FUV background 17

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APPENDIX

A. TWO-PHOTON CONTINUUM 2 An H atom in the 2s S1/2 radiatively decays by emitting two photons. The energies of the two photons add up to the energy of a Lyβ photon or 10.2 eV. The classical references for the two-photon spectrum is Spitzer & Greenstein (1951) where they invoked the two-photon process to explain the continuum of planetary nebulae. The standard modern reference for the two-photon spectrum is Drake et al.(1969). Here, for the A-coefficient, we use the fitting formula of Nussbaumer & Schmutz(1984):

A(y) = CY [1 − (4Y )γ ] + αCY β(4Y )γ (A1) where y = ν/ν0, Y = y/(1 − y), ν0 = c/λ0 and λ0 = 1215.67 A˚ is the wavelength of Lyα. The fit values are α = 0.88, β = 1.53, γ = 0.8 and C = 202.0 s−1. Since each de-excitation results in the emission of two photons, R 1 −1 0 A(y)dy = 2A21 = 16.4 s . Noting that A(y) is the probability of emitting a photon in the frequency interval dy = dν/ν0 we find the emissivity from a single decay to be 1 hν jν = A(y). (A2) 4π ν0

Traditionally, observers use the spectral intensity as a function of wavelength, jλ which can be shown to be jν ν/λ. The corresponding photon intensity is n(λ) = jλ/(hc/λ). While A(y) peaks at y = 1/2 corresponding to ν = ν0/2 the spectral intensity jλ peaks at 1400 A˚ (Figure1).

Table 3. GALEX bands & response to two-photon decay

Parameter unit FUV NUV Bandpass A˚ 1350–1750 1750–2800 FoV deg2 1.267 1.227 I cm2A˚ 9402 45008

∆λeq A˚ 255 730 n(2γ) count 0.27 0.44 CR phot s−1 309 838

B2γ CU 85.1 18.4 Note—The vital statistics of the FUV and NUV channels are summarized in the top half of the R table. Here, I = Aeff (λ)dλ and ∆λeq ≡ I/max(Aeff ). n(2γ) is the number of photons de- tected in each channel for a single two-photon de- cay. The last two lines are the response by the two GALEX channels to a uniform background from a column of 106 decays cm2 s−1. CR is the counting rate across the entire detector for a two- photon while B2γ is the inferred surface brightness (in CU).

The GALEX FUV band is formally 1350–1750 A˚ while the NUV band is 1750–2800 A˚ (see Table3). At one end of the probability distribution function (Equation A1), y → 1/2, the two photons have equal energy in which case the NUV band will register two photons. At the other end, y → 0, one photon will get registered in the FUV band and the other will be in the optical/infra-red (OIR) band. Integrating over A(y) we find that each decay results in 0.42 20 Kulkarni photons in the 1350–1750 A˚ band, 0.67 photons in the 1750–2800 A˚ band and 0.82 in the OIR band. [If the division is done by energy then the corresponding fractions are 33%, 37% and 30%]. However the GALEX passbands are not flat. Morrissey et al.(2007) provide a summary of the instrumental parameters14 as a function of λ. For surface brightness, the FoV also matters. Morrissey et al.(2007) only provide a single FoV value, Ω, for each of the two channels. So we assume that, within each band, the FoV is independent of wavelength. For each channel, the band-pass weighted quantities for an input two-photon spectrum, is computed as follows, 1 Z ∆λ Z j j = j A (λ)dλ, n(2γ) = eq λ λ A (λ)dλ (A3) λ ΩI λ eff I hc eff where I, the area-bandwidth product and ∆λeq, the equivalent width, are defined in Table3. The unit for n(2γ) (listed in Table3) is count, resulting from our choice of 1 sr, an area of 1 cm 2 and an integration time of 1 s. As can be seen from Table3, each decay results in 0.27 photon in the FUV band and 0.44 photon in the NUV band. Consider an astronomical object emitting only via the two-photon process. Integrating over the pixels of this object’s image, the ratio of FUV to NUV count rates is expected to be about 0.37. We end this section by noting Bracco et al.(2020) undertook a similar exercise but with a somewhat different approach. The two results agree to better than 10%.

