Inferring the Composition of Super-Jupiter Mass Companions of Pulsars with Radio Line Spectroscopy

Stanimirovic et al. (2003) and Weisberg et al. (2005) detected OH absorption in PSR B1849+00 and PSR B1641−45 using Arecibo and Parkes radio telescopes, respectively, out of a total of 25 pulsars. In addition, Minter (2008) detected OH absorption for PSR B1718−35 by using the Green Bank telescope (GBT) out of a sample of 16 pulsars. While PSR B1641−45 has a relatively large mean flux density of ${S}_{1400}=310\,\mathrm{mJy}$, the other two have 11 mJy (PSR B1718−35) and 2.2 mJy (PSR B1849+00) respectively. Thus, typically, only about 10% of the searched for targets show evidence of OH absorption (or stimulated emission) in pulsar lines of sight. The detections have been made only for low-galactic latitude pulsars (see galactic (l, b) in Table 2). However, almost all target pulsars listed in Table 1 turn out to be high galactic latitude pulsars. Since OH is generally strongly confined to the galactic plane, any detection of OH absorption (or stimulated emission) in these systems will be related to the binary system with high probability rather than being interstellar. The intrabinary molecular gas signature can be distinguished from circumbinary molecular gas by the orbital phase modulation (i.e., when the pulsar is near the inferior conjunction). Binning the recorded data with respect to the companion orbital phase would allow the exclusion of a low signal to noise ratio (S/N; for molecular gas) in high impact parameter data (when the companion orbital position is far away from the inferior conjunction). In the case of very high S/N of atomic/molecular gas in the background of a bright pulsar, it may even be possible to study the variation of absorption and emission with the impact parameter, which can then lead to information about the radial structure of the ablated wind from the companion. As only times close to the total eclipse would have substantial OH absorption in the trailing wind, only the fraction of the orbit near the eclipse ingress and egress have to be added from multiple orbital cycles.

Table 1.  Black Widow and Other Pulsar Targets for OH Line Spectroscopy^a

Pulsar	$l,b$	${P}_{\mathrm{spin}}$	W50	${P}_{B}$	Min. ${{\rm{m}}}_{c}$	Ageb	$\dot{E}$	D	S1400	References
	(deg)	(ms)	(ms)	(hr)	${{\rm{M}}}_{\mathrm{Jup}}$	(109 year)	${L}_{\odot }$	(kpc)	(mJy)
J2051−0827	39.2, −30.4	4.51	0.34	2.4	28.3	5.61	1.4	1.28	2.8	(1), (2), (3)
J2241-5236	237.5, −54.9	2.19	0.07	3.4	12.6	5.22	6.5	0.68	4.1	(4)
J0751+1807	202.7, −21.1	3.48	0.70	6.3	134d	7.08	1.9	0.40	3.2	(4), (8), (9)
J1012+5307	160.3, +50.8	5.25	0.69	14.4	112d	4.86	1.2	0.70	3.0	(4), (10), (11)
J1807−2459Ac	5.8, −2.2	3.06	0.33	1.7	9.4	⋯	⋯	2.79	1.1	(5), (6), (7)
B1259−63	304.2, −1.0	47	23	1236d	$10\,{M}_{\odot }$	$3\times {10}^{5}\,\mathrm{year}$	207	2.3	1.7	(12), (6), (13)
B1957+20	59.2, −4.7	1.61	0.035	9.1	22	2.2	40	1.53	0.4	(14), (3)
J1227−4853e	298.9, +13.8	1.69	⋯	6.8	221d	2.4	24	2.00	⋯e	(15)
J1723−2837	357.6, +4.26	1.86	0.20	14.8	247d	3.9	12	1.00	1.1	(16)

Notes.

^aPulsars with moderate mean flux density S1400; from ATNF Pulsar Database (Manchester et al. 2005): www.atnf.csiro.au/people/pulsar/psrcat/ PSR B1957+20 is included for comparing parameters. ^b $\mathrm{Age}={P}_{\mathrm{spin}}/2\dot{P};$ $\dot{P}$ uncorrected for the Shklovsky effect. ^cIn globular cluster NGC6544; pulsar spindown age and luminosity are uncertain due to acceleration effects. ^dThese companions correspond to red back pulsars. ^eEstimated flux in 1.4 GHz band is $1.6\mathrm{mJy}$ using the reported 607 MHz flux and spectral index −1.7 from (Roy et al. 2015).

References.  (1) Stappers et al. (1996), (2) Doroshenko et al. (2001), (3) Kramer et al. (1998), (4) Keith et al. (2011), (5) Ransom et al. (2001), (6) Hobbs et al. (2004), (7) Lynch et al. (2012), (8) Lundgren et al. (1995), (9) Nice et al. (2005), (10) Nicastro et al. (1995), (11) Lazaridis et al. (2009), (12) Johnston et al. (1992), (13) Melatos et al. (1995), (14) Fruchter et al. (1988), (15) Roy et al. (2015), (16) Crawford et al. (2013).

