THE ASTROPHYSICAL JOURNAL 2001. The …srk/Ay126/Lectures/Lecture... · No. 2, 2001 BALMER-DOMINATED SPECTRA OF NONRADIATIVE SHOCKS 997 uncertainty in broad component width, indicating

THE ASTROPHYSICAL JOURNAL, 547 :995È1009, 2001 February 1( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

BALMER-DOMINATED SPECTRA OF NONRADIATIVE SHOCKS IN THE CYGNUS LOOP,RCW 86, AND TYCHO SUPERNOVA REMNANTS

PARVIZ GHAVAMIAN,1,2,3 JOHN RAYMOND,4 R. CHRIS SMITH,5 AND PATRICK HARTIGAN1Received 2000 June 1 ; accepted 2000 September 25

ABSTRACTWe present an observational and theoretical study of the optical emission from nonradiative shocks in

three supernova remnants : the Cygnus Loop, RCW 86, and Tycho. The spectra of these shocks aredominated by collisionally excited hydrogen Balmer lines, which have both a broad component causedby proton-neutral charge exchange and a narrow component caused by excitation of cold neutrals enter-ing the shock. In each remnant, we have obtained the broad-to-narrow Ñux ratios of the Ha and Hblines and measured the Ha broad component width.

A new numerical shock code computes the broad and narrow Balmer line emission from nonradiativeshocks in partially neutral gas. The Balmer line Ñuxes are sensitive to Lyman line trapping and thedegree of electron-proton temperature equilibration. The code calculates the density, temperature, andsize of the postshock ionization layer and uses a Monte Carlo simulation to compute narrow Balmerline enhancement from Lyman line trapping. The initial fraction of the shock energy allocated to theelectrons and protons (the equilibration) is a free parameter. Our models show that variations inelectron-proton temperature equilibration and Lyman line trapping can reproduce the observed range ofbroad-to-narrow ratios. The results give 80%È100% equilibration in nonradiative portions of the north-east Cygnus Loop km s~1), 40%È50% equilibration in nonradiative portions of RCW 86(v

SD 300 (v

SD

km s~1), and equilibration in Tycho km s~1). Our results suggest an inverse600 [20% (vSD 2000

correlation between magnetosonic Mach number and equilibration in the observed remnants.Subject headings : radiative transfer È shock waves È supernova remnants

1. INTRODUCTION

Supernova explosions produce some of the strongestshocks in nature. During the event, the outer layers of a starare ejected at speeds as high as 30,000 km s~1. The denseejecta behave like a highly supersonic piston, producing astrong shock wave in the interstellar medium (ISM) aheadof the piston (commonly known as the forward shock).When the mass swept up by the forward shock begins toexceed the ejecta mass, the supernova remnant (SNR) entersthe blast-wave (Sedov-Taylor) phase of evolution (Truelove& McKee 1999). From the genesis of the SNR through theblast-wave stage, the forward shock is nonradiative,meaning that it loses a negligible fraction of its internalenergy to radiative cooling.

Due to the low density cm~3) and high Mach(n [ 1number of the forward shock, the heated inter-(M Z 200)stellar gas behaves like a collisionless plasma (Draine &McKee 1993). According to the strong shock jump condi-tions, electrons and protons are heated to temperatures in aminimum ratio (D1/2000). However,(T

e/T

p)\ (m

e/m

p)

plasma waves and MHD turbulence at the shock front maytransfer energy from protons to electrons (Tidman & Krall1971 ; Kennel, Edmiston, & Hada 1985 ; Cargill & Papa-dapoulos 1988), further equilibrating the two particle tem-

1 Department of Space Physics and Astronomy, Rice University, 6100South Main Street, Houston, TX 77005-1892 ; hartigan=sparky.rice.edu.

2 Current address : Department of Physics and Astronomy, RutgersUniversity, 136 Frelinghuysen Road, Piscataway, NJ 08854-8019 ;parviz=physics.rutgers.edu.

3 Visiting Astronomer, Cerro Tololo Inter-American Observatory,National Optical Astronomy Observatories. CTIO is operated by AURA,Inc., under contract to the National Science Foundation.

4 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,Cambridge, MA 02138 ; raymond=cfa.harvard.edu.

5 Cerro Tololo Inter-American Observatory, Casilla 603, Chile ;csmith=noao.edu.

peratures. Since the electrons and protons are a†ected by adi†erent range of plasma waves and the Coulomb equili-bration time downstream often exceeds the age of theremnant, the two particle temperatures can remain unequalthroughout the shock. The types of plasma waves excited atthe shock transition (and hence the amount of collisionlessheating) depend strongly on parameters such as the magne-tosonic Mach number and magnetic Ðeld orientation.M

SDue to this complicated dependence and the extreme diffi-culty of creating high Mach number collisionless shocks inthe laboratory, the properties of collisionless shocks remainpoorly understood.