B. ATOMIC PHYSICS: OSCILLATOR STRENGTH

I provide a brief summary of the essential atomic physical formulae. Astronomers prefer to use fλ, the spectrum as a function of wavelength, whereas books on atomic physics use fν . The transformation, though simple, more often than not leads to confusion and error. Furthermore, there are differing definitions for the Einstein A-coefficients and the oscillator strength, which can (and have) caused further confusion (see Hilborn 1982). The two basic equations are

2 2 2 8π e gl 0.6670 gl 2 −1 πe Aul = flu = 2 flu cm s , σlu(ν) = fluφν . (B4) mec gu λlu gu mec

Here, l (u) stands for lower (upper) levels and σlu(ν) is the absorption cross-section. φν is the probability distribution R as a function of frequency for the absorption process, φν dν = 1. The Doppler relation, v/c = 1 − ν/νlu links the probability distribution of the radial velocity, pv, and φν , namely, φν = pvdv/dν. The classical frequency-integrated 2 −2 −2 −1 cross-section is πe /(mec) = 2.659 × 10 cm s . This probability density function is the convolution of the natural broadening function (Lorentzian with a frequency width of Aul/(2π)) and a 1-D Gaussian distribution which accounts for Doppler-induced frequency shifts of moving atoms. In this paper, the columns are, at worst, mildly optical thick columns which allows us to entirely neglect the effect of natural broadening. Under this assumption the rate at which R an atom is excited by the incident flux of photons is R = σlu(ν)Fν dν where Fν is the photon spectral flux density, carrying the unit phot cm−2 s−1 Hz−1. If, as it happens to be the case, the photon intensity is constant over the spread in frequency due to thermal motions and that there is no relative radial velocity, v, between the absorbing photons and the Lyman series photons, then Fν = Fν (v = 0) = Fν (0) then Z 2 πe −1 R = Fν (0) σlu(ν)dν = Fν (0) flu s . (B5) mec

However, in aeronomy and planetary science it is conventional to use, Fλ, photon flux per unit A.˚ This quantity is related to F as follows: ν ν 108F = F (B6) λ λ ν 8 where the factor of 10 accounts for the fact that Fλ is the flux density per A˚ and not per cm.

C. ATOMIC PHYSICS: HYDROGEN DATA The A-coefficients for H I that are essential for this paper are given in Tables4. A critical element of the paper is 2 2 collisional perturbation of H atoms in the 2s S1/2 state. Recall that this level straddles P1/2 (−1058 MHz; famous

14 The tables for the effective area for FUV and NUV de- tectors were obtained from https://asd.gsfc.nasa.gov/archive/ galex/tools/Resolution Response/index.html The FUV background 21

Table 4. A-coefficients for selected lines of H I

−1 −1 −1 lower, l El cm upper, u Eu cm Aul (s ) 2 2 8 1s S1/2 0 2p P1/2 82 258.919 6.26 × 10 2 2 8 1s S1/2 ” 2p P3/2 ” 259.285 6.26 × 10 2 2 8 1s S1/2 ” 3p P1/2 97 492.211 1.67 × 10 2 2 8 1s S1/2 ” 3p P3/2 ” 492.320 1.67 × 10 2 2 7 1s S1/2 ” 4p P1/2 102 823.845 6.82 × 10 2 2 7 1s S1/2 ” 4p P3/2 ” 823.894 6.82 × 10 2 2 7 2p P1/2 82 258.919 3d D3/2 102 822.319 5.39 × 10 2 2 7 2s S1/2 ” 8.954 3p P3/2 ” 2.320 2.25 × 10 2 2 6 2p P1/2 ” 8.919 3s S1/2 ” 2.221 2.10 × 10 2 2 7 2s S1/2 ” 8.954 3p P1/2 ” 2.211 2.25 × 10 2 2 7 2p P3/2 ” 9.285 3d D5/2 ” 2.356 6.46 × 10 2 2 7 2p P3/2 ” 9.285 3d D3/2 ” 2.319 1.08 × 10 2 2 6 2p P3/2 ” 9.285 3s S1/2 ” 2.221 4.21 × 10 Note— The symbol ‘”’ stands for adopting significant dig- its from the same column but in the previous row. Thus, in the second row, El = 0 while Eu = 82 259.285. Mea- surements/data from https://physics.nist.gov/PhysRefData/ ASD/lines form.html.