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In the context of the BW PSR B1957+20, one can expect sufficiently large column depths of OH molecules near the eclipsed region, i.e., in orbital phases close to the inferior conjunction to show moderate line absorption with reasonable equivalent width (EW). This is because the gas can cool adiabatically with distance from the ablating companion star (see below) to enable molecule formation. Unfortunately, the mean flux density (${\rm{S}}1400=0.4\,\mathrm{mJy}$) for this pulsar is too low for an easy detection by existing telescopes. Table 1 lists the characteristics of pulsars that may be suitable for detection of molecular lines through radio spectroscopy. Note that we include here PSR B1957+20 for comparison—because of its very low mean flux density at 1400 MHz, it is not meant to be a target for OH detection with the current generation of telescopes. On the other hand, PSR B1259−63 has a massive Be star companion that does not belong to the BW or red back classes but whose high ${\rm{\Delta }}{DM}$ post- and pre-eclipse phases last for tens of days during which an OH line spectroscopy can be carried out. Table 2 lists pulsars from which OH line absorption has been observed in the ISM during the pulse "on" phase. For PSR J1641−45, not only has line absorption at 1612 MHz, 1665 MHz, and 1667 MHz been detected, but even stimulated emission at 1720 MHz driven by the pulsar during its pulse "on" phase has been detected by Weisberg et al. (2005). This is the only pulsar, an extremely bright one, from which the stimulated emission was discovered, not among the other fainter ones in Table 2. Therefore, the use of stimulated emission to probe the conditions of the companion's wind in a binary pulsar may be possible if such an extremely bright pulsar were to be discovered in a binary system. The examples in Table 2 give column depths of OH line absorption already detected against isolated and bright pulsars with existing telescopes in reasonable exposure times. They are "ordinary" pulsars with duty cycles ($=W50/{P}_{\mathrm{spin}}$) ranging from $1.8 \%$ to $10.8 \%$, moderate radio flux densities in the 1400 MHz band close to where the OH line absorption at 1612–1720 MHz band is expected and are young pulsars with spindown power comparable to the solar luminosity. The targets listed in Table 1 are binary, millisecond pulsars with high spindown power and moderate radio flux densities at 1400 MHz, with ultra-low-mass (or even comparable to Jupiter mass) companions in close orbits (${P}_{\mathrm{orb}}\sim 2\mbox{--}14\,\mathrm{hr}$). Note that their duty cycles are comparable to those in Table 2 even though they are much more rapidly spinning and their moderate mean flux densities at 1400 (1600) MHz and their small orbital dimensions make them good targets for H i/OH absorption studies against their pulsed flux. BWs among the entries in Table 1 (the first two) have ultralight companions that are partially degenerate and stripped due to ablation by the pulsar. X-ray studies of BW pulsars show that their non-thermal X-ray emission are orbitally modulated, which has been related to the intrabinary shocks close to the companion (Phinney et al. 1988; Roberts 2013).

Table 2.  Pulsars with OH Absorption Detected Against Pulsed Emission

Pulsar	l, b	${P}_{\mathrm{spin}}$	W50	Agea	$\dot{E}$	D	S1400	${N}_{\mathrm{OH}}/{T}_{\mathrm{ex}}$	Telescope	Exp.	References
	(°)	(ms)	(ms)	(105 year)	(${L}_{\odot }$)	(kpc)	(mJy)	(${10}^{14}\,{\mathrm{cm}}^{-2}\,{{\rm{K}}}^{-1}$)		(hr)
B1641−45	339.2, −0.2	455	8.2	3.59	2.2	4.5	310	0.28b	Parkes	5	(1)
B1718−35	351.7, +0.7	280	26	1.76	11.8	4.6	11.0	0.1–0.67	GBT	$\lt 10$	(2)
B1849+00	33.5, +0.02	2180	235	3.56	0.1	8.0	2.2	2.7	Arecibo	3.4	(3)

Notes.

^a $\mathrm{Age}={P}_{\mathrm{spin}}/2\dot{P}$. Note the different units of age used in Table 1. ^bCalculated from $\tau =0.03$ at 1667 MHz with FWHM ${\rm{\Delta }}v=2\,\mathrm{km}\,{{\rm{s}}}^{-1}$ reported in Reference (1).

References.  (1) Weisberg et al. (2005), (2) Minter (2008), (3) Stanimirovic et al. (2003).

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The eclipse duration in PSR B1957+20 is a function of observing radio frequency. Rasio et al. (1989) and Wasserman & Cordes (1988) argued that the radio radiation is being absorbed with a frequency dependent cross-section rather than refraction and reflection from ionized plasma wind (Phinney et al. 1988). Radio line absorption in the OH 1.6 GHz band against pulsed flux can be easily associated with interstellar versus in situ (planetary or intrabinary) origins by the independent evidence of binarity and a correlated change of OH absorption with orbital phase near eclipse ingress and egress.

The observed line intensity profile of OH absorption can be written in terms of the OH column density N_OH as (Hewitt et al. 2006)

$$\begin{eqnarray*}{I}_{\nu } & = & (1-f){I}_{0\nu }\\ & & +f\{{I}_{0\nu }\,\exp (-{\tau }_{\nu })+{B}_{\nu }({T}_{\mathrm{ex}})[1-\exp (-{\tau }_{\nu })]\}\end{eqnarray*}$$

where ${I}_{0\nu }$ is the background continuum intensity, represented by the pulsed radio flux during the pulsar "on" phase, ${B}_{\nu }({T}_{\mathrm{ex}})$ is the Planck function, T_ex is the excitation temperature of the four transitions that represent the OH molecular lines in the L band, ${\tau }_{\nu }$ is the optical depth at frequency ν, and f is a telescope beam filling factor (which we take to be unity as only the pulsar "on" phase is compared with the flux in between the pulses, i.e., "off-pulse"). This can be rewritten in terms of the brightness temperature, assuming that ${T}_{\mathrm{ex}}\gg {\rm{h}}{\nu }_{\mathrm{OH}}/{{\rm{k}}}_{{\rm{B}}}=0.08\,{\rm{K}}$, where the Rayleigh–Jeans approximation holds:

$$\begin{eqnarray*}&&T(v)-{T}_{0}=f({T}_{\mathrm{ex}}-{T}_{0})\{1-\exp \ [-\tau (v)]\}.\end{eqnarray*}$$