In a cold K), partially ionized medium, the([104charged and neutral distributions respond very di†erentlyto the passage of a collisionless shock. The cold neutrals areinitially una†ected by the shock transition, while chargedparticles are compressed by a factor of 4 and stronglyheated. Some of the cold neutrals entering the shock arecollisionally excited before being destroyed by collisionalionization or charge transfer. Radiative decay by theseexcited neutrals produces narrow-component Balmer emis-sion with a line width given by the preshock temperature. Incontrast, charge exchange between cold neutrals andprotons produces fast neutrals having the velocity distribu-tion of the postshock protons. Collisional excitation ofthe fast neutrals produces broad Balmer line emission(Chevalier & Raymond 1978 ; Chevalier, Kirshner, &Raymond 1980, hereafter CKR80). While the low tem-peratures in radiative shocks favor the emission of strongforbidden lines such as [O II], [O III], [N II], and [S II], thehigh temperatures behind nonradiative shocks producehydrogen Balmer line emission far more efficiently. Balmer-dominated shocks have been detected in Tycho (Kirshner,Winkler, & Chevalier 1987, hereafter KWC87; Smith et al.1991 ; Ghavamian et al. 2000), SN 1006 (KWC87; Smith et

995

996 GHAVAMIAN ET AL. Vol. 547

al. 1991 ; Winkler & Long 1997), RCW 86 (Long & Blair1990 ; Smith 1997), Kepler (Blair, Long, & Vancura 1991),portions of the Cygnus Loop (Raymond et al. 1983, here-after RBFG83; Fesen & Itoh 1985 ; Hester, Raymond, &Blair 1994, hereafter HRB94), four remnants in the LargeMagellanic Cloud (LMC; Tuohy et al. 1982 ; Smith et al.1991, 1994), and the bow shock surrounding the pulsar PSR1957]20 (Aldcroft, Romani, & Cordes 1992).

The optically emitting layer behind a nonradiative shockis extremely thin pc), so the proton temperature([10~3very close to the shock front can be measured from thewidth of a broad Balmer line. The proton temperature inturn depends on the shock velocity through the jump condi-tions, making the broad line proÐle a powerful diagnosticfor probing the global kinematics of young SNRs. Thebroad-to-narrow Ñux ratio is another important diagnosticbecause it also depends on the shock velocity (CKR80;Smith et al. 1991). Proper motion measurements of Balmer-dominated Ðlaments have been combined with shock veloc-ity estimates from the broad component to estimatedistances to several young SNRs (for example, see Long,Blair, & van den Bergh 1988 for SN 1006 and Hesser & vanden Bergh 1981 for Tycho).

Balmer-dominated spectra can be difficult to interpretbecause the broad component width and broad-to-narrowratio yield di†erent shock velocities depending on theassumed electron-proton equilibration. Since the equili-bration is not known a priori, spectroscopic observations inthe past have only yielded a range of shock velocities v

Sfrom the Balmer line proÐles : a minimum for novS(min)

equilibration and a maximum for full equilibration.vS(max)

In addition, collisional excitation behind the shock gener-ates both Lyman line photons and Balmer line photons.Lyman line trapping by slow neutrals partially convertsLyb and higher Lyman photons into Balmer line photons ;this enhances the Balmer line Ñux in the narrow componentand complicates the diagnostic interpretation of the broad-to-narrow ratio (CKR80; Smith et al. 1991 ; Ghavamian1999). Moreover, since the Lyb optical depth is larger thanthat of Lyc, the Ha broad-to-narrow ratio is more stronglya†ected than the Hb broad-to-narrow ratio. The Lymanline optical depth behind the shock typically lies between 0and 1, so that neither Case A nor Case B conditions apply.It is clear that disentangling the combined e†ects of Lymanline trapping and equilibration on a Balmer-dominatedspectrum requires (1) the acquisition of high signal-to-noiseratio (S/N) line proÐles in both Ha and Hb and (2) carefulmodeling of the atomic physics and radiative transfer.

In ° 2 of this paper, we present high S/N spectra of threenonradiative shocks which resolve the broad and narrow

components of both Ha and Hb. In ° 3 we present measure-ments of the Ha broad component width and broad-to-narrow ratios in Ha and Hb. We describe numerical modelsof nonradiative shocks in ° 4, including calculations of ion-ization structures for a range of equilibrations and MonteCarlo simulations of Lyb and Lyc trapping. In ° 5 wecompare the observed broad-to-narrow ratios with modelpredictions and simultaneously determine both the shockvelocity and equilibration for the observed SNRs. In ° 6 wediscuss our results, and in ° 7 we present our conclusions.

2. SPECTROSCOPIC OBSERVATIONS

The spectroscopic data sets presented here were acquiredover a 1 yr period between 1997 October and 1998 Septem-ber. We chose the instrumental setup for each observationto simultaneously maximize the number of detected pho-tons and provide enough spectral resolution to separatethe broad Balmer lines from the narrow Balmer lines. A logof our spectroscopic observations appears in Table 1. Thetelescope and spectrograph conÐgurations used for theobservations are described below.

2.1. Cygnus L oop and TychoOur spectroscopic observations of northeast Cygnus and

Tycho were performed on the nights of 1997 October 7È10(UT), using the 1.5 m Tillinghast reÑector telescope at theFred Lawrence Whipple Observatory. The Schmidt cameraCCD was a Loral 512] 2688 pixel chip, with an unbinnedplate scale of per 15 km pixel. To reduce readout noise0A.6during these observations, we binned the CCD chip by 4pixels along the spatial direction. The spectrograph wasequipped with a 1200 line mm~1 grating blazed at 5700 A� ,providing a dispersion of 0.38 pixel~1 and spectral cover-A�age of 1000 With this setup, only one Balmer line couldA� .be Ðt on the CCD chip, so we performed the Ha and Hbobservations at di†erent times. Our target in northeastCygnus Loop was Ðlament P7 from the list of positionsobserved by the Hopkins Ultraviolet Telescope (HUT)during the Astro-2 mission. We used a slit for Ha,1A.1 ] 3@yielding a resolution of 1.4 To obtain the Hb line proÐle,A� .we observed Cygnus P7 again on the night of 1998 Septem-ber 26 (UT). In this case, we used a 3@@] 3@ slit with the 1200line mm~1 grating, yielding a resolution of 1.5 AA� .Palomar Observatory Sky Survey (POSS) image of theCygnus P7 Ðlament appears in Figure 1, along with thetwo-dimensional FAST spectrum. The one-dimensional Haand Hb proÐles appear in Figure 2. From the Ha line, weÐnd a broad component width of 262 ^ 32 km s~1, with noevidence of forbidden line emission. The shift between thebroad and narrow component line centers is less than the