Table 5. Collisional Coefficients for H I 2s → 2p

3 −1 transition collider q0 (cm s ) γ 2 2 + −4 2s S1/2 → 2p P1/2 p 2.51 × 10 −0.27 2 2 −4 2s S1/2 → 2p P3/2 ” 2.23 × 10 −0.03 2 2 − −4 2s S1/2 → 2p P1/2 e 0.22 × 10 −0.37 2 2 −4 2s S1/2 → 2p P3/2 ” 0.35 × 10 −0.37 Note—Collisional coefficient at temperature T = 4 γ 10 T4 k is given by q = q0T4 . Condensed from Os- terbrock(1974).

2 Lamb shift) and P3/2 (+9911 MHz). Since the 2s state is metastable there was a hope that the Lamb shift could be observed in an astronomical setting (Wild 1952). The collisions here are different than other well-studied collisions in which energy differences are comparable to or larger than thermal energies. The dipole elements connecting the 2s to the 2p levels are large. Purcell(1952) showed that distant encounters with protons are more effective than electrons (which had been studied in earlier literature) in changing the angular momenta of H atoms. The computed collisional coefficients are given in Table5. The resulting modest critical density, ≈ 104 cm−3, significantly reduces the signal of the radio transitions from prime targets such as dense HII regions (Dennison et al. 2005).

D. THE THERMOSPHERE The troposphere (0–12 km) contains most of the atmosphere. The temperature within the stratosphere (12–50 km) increases with height owing to absorption of solar UV by ozone. Once the ozone is dissociated temperature starts to decrease again with height until dissociation of oxygen and other molecules begins in the thermosphere. The mesosphere (50–85 km) is the layer between the stratosphere and the thermosphere. It is in this layer in which meteors burn, providing a convenient layer of Na I for laser guide-star adaptive astronomy. 22 Kulkarni

Figure 12. (Left) MSIS-E-90 atmosphere model for latitude of 30◦ and longitude of 0◦ on 01-January-2000 at a local time of 0100. From https://ccmc.gsfc.nasa.gov/modelweb/models/msis vitmo.php (Right). The run, with height, of the mean time P −1 −16 2 between collisions between an H atom and a neutral atom, τ = (σ0 nivi) where σ0 = 10 cm is the characteristic “hard p i sphere” approximation for the cross section and ni and vi = 3kT/mi are the number density and thermal velocity of species i.

The thermosphere (80–700 km) is the region in which low-earth satellites are located (e.g., International Space Station: 420 km; HST 540 km; Swift Observatory: 550 km; GALEX: 685 km; FUSE, 750 km). The temperature in the thermosphere increases with height, owing to the absorption of solar EUV radiation by majority molecular species. Thus atoms are the primary constituents of the thermosphere. The temperature is strongly dependent on the solar EUV irradiance, ranging from 800 K to 2000 K (with strong day/night dependence). The radial extent of the thermosphere is quite sensitive to solar EUV, puffing up to 1000 km during solar maximum and receding to 500 km during solar minimum.

+ + + + + + Figure 13. (Top) The run of electrons with altitude. (Bottom) The run of H , He ,O and other ions (“M ”: O2 ,N and NO+), but expressed as a percentage of the electron density. The FUV background 23

The density, ionization fraction and temperature of the thermosphere were obtained from the Community Coordinat- ing Modeling Center (CCMC) portal15: MSIS-E-90 (“Mass Spectrometer and Incoherent Scatter radar - Exosphere- [19]90”) for the run of neutral particles (Figure 12) and IRI-2016 (International Reference Ionosphere – 2016).