Here, $\tau (v)$ can be written in terms of the optical depth at the line center ${\tau }_{0}$ with a Gaussian profile:

$$\begin{eqnarray*}&&\tau (v)={\tau }_{0}\,\exp [-{(v-{v}_{0})}^{2}/(2{\sigma }^{2})]\end{eqnarray*}$$

where the line width σ is related to the velocity dispersion ${\rm{\Delta }}v$ through $\sigma ={\rm{\Delta }}v/{(8\mathrm{ln}2)}^{1/2}$. The optical depth at the line center depends upon the OH molecular column density N_OH, the corresponding Einstein A-coefficient, the degeneracies of the upper and lower states and the occupation fraction of the initial state, and the excitation temperature T_ex, and it can be collectively written as

$$\begin{eqnarray}&&{\tau }_{0}=a\displaystyle \frac{{N}_{\mathrm{OH}}({10}^{15}\,{\mathrm{cm}}^{-2})}{{T}_{\mathrm{ex}}({\rm{K}}){\rm{\Delta }}v(\mathrm{km}\,{{\rm{s}}}^{-1})}\end{eqnarray} \tag{ 1 }$$

with the constant a = 0.454, 2.345, 4.266, and 0.485 for the 1612, 1665, 1667, and 1720 MHz lines respectively (see Hewitt et al. 2006). For the strongest "main" line at 1667 MHz, this leads to the column depth:

$$\begin{eqnarray}&&{N}_{\mathrm{OH}}=2.3\times {10}^{14}\,{\mathrm{cm}}^{-2}\,({T}_{\mathrm{ex}}/{\rm{K}})\int \tau (v){dv}/(\mathrm{km}\,{{\rm{s}}}^{-1}).\end{eqnarray} \tag{ 2 }$$

As an example, Stanimirovic et al. (2003) obtained for interstellar OH absorption against PSR B1849+00, ${\tau }_{\max }\,=0.4\pm 0.1$ at 1665 MHz and ${\tau }_{\max }=0.9\pm 0.1$ at 1667 MHz with corresponding line-width FWHM of $1.5\pm 0.4\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and $1.1\pm 0.2\,\mathrm{km}\,{{\rm{s}}}^{-1}$. The corresponding ${N}_{\mathrm{OH}}/{T}_{\mathrm{ex}}$ are tabulated in Table 2.

4.2.1. Eclipsing Gas Temperature

4.2.2. OH Column Depths

To estimate the column depth of N_OH from beyond the eclipsing region, we may adopt ${T}_{\mathrm{ex}}\sim 100\,{\rm{K}}$ and a line-width of FWHM $\sim \,3\,\mathrm{km}\ {{\rm{s}}}^{-1}$ (see Section 5.1). Since the ablated wind from the companion is primarily in the tangential direction, or may reside for long periods in a Keplerian disk centered on the neutron star, the effective velocity width of the radiatively connected region in the line of sight along the pulsar beam may not exceed the thermal speed by much. Therefore, a velocity resolution of $1.0\,\mathrm{km}\,{{\rm{s}}}^{-1}$ would sufficiently resolve the absorption profile in several bins. Adopting ${\tau }_{0}=0.3$ at 1667 MHz (similar to that observed in interstellar OH absorption against PSR B1718−35 with a comparable spectral resolution as in Minter 2008), one requires an intra-system column depth of ${N}_{\mathrm{OH}}=2.28\times {10}^{16}\times 0.9\sim {10}^{16}\,{\mathrm{cm}}^{-2}$.

We argue that this column density is comparable to what is derived from the mass-loss rate from the pulsar companion in a wind outflow:

$$\begin{eqnarray*}&&{\dot{M}}_{w}={f}_{{\rm{\Omega }}}4\pi \rho {r}^{2}{v}_{w}.\end{eqnarray*}$$

Here ${f}_{{\rm{\Omega }}}$ is the fraction of the total solid angle covered by the wind ablated from the companion, ρ is the mass density in the wind at a distance r from the pulsar, and v_w is the speed of the wind being blown out by the pulsar energetic outflow in photons and particles. The column density N_OH that radio radiation from the pulsar encounters is

$$\begin{eqnarray*}&&{N}_{\mathrm{OH}}\sim ({X}_{\mathrm{OH}}/{m}_{\mathrm{OH}})\rho \times s.\end{eqnarray*}$$