TABLE 1

JOURNAL OF SPECTROSCOPIC OBSERVATIONS

a d Date Range Position Anglea ExposureTarget (2000) (2000) (UT) (A� ) (deg) (s)

Cygnus P7 . . . . . . . . . . . . . . . . . . . . 20 54 32.5 32 17 32.8 1997 Oct 7 6050È7050 90 6 ] 90020 54 32.5 32 17 32.8 1998 Sep 26 4370È5370 90 11 ] 1800

Southwestern RCW 86 . . . . . . 14 41 08.1 [62 43 54.5 1998 Apr 6 5800È7430 128.7 4] 90014 41 08.1 [62 43 54.5 1998 Apr 8 4610È5110 128.7 3] 1800

Tycho knot g . . . . . . . . . . . . . . . . 00 25 50.9 64 09 19.2 1997 Oct 10 3600È7600 [10 6 ] 1800

NOTE.ÈUnits of right ascension are hours, minutes, and seconds, and units of declination are degrees, arcminutes, and arcse-conds.

a Measured east of north.

No. 2, 2001 BALMER-DOMINATED SPECTRA OF NONRADIATIVE SHOCKS 997

uncertainty in broad component width, indicating a nearlyedge-on viewing geometry.

The 1997 Tycho observations focused on the bright non-radiative shock knot g. In these observations, we used the300 line mm~1 grating with a 3A slit, centered on 5545 A�(4000 coverage). This convenient combination put bothA�Ha and Hb on the CCD chip and allowed simultaneousdetection of the Ha and Hb broad components. As in theprevious observations, we binned the CCD chip by 4 pixels.The resolution for this setup was 275 km s~1 (6 at Ha). ItA�is evident from the two-dimensional spectrum (Fig. 3) thatthe Ha surface brightness and broad component width varywith position along the knot g Ðlament, reÑecting variationsin preshock density, shock velocity, and viewing angle. Theextraction aperture shown in Figure 3 was centered on theupper part of knot g, where the broad component is bright-est and its width most nearly constant (hereafter labeledknot g1). The extracted Ha and Hb line proÐles appear inFigure 4. Knot g1 is the brightest portion of the Ðlament(broad component FWHM of 1765^ 110 km s~1), whilethe lower part knot g2 is fainter (broad FWHM of

2105 ^ 130 km s~1). The broad component center of knotg1 is redshifted 132 ^ 35 km s~1 from the center of thenarrow component, indicating that the plane of the shock istilted slightly into the plane of the sky.

In the 1997 Tycho spectra, we detected faint di†use Haemission above knot g. We did not subtract the one-dimensional di†use spectrum directly from that of knot g1because then the resulting one-dimensional spectrumbecomes too noisy to detect the Hb broad component.Instead, we measured the Ha surface brightnesses of knot gand the di†use emission separately, using FAST spectraacquired in 1998 (using the same detector setup as the 1997observations, see Ghavamian et al. 2000). From the 1998data, we found that the di†use Ha emission contributesapproximately 12% of the narrow Ñux from knot g1. Wehave used this information to correct the Ha broad-to-narrow ratio of knot g1. The di†use Hb emission is veryfaint and is almost lost in the noise, so we have not cor-rected the Hb broad-to-narrow ratio. In a separate paper(Ghavamian et al. 2000), we show that the di†use emissionis associated with a photoionization precursor from the

FIG. 1.ÈTop : Red image from the Palomar Optical Sky Survey, showing the P7 nonradiative shock in northeast Cygnus Loop. The approximate positionand width of the FAST spectrograph slit is marked. Bottom : Two-dimensional sky subtracted spectrum of position P7, showing the Ha broad and narrowcomponents. Extraction aperture for the one-dimensional spectrum is indicated by a bracket. East is at the top.


FIG. 2.ÈExtracted one-dimensional Ha and Hb proÐles of CygnusLoop P7.

supernova blast wave. Since the width of the broad Hb lineand its shift from zero velocity must be the same as from thehigh S/N Ha measurement, we hold these quantities Ðxedwhile simultaneously Ðtting the broad and narrow Hb lines.Constraining the Ðt in this manner is useful because the S/Nof the Hb broad component is signiÐcantly lower than thatof Ha.

Prior to our observations, the most recent spectroscopicstudy of knot g was that of Smith et al. (1991), who found abroad FWHM of 1900 ^ 300 in Ha. This value wasobtained by averaging the broad component width over thelength of knot g. We note that this velocity lies roughlymidway between our separately measured broad com-ponent widths for sections 1 and 2. In addition, the averageHa broad-to-narrow ratio of sections 1 and 2 lies within therange quoted by Smith et al. (1991). However, in our spectrawe Ðnd a signiÐcantly smaller shift in the broad componentline center than Smith et al. (1991). This may be due to thestrong variations in shock viewing angle along the length ofthe Ðlament. In Table 2 we present our observational resultsfor knot g side by side with those of earlier papers.