Figure 14. Radial profile of electrons (equatorial plane). Note the unit for electron density is m−3. Figure supplied by Matthew D. Zettergren.

D.1. The Ionosphere As can be seen from Figure 13 the IRI-2016 model for electrons stops at 2000 km. The higher altitude profile was generated, at my request, by Matthew Zettergren, Embry-Riddle Aeronautical University. The GEMINI open-source ionospheric model16 was used in a 2-D meridional, dipole configuration to simulate plasma density evolution over several days (e.g., Zettergren & Snively 2015). The grid used covers ±60◦ in latitude, corresponding to apex altitudes of about 32,000 km (altitude of the model grid above magnetic equator). GEMINI solves conservation of mass, momentum, and energy equations for the ionospheric plasma for 6 ion species relevant to the terrestrial ionosphere, including protons. The model was run moderate for high solar and geomagnetic activity levels of F10.7=129.5, F10.7a=104.7, and solar index Ap=37. The date of the simulation is 10/6/2011 (near equinox) and the UT is about 5:45 (corresponding roughly to noon local time), representing a typical daytime plasma density state. Figure 14 shows a profile extracted from the geomagnetic equator. The results are meant to be illustrative of plasmasphere conditions during geomagnetically quiet times.

D.2. Bowen Fluorescence of O I by Lyβ The reason that the measurements in Table1 are listed as Ly β+OI is because there happens to be a near coin- cidence between Lyβ and an excited state of O I (Meier et al. 1987; see Figure 15 for a partial Grotrian diagram of O I). The atomic parameters for the resulting six transitions between the ground state and the excited state, 2 2 3 4 o 3 1s 2s 2p ( S )3d D1,2,3, are given in Table6. Next, as can be gathered from Table6, the excited OI atom has a probability of 5/8 to return to ground state and 3/8 probability of decaying to 1s22s22p3(4So)4s with subsequent cascade to ground state with the last lap involving the famous O I λ1304 A˚ triplet. This Bowen fluorescence explains the brilliance of the airglow O I triplet.

E. THE EXOSPHERE The exosphere is defined as the region in which the collisions of neutral particles with each other ceases to be important. If so, neutral particles are on ballistic trajectories with paths determined by initial conditions. The base of the exosphere (“exobase”) depends on solar activity but a typical value is 500 km. Three families of particles are defined as follows: “ballistic” – particles that lack sufficient speed and so fall back; “escapers” – particles which have

15 https://ccmc.gsfc.nasa.gov/about.php 16 https://github.com/gemini3d 24 Kulkarni

Figure 15. Partial Grotrian diagram for O I. (Right): Solar Lyβ photons excite O I atoms from the ground 2 2 4 3 2 2 3 4 0 3 o state,1s 2s 2p P0,1,2, to the 1s 2s 2p ( S )3d D state. The O I atom can decay back to the ground state or decay to the 1s2s222p3(4So)3p state which then decays to ground state emitting, along the way, the famous O I triplet λ1302.17, 1304.86, 1306.0 A˚ photons. (Left): Grotrian diagram (not to scale) restricted to the six allowed transitions between the ground state and 1s22s22p3(4S0)3d 3Do. The wavelength for each transition is converted to a velocity w.r.t. the rest wavelength of Lyβ. The rightmost three lines (black color) have sufficiently small velocity shifts, 8–9 km s−1, that they can be excited by solar Lyβ photons. These three lines are referred to as the “trio” in the main text. The remaining three (gray) have large velocity shifts, 500–700 km s−1, and so cannot be excited by solar Lyβ.