Here, X_OH is the OH fraction⁷ in the wind (the fraction of the nucleons locked up in OH), whose typical value is $\sim {10}^{-3}$, n_X's are the number densities of the element X, and s is the absorption lengthscale encountered by the pulsar beam in the intrabinary space typically of dimensions comparable to that of the eclipsing region, and m_OH is the mass of an OH molecule. For the case of PSR B1957+20, Phinney et al. (1988) has argued that the eclipsing region is $\sim 0.7\,{R}_{\odot }$, and so we adopt in our calculations $s\sim r$. The expression above can be further reduced in terms of parameters of the pulsar system under consideration by the equating the timescale of complete ablation of the planetary mass companion to the spindown timescale for the pulsar:

$$\begin{eqnarray}&&{N}_{\mathrm{OH}}=\displaystyle \frac{{\dot{M}}_{w}{X}_{\mathrm{OH}}s}{{f}_{{\rm{\Omega }}}4\pi \rho {r}^{2}{v}_{w}{m}_{\mathrm{OH}}}.\end{eqnarray} \tag{ 3 }$$

In terms of the parameters of PSR B1957+20 with an assumed evaporation timescale⁸ yields a column density of

$$\begin{eqnarray}&&{N}_{\mathrm{OH}}=4.8\times {10}^{16}\,{\mathrm{cm}}^{-2}\left(\displaystyle \frac{{m}_{c}}{0.02{M}_{\odot }}\right)\left(\displaystyle \frac{{X}_{\mathrm{OH}}}{{10}^{-3}}\right)\left(\displaystyle \frac{s}{r}\right)\\ &&\,\times {\left[\left(\displaystyle \frac{{\tau }_{\mathrm{evap}}}{2\times {10}^{8}\mathrm{year}}\right)\left(\displaystyle \frac{{f}_{{\rm{\Omega }}}}{0.25}\right)\left(\displaystyle \frac{r}{2.6{R}_{\odot }}\right)\left(\displaystyle \frac{{v}_{w}}{200\mathrm{km}{{\rm{s}}}^{-1}}\right)\right]}^{-1}.\end{eqnarray} \tag{ 4 }$$

This column density is indeed comparable to the values measured from the observed interstellar OH absorption toward pulsars (see Table 2).

To assess the column depth of N_OH in other binary pulsar systems listed in Table 1, we note that variables in Equation (3) can be scaled to observable or listed parameters in Table 1. As before, we assume that $(s/r)\sim \mathrm{constant}$. Furthermore, Kluzniak et al. (1988) derive ${\dot{M}}_{w}$ as well as the evaporative wind velocity v_w driven by total γ-ray radiation and TeV ${e}^{\pm }$ pairs radiated by the pulsar and intercepted by the secondary when the binary pulsar enters the present phase of radio pulsar driven evaporation of the secondary in PSR B1957+20. Using these expressions, we estimate: ${N}_{\mathrm{OH}}\propto {L}_{\gamma }^{2}{r}^{-5}{{hR}}_{c}^{2}$, where ${L}_{\gamma }$ is the Mev γ-ray luminosity, R_c is the companion radius intercepting a fraction of that luminosity at an orbital distance r, and h is the scale height of the companion's (inflated) atmosphere, which may extend to a significant fraction of the Roche lobe. Since the engine of both the gamma-ray luminosity as well as the TeV ${e}^{\pm }$ luminosity is the spindown power of the neutron star, we can scale the column density of molecular OH in terms of the pulsar parameters as ${N}_{\mathrm{OH}}\propto {\dot{E}}^{2}/{P}_{\mathrm{orb}}^{10/3}$. For parameters listed in Table 1, the highest N_OH is predicted for the original BW pulsar PSR B1957+20, but the N_OH for the first three entries of Table 1 are within an order of magnitude of that predicted for PSR B1957+20.

The red back "candidate transitional" pulsar PSR J1723−2837 has a mean flux density (at 1400 MHz) of 1.1 mJy as listed in the ATNF catalog. However, with an orbital period of 14.76 hr the column density of N_OH could be low. This pulsar may be a good candidate in the upcoming SKA era for attempts to detect OH/H i lines, but is likely to be challenging with present telescopes if its flux remains steady. The transitional MSP PSR J1227−4853 (Roy et al. 2015) with shorter orbital period and significant $\dot{E}$ has been found to eclipse at 1420 GHz (Parkes) and at 607 MHz (GMRT). Its continuum flux density at 607 MHz is 6.6 mJy; assuming a spectral index of −1.7 used by Roy et al. (2015), the estimated flux at 1420 MHz would be 1.6 mJy, which may also pose a challenge for atomic and molecular line detections even if its column depth of molecular OH could be significant.

If both molecular OH and neutral atomic hydrogen are present in the gas ablated from the ultra-low-mass companion of the energetic pulsar, one can also estimate the possible column depths. If the X_OH fraction is slightly subsolar, e.g., $\sim {10}^{-3}$ as we have assumed above, the column depth of H i in the intrabinary gas, assuming much of the gas is made of hydrogen and hydrogen is locked up mostly in atomic form (H i) once the gas cools down to a temperature of $\sim 100\,{\rm{K}}$, and scaling the N_OH column depth in Equation (3) yields ${N}_{{\rm{H}}{\rm{I}}}\sim (5-10)\,\times {10}^{19}\,{\mathrm{cm}}^{-2}$. Given the relation between column depth and the EW for H i absorption (Kulkarni & Heiles 1988, p. 95): ${N}_{{\rm{H}}{\rm{I}}}=1.8\times {10}^{18}\langle 1/{T}_{s}{\rangle }^{-1}\mathrm{EW}$, where $\langle 1/{T}_{s}{\rangle }^{-1}$ is harmonically weighted spin temperature along the path, whose value may be 50–100 K, one finds $\mathrm{EW}\sim 2\,\mathrm{km}\,{{\rm{s}}}^{-1}$. One can detect such EW's in the pulsar spectrum easily. Note that variations in column densities of $\sim {10}^{18-19}\,{\mathrm{cm}}^{-2}$ have been detected on occasions toward strong pulsars like PSR B1929+10 (Weisberg & Stanimirović 2007; Stanimirović et al. 2010). This gives an indication of the scale of measurable H i column depths in ISM and the estimated ${N}_{{\rm{H}}{\rm{I}}}\sim (5-10)\times {10}^{19}\,{\mathrm{cm}}^{-2}$ derived from Equation (3) can therefore be detected.