Since the Ha and Hb lines were recorded simultaneously,we were able to measure the broad and narrow-componentBalmer decrements separately for knot g1. AlthoughIHa/IHb

FIG. 3.ÈTop : Narrowband Ha image of Tycho, acquired with the direct imager at the 4 m telescope at KPNO. The FAST slit position is indicated.Bottom : Two-dimensional sky subtracted spectrum of knot g, showing broad and narrow Ha emission. The broad component is lost in the noise near thebottom of the slit, where the emission is faintest. Extraction aperture for the one-dimensional spectrum (position 1 in the text) is indicated by a bracket. Northis at the top.


FIG. 4.ÈOne-dimensional Ha and Hb proÐles of knot g1. The spec-trum has not been corrected for interstellar reddening.

the S/N of the broad Hb line is signiÐcantly lower than thatof the broad Ha line, we were able to estimate the broad HbÑux by setting the width and center of this line equal to thatof Ha during Ðtting. The Balmer decrements are shown inTable 3 for the broad and narrow components of Tychoknot g1. For comparison, we have computed forIHa/IHbboth upper and lower limits of the visual extinction param-eter determined by CKR80. It is evident(1.6¹ A

V¹ 2.6)

that the narrow Balmer decrement is considerably larger

TABLE 2

PRESENT AND PRIOR Ha MEASUREMENTS OF THE KNOT gBROAD COMPONENT

FWHM *Vbn

Paper (km s~1) (km s~1) a Ib/I

n

Chevalier et al. 1980 . . . . . . 1800 ^ 200 . . . 0.4È1.3Kirshner et al. 1987 . . . . . . 1800 ^ 100 238 ^ 18 1.08^ 0.16Smith et al. 1991 . . . . . . . . . . 1900 ^ 300 240 ^ 60 0.77^ 0.09Present work :

Knot g1 . . . . . . . . . . . . . . . . . 1765 ^ 110 132 ^ 35 0.67 ^ 0.1bKnot g2 . . . . . . . . . . . . . . . . . 2105 ^ 130 \130 0.75^ 0.1b

a Velocity shift between broad component line center and narrow com-ponent.

b Broad-to-narrow ratio corrected for contribution from di†use emis-sion.

TABLE 3

TYCHO KNOT g1 BALMER DECREMENTS

Ha/Hb Ha/Hb Ha/HbA

Va (Broad) (Narrow) (Total)

Undereddened . . . . . . 6.11^ 2.518.0 9.93^ 1.231.57 7.97^2.274.471.6 . . . . . . . . . . . . . . . . . . . 3.67^ 1.54.83 5.96^ 0.730.93 4.78^1.362.72.6 . . . . . . . . . . . . . . . . . . . 2.66^ 1.083.52 4.33^ 0.530.68 3.48^0.991.95

and 2.6 correspond to the range of visual extinctiona AV

\ 1.6parameters inferred for Tycho (Chevalier et al. 1980).

than the broad Balmer decrement. Interestingly, usingyields a broad Balmer decrement of 3.67, consis-A

V\ 1.6

tent with pure collisional excitation in a high-temperatureK) gas. Adopting the smaller value of also results(Z106 A

Vin better agreement between observations and models of thephotoionization precursor (Ghavamian et al. 2000).

2.2. RCW 86We observed the galactic SNR RCW 86 on 1998 April

5È8 (UT), using the RC Spectrograph at the f/7.8 focus ofthe Cerro Tololo Inter-American Observatory (CTIO) 4 mtelescope. The RC Spectrograph features a Loral 3k CCDwith 15 km pixels connected to a Blue Air Schmidt camera.This combination gives a plate scale of pixel~1. A0A.5decker allows for an adjustable slit length, with a maximumunvignetted slit size of 5@. Our observations targeted fourpositions around the remnant and included moderateresolution spectra of the Ha and Hb lines. In the Ha obser-vations, we used CTIO 1200 line mm~1 grating (0.5 A�pixel~1, blazed at 8000 with a Ðlter inserted to excludeA� )higher order emission lines. With a 3A slit, the spectralresolution of the Ha observations was 2.2 with spectralA� ,coverage of 1565 To obtain the Hb line proÐles, we usedA� .the 860 line mm~1 grating in second order (0.34 pixel~1,A�blazed at 5500 with Ðlter bg39 to exclude higher orderA� ),emission lines. Combined with a slit size of 3A, theresolution of the Hb setup was 1.4 with spectral coverageA� ,of 1135 A� .

The brightest nonradiative shocks in RCW 86 are locatedin the southwestern corner of the remnant, Ñanking thebright radiative shocks (Fig. 5). From the two-dimensionalspectrum of SW RCW 86, it is evident that some of theshocks at this location have become radiative (marked bythe absence of broad Balmer emission and the presence of[N II] jj6548, 6583 and [S II] jj6716, 6731 lines). We cen-tered the extraction aperture as shown in Figure 5 to avoidthe radiative emission above and bright stellar continuumbelow the aperture. Both optical (Leibowitz & Danziger1983 ; Rosado et al. 1996 ; Smith 1997) and X-ray (Kaastraet al. 1992 ; Smith 1997 ; Vink, Kaastra, & Bleeker 1998 ;Petruk 1999) observations have shown that the radiativeemission arises from a dense cloud overrun by the super-nova blast wave. The high neutral density is alsoresponsible for the brightness of the broad and narrow Halines in the one-dimensional spectrum (Fig. 6).