Table 6. Wavelength & Oscillator Strengths for selected O I levels

−1 −1 −1 λ(A)˚ lower upper Aul(s ) flu El(cm ) Eu(cm ) 3 3 o 7 1025.762 P2 D3 7.66 × 10 0.0169 0 97 488.538 3 3 o 6 −4 1025.763 P2 D1 2.11 × 10 2 × 10 ” 97 488.378 3 3 o 7 1025.763 P2 D2 1.91 × 10 0.0030 ” 97 488.448 3 3 o 7 1027.431 P1 D2 5.71 × 10 0.0151 158.265 97 488.448 3 3 o 7 1027.431 P1 D1 3.17 × 10 0.0050 ” 97 488.378 3 3 o 7 1028.157 P0 D1 4.22 × 10 0.0200 226.977 97 488.378 3 3 o 7 1,1286.34 P1 D2 2.32 × 10 0.2833 88 630.587 97 488.448 3 3 o 7 1,1286.40 P1 D1 1.29 × 10 0.2380 ” 630.587 97 488.378 3 3 o 7 1,1286.91 P2 D3 3.09 × 10 0.1700 ” 631.146 97 488.538 3 3 o 6 1,1287.02 P2 D2 7.74 × 10 0.1700 ” 631.146 97 488.448 3 3 o 5 1,1287.11 P2 D1 8.60 × 10 0.1020 ” 631.146 97 488.378 3 3 o 7 1,1283.72 P0 D1 1.72 × 10 0.5097 ” 631.303 97 488.378

Note— For El and Eu ‘”’ stands for adopting significant digits above the pre- vious row. The table has two sections separated by a horizontal line. For the 2 2 4 3 top part, the lower level is the ground state, 1s 2s 2p P2,1,0 and the upper 2 2 3 4 o 3 level is the excited state 1s 2s 2p ( S )3d D1,2,3. For the lower part, the up- per level is 1s22s22p3(4So)3p and the lower level is 1s22s22p3(4So)4s. See also Figure 15. The atomic data are from https://physics.nist.gov/PhysRefData/ ASD/lines form.html.

sufficient speed to escape; and “satellite” — particles which do undergo a rare collision (R . 2.5 RE) which sends them back down. Thus the particle density of the exosphere is not a simple power law. The temperature in the exosphere decreases to one third of the base value at 4 RE and two-fifth at 10 RE; here, RE is the radius of Earth (≈ 6, 400 km). We adopt the “standard” temperature of 1025 K (cf. ØStgaard et al. 2003). The corresponding thermal rms velocity is 2.9 km s−1. The FUV background 25

E.1. IMAGE IMAGE (Imager for Magnetopause-to-Aurora Global Exploration) was a NASA Medium Explorer class mission that was designed to study the response of Earth’s magnetosphere to changes in the solar wind. It was launched into a highly elliptical orbit (1,000 km×46,000 km) with an inclination of 90.01◦ and an orbital period of about 14 hours. Its payload included an FUV imaging system which included the “GEO” photometer (Mende et al. 2000). The three photometers, oriented differently, respond to radiation coming from within their 1-degree FoV in the wavelength range 1150–1500 A.˚ This instrument was designed to measure the brightness of the geo-coronal Lyα. ØStgaard et al.(2003) provide a two-exponential model fit to the observed brightness. An example fit is

I(r) = 16.9 exp(−r/1.03) + 1.06 exp(−r/8.25) kR (E7) where I(r) is the Lyα intensity and the radius r is in units of RE. Under the (admittedly simplistic) assumption of the medium being optically thin the authors invert the observations and provide density profile for H atoms.

E.2. SWAN-SOHO The Solar & Heliospheric Observatory (SOHO) is a ESA-NASA mission that is located in the vicinity of the Earth- Sun L1 region and focused on the studies of the atmosphere of the Sun, the solar wind and helio-seismology. It carries Solar Wind ANistoropies (SWAN) instrument whose primary goal is to study the structure of the solar wind through its interaction with the IPM. A hydrogen cell acts by absorbing the incident solar light at the rest wavelength of Lyα. In effect, the cell provides a spectral resolution of 105 (Bertaux et al. 1995). SWAN data has also been use to study the distribution of geo-coronal H atoms (Baliukin et al. 2019).