Measurement of absorption spectra of interstellar H i and OH against the pulsed radiation has been described in Koribalski et al. (1995), Stanimirovic et al. (2003), and Weisberg et al. (2005; see especially the supporting online material of the last reference). Briefly, a correlation spectrometer is used in the pulsar binning mode wherein each correlation function was recording into one of ${2}^{N}$ (e.g., 32) pulse phase bins that resulted in ${2}^{N}$ phase binned spectra covering a radio frequency bandwidth (e.g., 4 MHz for Parkes telescope, subdivided into 2048 spectral channels typically $\sim 2\,\mathrm{kHz}\,\mathrm{or}\sim \,0.4\,\mathrm{km}\,{{\rm{s}}}^{-1}$ wide). The spectra obtained in the phase bins that had the pulsar pulse were collapsed into a single "pulsar-on" spectrum. Similarly, the spectra recorded for off-pulse phase bins were integrated into a single "pulsar-off" spectrum. The so-called pulsar spectrum was formed from the difference of pulsar-on and pulsar-off spectra and normalized by the mean pulsar flux. A frequency switching that takes the central radio frequency away from the line was used to flatten the baseline. This method of constructing the pulsar spectrum removes any in-beam molecular or atomic line emission due to the broad telescope beam and measures the pulsar signal alone absorbed by any intervening OH in intrabinary or ISM.

We estimate the typical exposure requirements for GBT if it were to be used for OH line detection from pulsars similar to those listed in Table 1. We note that all pulsars listed in Table 1 have typical mean S1400 flux densities of 3–4 mJy (except the one in globular cluster NGC6544, namely PSR J1807−2459A). These pulsars also have a 10% duty cycle, which implies that the peak flux during the pulse on phases will be approximately 10 times higher, though the pulse on phase is only one-tenth the pulsar full period. The radiometer equation gives the sensitivity of a radio telescope and receiver system to pulsed signals in terms of a threshold flux density S_mean (Lorimer & Kramer 2012):

$$\begin{eqnarray}&&{S}_{\mathrm{mean}}=\beta \displaystyle \frac{{({\rm{S}}/{\rm{N}})}_{\mathrm{mean}}{T}_{\mathrm{sys}}}{G\sqrt{{n}_{p}{t}_{\mathrm{int}}{\rm{\Delta }}f}}\sqrt{\displaystyle \frac{W}{P-W}},\end{eqnarray} \tag{ 5 }$$

where T_sys is the system noise temperature (including sky and receiver noise), G is the telescope gain to convert between temperature and flux density (in the Rayleigh–Jeans limit), n_p is the number of orthogonal polarizations summed in the signal, t is the integration time for the pulsar observation, ${\rm{\Delta }}f$ is the observing bandwidth, ${({\rm{S}}/{\rm{N}})}_{\mathrm{mean}}$ is the threshold S/N for detection, W is the "pulse on" duration, P is the pulse period (rotational period of the neutron star), and $\beta \simeq 1$ denotes a factor of imperfections due to the digitization of the signal in the receiver system and other effects.

For a pulsar with a mean flux $3\,\mathrm{mJy}$ and a ${({\rm{S}}/{\rm{N}})}_{\mathrm{mean}}\sim 9$ (see discussion after Equation (8)), observed with GBT¹⁰ , Equation (5) can be rewritten as

$$\begin{eqnarray}\displaystyle \frac{{t}_{\mathrm{int}}}{5\,\mathrm{hr}} & = & {\left(\displaystyle \frac{{T}_{\mathrm{sys}}}{16{\rm{K}}}\right)}^{2}{\left(\displaystyle \frac{G}{2{\rm{K}}{\mathrm{Jy}}^{-1}}\right)}^{-2}{\left(\displaystyle \frac{{S}_{\mathrm{mean}}}{0.2\mathrm{mJy}}\right)}^{-2}\left(\displaystyle \frac{W/P}{0.1}\right)\\ & & \times {\left[\left(\displaystyle \frac{{n}_{p}}{2}\right)\left(\displaystyle \frac{{\rm{\Delta }}f}{0.4\mathrm{km}{{\rm{s}}}^{-1}}\right)\right]}^{-1}.\end{eqnarray} \tag{ 6 }$$

The minimum detectable flux for a given pulsar (with P and W) is usually set by the threshold S/N. However, since we are interested in the absorption line profile in the pulsar spectrum, which flux are we referring to? If we set ${S}_{\mathrm{mean}}$ to be the continuum flux of the pulsar spectrum outside the channels occupied by the absorption line, the flux there is relatively high. However, what matters for line profile determination is that our flux uncertainty should be low enough to measure the residual between the continuum S_c (assumed to be constant over a narrow line width, say of Gaussian profile¹¹ ) and the flux density at the line center (where the optical depth is ${\tau }_{0}$) with a high enough significance. Since the absorption of pulsed radiation in the OH band is variable with time due to the pulsar orbital motion, the determination of its magnitude can be limited by the errors arising out of short exposure times. The EW for the absorption line itself (for a Gaussian line) can be written as

$$\begin{eqnarray*}&&\mathrm{EW}=1.06\,{\tau }_{0}\,\mathrm{FWHM}.\end{eqnarray*}$$