We reduced all spectroscopic data using standard rou-tines in IRAF.6 After applying overscan, bias, Ñat-Ðeld,response, and dark count corrections to all two-dimensional spectra, we corrected for the slit function in thetwo-dimensional spectra using twilight sky Ñights. Finally,we untilted the emission lines using wavelength solutionsfrom calibration lamp spectra, then subtracted the skybackground using night sky emission adjacent to eachtarget object.

The spectra presented in this paper were acquiredunder varying photometric conditions. Skies were non-photometric during the 1997 observations, so we onlyapplied a sensitivity correction to the Cygnus Loop andTycho data. Throughout the 1998 observations, however,conditions at Mount Hopkins were nearly photometric.

6 IRAF is distributed by the National Optical Astronomy Observa-tories, which is operated by the AURA, Inc., under cooperative agreementwith the National Science Foundation.


Comparison of the spectra from standard stars Hiltner 102,LB 1240, HD 192281, and BD 284211 with one anotherindicates that the 1998 Cygnus Loop spectrum is photo-metrically accurate to within 20%. The Ha observations ofRCW 86 were performed under nearly photometric condi-tions, while the Hb observations were performed underpartial cirrus. From an examination of spectra from stan-dard stars LTT 3218, LTT 7379, and LTT 7987, we estimatethat the Ha Ñux shown in Figure 6 is accurate to within10%.

3. LINE PROFILE FITS

To extract the broad and narrow component Ñuxes andmeasure the width of the broad component, we Ðt eachBalmer line proÐle with two independent Gaussians plusa linear background. Using the IRAF deblending taskSPLOT and self-written routines for s2 minimizationÐtting, we obtained quantitative estimates of the line Ñux,width, and center of the broad and narrow Balmer lines.Since the position of the zero level background a†ects the

estimated Ñux of the broad component, this baseline uncer-tainty is the dominant source of nonrandom error in caseswhere the broad component is very wide and faint (as inTycho, for example). Both the baseline uncertainty and sta-tistical uncertainty have been included in the quoted broad-to-narrow ratios.

We measured the broad component widths from the Haline proÐles, then corrected for the instrumental response bysubtracting the narrow component width in quadraturefrom the broad component width. In each case, the narrowcomponent widths were equal to the instrumental res-olution (they were unresolved). The broad and narrow com-ponents are most tightly blended in the Cygnus P7 lineproÐles. The error bars on the Ha and Hb broad-to-narrowratios are correspondingly larger.

4. THE NUMERICAL SHOCK MODELS

The goal of the numerical models is to calculate the Haand Hb broad-to-narrow ratios for arbitrary shock speed,equilibration, and preshock neutral density. Comparing the

FIG. 5.ÈTop : Southwestern portion of RCW 86, from Smith (1997). The Balmer-dominated Ðlaments covered by the slit are among the brightestnonradiative shocks observed to date. Radiative emission can be seen in the upper right side of the image. Bottom : Two-dimensional sky subtracted spectrumof the southwestern Ðlament. The extraction aperture of the one-dimensional spectrum is indicated by a bracket. Northwest is located at the top.


FIG. 6.ÈHa and Hb line proÐles for southwestern RCW 86. Noreddening correction has been applied.

predicted broad-to-narrow ratios with the observed valuesshould then allow us to constrain the shock velocity andequilibration of the observed SNRs. To compute theBalmer line Ñux from a nonradiative shock, we have com-puted the density and temperature of the postshock gas as afunction of position. We have used the results of these calcu-lations to compute the Lyb and Lyc optical depths behindthe shock. The Lyman optical depth behind the shock typi-cally lies between 0 and 1, meaning that neither Case A norCase B conditions apply.

4.1. Cross SectionsWe have consulted the following sources for collision

cross sections and strengths :

1. Collisional ionization and charge exchange.ÈIn theshock models we use the polynomial Ðt to the electron-hydrogen ionization cross section from Janev et al. (1987).The proton ionization cross section is from a numerical Ðtby Janev et al. that reproduces the experimental data ofShah et al. (1998), Shah, Elliott, & Gilbody (1987), and Shah& Gilbody (1981) to within their uncertainties. The H-H`charge exchange rate used in the shock code has been takenfrom the analytic Ðt of Freeman & Jones (1974).

2. Collisional excitation.ÈFor electron temperaturesbelow 5] 105 K, we use the polynomial Ðts of Giovanardi,Natta, & Palla (1987) to the 3s, p, d and 4s, p, d, f collisionstrengths. Above 5 ] 105 K, we compute the collision ratesdirectly using the modiÐed Born approximation cross sec-tions of Whelan (1986). The available data on proton excita-tion are relatively sparse in the literature, and theoreticalcross sections to n \ 3 and 4 have only recently becomeavailable. The close-coupling calculations by (1999)Mart•� ninclude cross sections to Ðne structure levels of n \ 3 and 4for proton energies above 40 keV. The calculations of

include charge exchange into excited states, aMart•� nprocess which contributes as much as 20% to the Balmer

line Ñux at high shock velocities km s~1). The(vSZ 1500

theoretical values agree reasonably well with the experi-mental measurements of Detle†sen et al. (1994), whichcovered proton energies in the range 40 keV¹ E¹ 800keV. The shock code utilizes the cross sections of Mart•� n(1999) and uses the calculations of McLaughlin, Winter, &McCann (1998) to estimate excitation cross sectionsbetween 10 and 40 keV.