The EW can be written in terms of the flux density in the continuum ${F}_{c}(\lambda )$ and in the line $F(\lambda )$ in a wavelength interval ${\lambda }_{1}\lt \lambda \lt {\lambda }_{2}$ (outside this regime $F(\lambda )={F}_{c}(\lambda )$) as

$$\begin{eqnarray*}&&{\mathrm{EW}}_{\lambda }={\rm{\Delta }}\lambda -{\int }_{{\lambda }_{1}}^{{\lambda }_{2}}\displaystyle \frac{F(\lambda )}{{F}_{c}(\lambda )}d\lambda \end{eqnarray*}$$

where ${\rm{\Delta }}\lambda ={\lambda }_{2}-{\lambda }_{1}$. By applying the mean value theorem this can be re-expressed as (Vollmann & Eversberg 2006)

$$\begin{eqnarray*}&&{\mathrm{EW}}_{\lambda }={\rm{\Delta }}\lambda \left[1-\displaystyle \frac{\overline{F(\lambda )}}{\overline{{F}_{c}(\lambda )}}\right]\end{eqnarray*}$$

where ${F}_{c}(\lambda )$ is measured outside the line region and interpolated across the line where the absorption is taking place and the overbars denote the average value of a variable. Both $F(\lambda )$ and ${F}_{c}(\lambda )$ have statistical errors. If their errors are not correlated, then their standard deviations can be determined separately. On the other hand, the error estimate of the EW of a spectral line has been derived by Vollmann & Eversberg (2006). The overall uncertainty of the EW is determined by two factors: the photometric uncertainty of the system and the uncertainty of the continuum estimation over the line. For low absorption line fluxes Vollmann & Eversberg (2006) show that the standard error of the EW $\sigma ({\mathrm{EW}}_{\lambda })$ can be written as

$$\begin{eqnarray*}&&\sigma ({\mathrm{EW}}_{\lambda })=\displaystyle \frac{({\rm{\Delta }}\lambda -{\mathrm{EW}}_{\lambda })}{({\rm{S}}/{\rm{N}})}\times {\left[1+\displaystyle \frac{\overline{{F}_{c}}}{\overline{F}}\right]}^{1/2}\end{eqnarray*}$$

where the relevant S/N ratio is the S/N in the undisturbed continuum and ${\rm{\Delta }}\lambda$ is a measure of the line width in wavelength units. For weak lines, i.e., where the depth of the line is very small, $\overline{F}\sim \overline{{F}_{c}}$ and the above relation reduces to

$$\begin{eqnarray*}&&\sigma ({\mathrm{EW}}_{\lambda })=\sqrt{2}\displaystyle \frac{({\rm{\Delta }}\lambda -{\mathrm{EW}}_{\lambda })}{({\rm{S}}/{\rm{N}})}.\end{eqnarray*}$$

This then leads to, using ${\rm{\Delta }}\lambda -{\mathrm{EW}}_{\lambda }={\rm{\Delta }}\lambda (\overline{F}/\overline{{F}_{c}})$,

$$\begin{eqnarray}&&\displaystyle \frac{{\mathrm{EW}}_{\nu }}{\sigma ({\mathrm{EW}}_{\nu })}=({\rm{S}}/{\rm{N}})\displaystyle \frac{{\mathrm{EW}}_{\nu }}{{\rm{\Delta }}\nu }\times {\left[\displaystyle \frac{\overline{F}}{\overline{{F}_{c}}}\left(1+\displaystyle \frac{\overline{F}}{\overline{{F}_{c}}}\right)\right]}^{-1/2}\end{eqnarray} \tag{ 7 }$$

where ${\rm{\Delta }}\nu =(c/{\lambda }^{2})({\lambda }_{2}-{\lambda }_{1})$. Here $({\rm{S}}/{\rm{N}})$ ${(=({\rm{S}}/{\rm{N}})}_{\mathrm{mean}}$) is determined by the observational and telescope parameters as in Equation (5). For weak or low absorption line strengths, one has $\overline{F}\sim \overline{{F}_{c}}$, and the flux dependent factor in square brackets in Equation (7) is about $1/\sqrt{2}$. Similarly, the ratio of FWHM to ${\rm{\Delta }}\nu$ that arises out of the ratio $({\mathrm{EW}}_{\nu }/{\rm{\Delta }}\nu )$ is also of the order of unity. Therefore, given a ${\tau }_{0}$, the significance of the line detection in terms of its EW is

$$\begin{eqnarray}&&\displaystyle \frac{{\mathrm{EW}}_{\lambda }}{\sigma ({\mathrm{EW}}_{\lambda })}\approx 0.3\left(\displaystyle \frac{{\tau }_{0}}{0.3}\right){({\rm{S}}/{\rm{N}})}_{\mathrm{mean}}.\end{eqnarray} \tag{ 8 }$$

Against the very bright pulsar-on phase, it will be relatively easier to measure the changes in the flux density in the OH-line profile, since the high flux density at a specific frequency channel can be relatively easily determined. The local continuum of the pulsar spectrum can be determined much more accurately than in the line absorption frequency channels, since many more channels (compared to where the line is located) can be averaged together in the continuum during data processing to get a good ${({\rm{S}}/{\rm{N}})}_{\mathrm{mean}}$. If the absorption in the 1667 MHz band has, for example, an optical depth at the line center ${\tau }_{0}=0.3$ then the significance of the EW measurement would be lower than that of pulse detection in the continuum part of the pulsar spectrum. With increasing ${\tau }_{0}$, the EW will be determined at a significance ${({\rm{S}}/{\rm{N}})}_{\mathrm{mean}}$ approaching the pulsar signal detection itself (in the continuum). Thus to effect a detection of the absorption line with a similar ${\tau }_{0}$ (and with $(\mathrm{EW}/{\rm{\Delta }}\mathrm{EW})=3$), one requires a significance of detection of the pulsar spectrum of ${({\rm{S}}/{\rm{N}})}_{\mathrm{mean}}\sim 9$, while for deeper absorption lines (with higher ${\tau }_{0}$), a significant detection of the OH absorption line may be effected at an even lower ${({\rm{S}}/{\rm{N}})}_{\mathrm{mean}}\sim 3\mbox{--}5$. Our estimate of the telescope resource requirements before (e.g., Equation (6)), however, is based on the conservative case of moderate absorption depths and line-widths.

The dip in the flux density would decrease toward the absorption line wings over a velocity range whose magnitude could be similar or somewhat larger than the thermal speed (about $0.4\,\mathrm{km}\,{{\rm{s}}}^{-1}$) of the OH gas, i.e., about $2\mbox{--}3\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (we note that Minter 2008 gives the FWHM for PSR B1718−35 OH absorption in the ISM to be about $3\,\mathrm{km}\,{{\rm{s}}}^{-1}$ at 1667 MHz; in the binary system, the wind speed from the companion and the OH line width is likely to be larger). The orbital velocity scale is typically several $100\,\mathrm{km}\,{{\rm{s}}}^{-1}$ for a system like PSR B1957+20, though because of the mainly tangential swept back wind, the velocity width in the radiatively connected region along the line of sight is likely to be substantially smaller. Thus, the OH absorption at orbital phases near the eclipse may be distributed over a velocity width $\leqslant 10\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Note that since only times close to the total eclipse would have substantial OH absorption, only that fraction of the orbital period (e.g., the $\sim {50}^{{\rm{m}}}$ periods around ingress and egress for PSR B1957+20) are relevant and the required total exposures have to be distributed over these epochs. Since these periods last together about 10%–15% of the full orbital phase (for the case of PSR B1957+20), the total exposure time has to be spread over many orbital cycles. Finally, it will be efficient to use a sufficiently large bandwidth of the spectrometer in several spectral windows to cover all L-band OH lines (1612–1720 MHz) and H i line simultaneously.

Thus, each pulsar listed in Table 1 (first four rows) requires about 5 hr of telescope time with the typical sensitivity of GBT. Pulsars in the declination range $0^\circ \mbox{--}40^\circ$ are accessible to Arecibo, which is the most sensitive existing telescope. In fact, GBT, Arecibo, and the Parkes Radio Telescope each detected one of the pulsars listed in Table 2. Future telescopes like the SKA will have typically 10 (SKA1-Mid) to 100 times (full SKA) the flux sensitivity compared to GBT. With SKA-1 increased sensitivity in L band (Band3) using low noise amplifiers, it will be possible to carry out such molecular line detections down to a pulsar mean flux density level of $0.7\,\mathrm{mJy}$ or lower, opening up several other known pulsar targets for similar studies. In addition, many newly discovered pulsar targets will become available for studies of composition of the winds from the companion.

So far, our discussion of absorption spectroscopy in the pulsar spectrum has been in the time (pulsar rotational phase) versus radio frequency domain and in the context of single dish telescopes. In Section 4.1, we have discussed that detection of orbital phase modulation can distinguish the intrabinary atomic/molecular gas from the circumbinary gas or that in the ISM. By employing multiple antenna dishes in an interferometer, it is also possible to add a third domain, namely spatial information in the plane of the sky. While it is not possible at present to spatially resolve a binary system at the relevant distances of pulsars, any detection of molecular or atomic gas (through absorption or stimulated emission) coincident with the pulsar position can be expected to be associated with the binary system if it is point like and spatially coincident. This is especially so if the pulsar occurs out of the galactic plane where the occurrence of such gas clouds is rarer in the ISM. The primary advantage offered by interferometric line absorption studies is the ability to resolve out the foreground neutral atomic/molecular line emission and thus to yield an uncontaminated measure of the absorption line profile that traces gas in the narrow beam subtended by the background source (in this case, the pulse-on phase of the pulsar radiation). Moreover, it is possible with interferometric systems to achieve a high spectral dynamic range, which opens up the possibility of measuring small optical depths, which can probe a Warm Neutral Medium (WNM; Roy et al. 2013). Weisberg & Stanimirović (2007) showed that for PSR B1849+00, whose line of sight passes near the edge of the SNR Kes 79, had far smaller optical depth for OH absorption in the pulsar-off phases than in the pulsar-on OH spectra ($\tau (1667\,\mathrm{MHz})=0.02\,\mathrm{versus}\,0.9$), which they explained as the pulsar-earth pencil beam of radiation having intercepted dense ($\gt {10}^{5}\,{\mathrm{cm}}^{-3}$) and small (typical angular size $\lt 15^{\prime\prime}$) cloudlets in the pulsar ISM spectra. In the pulsar-off spectrum, the sampling of absorption is across the full telescope beam that represents a larger solid angle average across a clumpy ISM consisting of high optical depth, small molecular cloudlets embedded in lower density medium.