4.2. Ionization StructuresThe shock structure calculation is initiated by using the

jump conditions to set the temperature and density at theÐrst time step. The equations describing the number den-sities of slow neutrals, fast neutrals, electrons, and protonsform a set of coupled, linear di†erential equations whichcan be solved together to calculate the density of eachspecies behind the shock. Assuming a pure H plasma (n

e\

the equation for the slow neutrals isnp),

dnH0(s)dt

\ [nenH0(s)(Sp

ivT

es ] Sp

ivT

ps ] Sp

cxvTs) , (1)

where and are the electron/proton ioniza-SpivT

e,ps Spcx

vTstion coefficients and H-H` charge exchange coefficients forslow neutrals (in cm3 s~1), and n is the number density(cm~3) of a given particle species. Solving this equation for

the slow neutral density at time stepnH0(s), tj(\ t

j~1] *t)is

nH0j (s) \ nH0j~1(s)e~nej~1(Wpi vXs`Wpcx vXs)*t , (2)

where is the total ionization coefficientSpivTs\ ST

e] ST

pdue to protons and electrons evaluated at The relationtj.

for fast neutrals is

nH0j ( f ) \ nH0j~1( f )e~(nej~1Wpi vXf*t)] 4nH0j~1(s)e~(nej~1Wpi vXs*t)[1[ e~(nej~1Wpcx vXs*t)] , (3)

where STf represents the collision rate coefficient for elec-trons or protons and fast neutrals. In the above equation,the Ðrst term represents the loss of fast neutrals by ioniza-tion, while the second term represents the increase of fastneutrals by charge exchange (the factor of 4 takes intoaccount the compression behind a strong shock). Mass con-servation requires that the electron and proton numberdensities increase in proportion to the ionization rate ;therefore, the equations for and can be used tonH0(s) nH0( f )calculate the proton density

npj \ n

ej \ n

pj~1] 4nH0j~1(s)[1[ e~(nej~1Wpi vXs*t)]

] nH0j~1( f )[1[ e~(nej~1Wpi vXf*t)] . (4)

Two important complications arise in the collision ratecalculations for a nonradiative shock. First, justT

eD T

pbehind the shock so that the electron and proton tem-peratures change continually with position as Coulomb col-lisions attempt to establish equilibrium. Hence, the collisionrate coefficients also evolve with position and must be com-puted for a range of electron and proton temperatures.Second, since the slow neutrals are una†ected by the shock,they encounter an anisotropic proton distribution which isbulk shifted to (CKR80; Smith et al. 1991). Slow(3/4)v

Sneutrals can also encounter an anisotropic electron dis-tribution if the plasma is far from temperature equilibriumand the postshock bulk velocity is comparable to the elec-tron thermal speed.


Another important complication occurs when the tem-peratures of the electrons and fast neutrals are unequal. Forthe case (equilibrated plasma), the electron thermalT

e\ T

pspeed is 43 times higher than the fast neutral thermal speed(\ proton thermal speed). The fast neutrals are e†ectivelyat rest relative to the electrons, and the ionization rate inte-gral is simple to evaluate. However, for an unequilibratedcase, the collision rate integral involves the distributionfunctions of both electrons and fast neutrals and becomes acomplicated six-dimensional integral over the velocities ofboth species (Smith et al. 1991). Fortunately, the electronÈfast neutral collision rate integrals can be reduced to one-dimensional form involving the relative speed of the twoparticle species (R. Bandiera 1998, private communication ;J. Weisheit 1998, private communication). Throughout theshock code, we have computed the electronÈfast neutral andelectronÈslow neutral rates using the reduced integral.

As mentioned earlier, electrons and protons behind anonradiative shock can be heated to very di†erent fractionsof the bulk Ñow energy. The total thermal energy Etotacquired by the postshock gas, however, is constant :

Etot\ 316(me] m

p)v

S2B 316mp

vS2 . (5)

The initial fraction of this energy going into electrons andprotons is controlled by collisionless heating at the shockfront and is therefore a free parameter. Let us deÐne theparameter such that for no initial electron-protonfeq feq\ 0equilibration and for full initial electron-protonfeq \ 1equilibration. The postshock temperatures for arbitrary feqare then

Tp\ T0p ] 1

2316

[me

feq] (2[ feq)mp]

vS2k

, (6)

Te\ T0e ] 1

2316

[mp

feq] (2[ feq)me]

vS2k

, (7)

where are the preshock proton and electron tem-T0p,eperatures, respectively. For the extreme cases of no equili-bration and full equilibration,

Tp,e \ T0p,e]

316

mp,ek

vS2 ( feq \ 0) , (8)

Tp,e\ T0p,e ] 1

2316

(me] m

p)

kvS2B T0p,e

] 12

316

mp

kvS2 ( feq \ 1) . (9)

The temperature of the preshock medium is generally lowenough so that The initial ratio ofT0p \T0e(\ T0). T

e/T

pbehind the shock can be obtained by dividing equations (7)and (6). Assuming that and are small enough toT0p T0eignore,

Te

TpB

feq2 [ feq

, (10)

where terms involving have been dropped. A diagramme/m

pshowing the dependence of and on and appearsTe

Tp

vS

feqin Figure 7.To illustrate the dependence of postshock ionization

structures on the shock velocity and equilibration, we now

FIG. 7.ÈElectron and proton temperatures predicted by the jump con-ditions, shown for a range of shock velocities and equilibrations. Thecalculation assumes a pure H gas.