As is well known, the significance of detection of a pulsating point source can be largely improved by removing the off-pulse noise. A background sky subtraction procedure for the pulse on—pulse off "gated" image constructed by an interferometer (such as the ATCA, Camilo et al. 2000) can unambiguously identify the location of the pulsed emission. Millisecond pulsars (MSPs, discovered by Fermi with poor spatial information), which require high time resolution for the gating procedure have been localized to an accuracy of $\pm 1^{\prime\prime}$ in the on–off gated image plane at GMRT. GMRT observations use a coherently de-dispersed gating correlator of the multiple antenna outputs that accounts for, at the same time, orbital motions of the MSPs while interferometer visibilities are folded with a topocentric rotational model derived from periodicity search simultaneously with the beamformer output (Roy & Bhattacharyya 2013). The positional accuracy of even the faint MSPs obtained from the on–off gated image improve with the S/N of the pulsar detection. Even though at 322 MHz, the GMRT synthesized beam has a positional accuracy of FWHM $10^{\prime\prime}$, an accuracy of $\pm 1^{\prime\prime}$ has been obtained for the timing position of the pulsar with a S/N of 5, and the astrometric accuracy accelerates the convergence in pulsar timing models for newly discovered pulsars such as the Fermi MSPs (Roy & Bhattacharyya 2013). This reduces the telescope time requirements for subsequent follow-up timing observations and reduces the effect of large covariances in the timing fit between pulsar position and spin period derivative, $\dot{P}$. While GMRT cannot observe in the OH bands, it can do so in the H i (1420 MHz) band. For example, GMRT interferometric observations of Galactic H i 21 cm absorption spectroscopy toward 32 compact bright extragalactic radio sources (with L-band flux greater than 3 Jy) have led to the detection of WNM of spin temperature of several thousand degrees K (Roy et al. 2013). In the H i band these observations typically achieved a velocity resolution of $\sim 0.4\,\mathrm{km}\,{{\rm{s}}}^{-1}$ and a velocity coverage of $\sim 105\,\mathrm{km}\,{{\rm{s}}}^{-1}$ by using a single IF band with two polarizations and a baseband bandwidth of 0.5 MHz subdivided into 256 channels.

At the VLA, THOR—The H i, OH, Recombination Line Survey of the Milky Way has been undertaken to study atomic, molecular, and ionized emission of Giant Molecular Clouds (GMCs) in our galaxy (Bihr et al. 2015). These observations are interferometric (e.g., the Pilot survey for H i from GMC W43 was in the C-array of VLA) and have high spectral resolution (channel width of 1.953 kHz, with ${\rm{\Delta }}v\sim 0.41\,\mathrm{km}\,{{\rm{s}}}^{-1}$ in the H i 21 cm line, while that for OH lines for the region around the same target the channel spacing was $0.73\,\mathrm{km}\,{{\rm{s}}}^{-1}$ at 1612 MHz (Walsh et al. (2016))). The synthesized beam size was of the order of $20^{\prime\prime} \times 20^{\prime\prime}$ but the absolute positional accuracy of a point source with a $5\sigma$ detection would be ${{\rm{\Theta }}}_{\mathrm{Beam}}/2\sigma =2^{\prime\prime}$. Thus, the VLA C-array and GMRT would achieve similar spatial resolutions for OH/H i bands centered around a target binary MSP at about 1 kpc, with $\sim 0.01\,\mathrm{pc}$ spatial scale. Since spatial densities of molecular or H i clouds along a narrowly defined line of sight toward high galactic latitude are likely to be small, a detection of absorbing gas is likely to be physically associated with the pulsar system itself, especially if the column densities are high.

The typical rms noise in the THOR Pilot survey was 19 mJy/beam for the OH lines and 9 mJy/beam for the H i band, which was achieved in an eight-minute pointing of the VLA. The survey has not exploited any underlying time dependent structure (e.g., a pulsar) of the spectral line signal. As we argue in Section 5.1, for a pulsar of mean flux density of 3 mJy being obscured by companion wind of optical depth $\tau \sim 0.3$, a typical $10\sigma$ signal for the pulsar continuum required for adequate line detection would lead to $1\sigma$ rms of 0.3 mJy. If there were no pulsed signal, such a low rms could be achieved with the VLA if we scale the THOR exposure ($8\,\mathrm{minutes}$) by the ratio of the square of the respective flux rms, to obtain an overall exposure of ${(19\mathrm{mJy}/0.3\mathrm{mJy})}^{2}\ \times \ 8\,$minutes = 534 hr. However, for a MSP with a duty cycle that is 10% of its spin period, if we can (phase)gate the pulsar signal and construct the pulsar on–off spectra as described before we will gain a factor of one-tenth, thus reducing the exposure requirement to about $53\,\mathrm{hr}$. This is still too large for a single pulsar target. While such a capability may not exist at present for the OH band, the H i line transitions may be more promising, requiring about $12\,\mathrm{hr}$ of exposure (with similar scaling and assumptions of optical depth in the H i line absorption), provided adequate hardware and computational resources are available.