describe four numerical models (assuming preshock param-eters cm~3, K) :n0\ 1 fH0 \ 0.5, T0\ 5000

1. 250 and 500 km s~1.ÈThe postshock electron andproton temperatures equilibrate rapidly in the 250 km s~1model, even when The protons are too cold tofeq \ 0.appreciably ionize H, so charge transfer and electron colli-sions dominate the ionization balance of the postshock gas.The slow and fast neutral densities are shown as a functionof position behind the shock in Figure 8. Coulomb colli-sions in the 500 km s~1, model only produce 30%feq\ 0equilibration by the time the neutrals fully ionize, and, onceagain, charge exchange is the dominant interaction betweenprotons and neutral atoms. In this model, the broad andnarrow Balmer emission is produced well before equili-bration is complete. Note that the ionization zones for both250 km s~1 and 500 km s~1 models are more extended forlow equilibration than for high equilibration. At higherequilibrations, the electrons are hotter, ionizing the gasmore efficiently and reducing the thickness of the postshockneutral layer. At Ðxed equilibration, the size of the neutrallayer is proportional to the shock velocity. Therefore, thefaster the Ñow, the larger the neutral layer.

2. 1500 and 2500 km s~1.ÈThere are two importantdi†erences between these models and those at lower shockvelocity. First, in a high-velocity shock starts o†T

pfeq \ 0

so much higher than that electron-proton equilibration isTeonly a few percent complete by the time all the neutrals

disappear and all the Balmer emission is produced. Second,the proton ionization rate is now comparable to the elec-tron ionization rate, and a sizable fraction of the Balmerline emission is produced by proton excitation. At theseshock velocities, the proton-slow neutral collision rate is afactor of 2 or more higher than the proton-fast neutral colli-sion rate, due to the bulk velocity-shifted proton distribu-tion encountered by the slow neutrals. At Ðxed shockvelocity, the size of the postshock ionization layer is larger


FIG. 8.ÈIonization structures of four nonradiative shocks, shown for a 50% preionized medium with total number density of 1 cm~3. Densities of fast andslow neutrals are plotted vs. postshock distance for the cases of no equilibration and full equilibration at the shock front.( feq \ 0) ( feq \ 1)

for high equilibration than low equilibration (Fig. 8), theopposite of the 250 and 500 km s~1 models.

4.3. Calculation of Broad FW HM versus Shock VelocityTo make the best use of the observational data, we have

used the broad component Ha widths in each SNR tobracket the range of possible shock velocities. For a givenbroad component width, no equilibration yields( feq \ 0)the smallest shock velocity while full equilibrationv

S(min),

yields the largest shock velocity (CKR80;( feq\ 1) vS(max)Smith et al. 1991). Intermediate equilibrations yield inter-mediate For each observed shock, we ran numericalv

S.

models sampling a range of between 0 and 1, with eachfeqvalue of mapping onto a unique We then comparedfeq vS.

the output Ha and Hb broad-to-narrow ratios with theobserved values. In this manner, our models self consistent-ly utilized all of the relevant observables from each Balmerline spectrum.

For a given broad FWHM, the map between and isfeq vSthe line proÐle function (Smith et al. 1991 ; CKR80)./(v

x)

For a plane parallel shock viewed edgewise, is deÐned/(vx)

as the number of fast neutrals with velocity along the linevxof sight :

/(vx) \ l

p3

n3@2(Spcx

vTs ] SpivTs)

e~l2v02

]P~=

= P~=

=vp

cx(v)e~l2(v2~2v0 vz)dv

ydv

z, (11)

where andv2\ vx2] v

y2] (v

z[ v0)2, l

p\ (m

p/2kT

p)1@2,

is the bulk velocity of the shocked gas, taken tov0\ 3/4vSbe along the z-axis. We have computed numerically/(v

x)

for a range of and In Figure 9 we present plots of thefeq vS.

expected FWHM versus shock velocity for four equili-brations. We have used these results to estimate for thev

Sobserved nonradiative shocks, listed in Table 4 for the limitsof no equilibration and full equilibration.


FIG. 9.ÈVariation of broad component FWHM with shock velocity,computed for di†erent equilibrations using eq. [11]. The numerical valuesin the Ðgure have been computed for a nonradiative shock viewed edge-on.

4.4. Monte Carlo Models of L yb and L yc TrappingWith the densities and temperatures of di†erent particle

species known for a given and we can now quantitat-vS

feq,ively estimate the contribution of Lyman line trapping tothe Ha and Hb lines. Since Lyman photons are emitted withDoppler shifts randomly distributed over the line proÐle, itis possible for a Lyman photon generated near line center ofa fast neutral to be absorbed by a slow neutral, and viceversa. The conversion of Lyman line photons to Balmer linephotons depends on the likelihood of absorption. Due tothe velocity shift between fast and slow neutrals, the(3/4)v

Soptical depth depends on both the frequency and directionof an emitted Lyman photon. Therefore, our radiativetransfer calculation follows the propagation, absorption,and conversion of individual Lyman photons behind theshock.

We model the conversion of Lyman photons into Balmerphotons using a Monte Carlo simulation. Lyman photonsare propagated through the shock until they either are con-verted into Balmer photons or escape from the grid. Due tothe large Doppler motions of fast neutrals, most of thebroad Lyman photons escape from the shock. NarrowLyman photons, on the other hand, can be absorbed bothbehind and ahead of the shock. There is e†ectively an inÐn-

ite optical depth to narrow-component Lyman photons inthe preshock region ; some absorption occurs even if thepreshock neutral density is very small (CKR80). For thisreason, the radiative transfer calculation includes the pre-shock region, where the temperature and density areassumed constant.

The Monte Carlo program used in this work is based onan earlier code used by HRB94, with atomic data updatedto include Hb emission and proton excitation. The prob-ability that a Lyman photon will be absorbed at each pointin the grid depends on the optical depth at each point.qlThe optical depth, in turn, depends on the random velocityof the emitting atom along the direction of emission andwhether the atom is a fast or slow neutral. In the MonteCarlo program, 10,000 individual photons are generatedwith random Doppler shifts, randomly distributed behindthe shock according to the emissivity. If a given excitationproduces a Lyman photon, the photon is followed along arandomly oriented ray until it is either converted into aBalmer photon or escapes from the grid. After 10,000 exci-tations, the number of accumulated broad and narrowBalmer photons is divided to obtain the broad-to-narrowintensity ratio.

Before we compare our models with the observations, wenote that our predicted broad-to-narrow ratios o†er anumber of improvements over previous calculations (suchas those of HRB94). The current models utilize more recentatomic rates and compute the broad-to-narrow ratios forboth Ha and Hb. Our new models also include direct col-lisional excitation of hydrogen by protons. Finally, byarranging our models into a grid, we have been able topredict broad-to-narrow ratios for a single set of modelscovering a wide range of shock velocities and equilibrations.

5. COMPARISON OF MODELS AND OBSERVATIONS

With the numerical models available, we can now esti-mate and for Cygnus P7, SW RCW 86, and Tychov

Sfeqknot g1. There is observational evidence that the tem-

perature ahead of nonradiative shocks can be as high as40,000 K (Ghavamian et al. 2000 ; Smith et al. 1994 ;HRB94). The model predictions are rather insensitive to thepreshock temperature (the Lyman optical depth at linecenter P T ~1@2), so in the following sections we have set

K in all models. It should also be noted thatT0\ 5000since the preshock neutral density is equal to the neutralfraction times the total preshock density the(nH0 \ fH0 n0),predicted broad-to-narrow ratios from models with high fH0and low are similar to models with low and highnH0 fH0 n0.The reasons for this property are that (1) the total preshockdensity scales out of the ratio of the broad-to-narrowLyman line optical depths and (2) the equilibration, ioniza-

TABLE 4

LINE PROFILE FITS

Ib/I

nIb/I

nBroad FWHM v

STarget (Ha) (Hb) (km s~1) (km s~1)a

Cygnus P7 . . . . . . . . . . . . . . . . . . . . 0.59^ 0.3 0.99 ^ 0.3 262^ 32 235È395Southwestern RCW 86 . . . . . . 1.18^ 0.03 1.54 ^ 0.17 562^ 18 545È793Tycho knot g . . . . . . . . . . . . . . . . 0.67^ 0.1b 1.15^ 0.3 1765^ 110 1940È3010

a Extrema in shock velocity correspond to the equilibrated and unequilibrated cases, respec-vStively. Quoted includes uncertainty in broad FWHM.v

Sb Corrected for contribution from di†use emission.


tion, and charge exchange times behind the shock scale aso†setting the inÑuence of varying on the excitation/1/n0, n0ionization rates. Along with and the preshock neutralv

Sfeq,fraction is the most important quantity a†ecting thefH0

broad-to-narrow ratios. For this reason, we set n0\ 1cm~3 in the models that follow.

5.1. Cygnus L oop P7To interpret our results, we ran numerical shock models

using the fractional equilibrations 0.03, 0.05, 0.2, 0.5,feq\ 0,0.8, and 1.0, along with their corresponding shock velocitiesof 265, 268, 270, 273, 296, 332, and 365 km s~1. The predict-ed Ha and Hb broad-to-narrow ratios are shown in Figure10 versus equilibration for preshock ionization fractions of0.1, 0.5, and 0.9. In these models, the total preshock densityis 1 cm~3. Each curve in the Ðgure corresponds to a di†er-ent preshock neutral fraction Each point along a givenfH0.curve corresponds to a di†erent equilibration and mapsto a di†erent shock velocity. An outstanding feature ofthe models is that the broad-to-narrow ratios reach amaximum at low equilibration. The peak occurs becausethe broad-to-narrow ratios are proportional to the ratio ofthe charge exchange rate to the collisional ionization rate.In these models, the ratio peaks in the range 0.03 [ feq[

At higher equilibrations, the collisional ionization by0.05.electrons is increasingly e†ective, reducing the size of theneutral layer and reducing the broad-to-narrow ratio.

Although variations in do a†ect the shape and posi-fH0tion of the broad-to-narrow curves, the most conspicuousfeature in Figure 10 is that the broad-to-narrow ratios arepredominantly sensitive to (and hence The Ðgurefeq v

S).

shows that the higher equilibration models match theobservations best. In Ha, the predicted broad-to-narrow

FIG. 10.ÈPredicted broad-to-narrow ratios for Cygnus Loop P7(broad FWHM\ 262 ^ 32 km s~1), shown vs. equilibration fraction feqfor three di†erent preshock neutral fractions cm~3). Each equili-(n0\ 1bration combines with a unique shock velocity to yield the observed broadFWHM. Horizontal dashed lines mark the range of broad-to-narrowratios derived from observations.

ratios fall slightly above the observed values. The bestmatch occurs for full equilibration, where the model valuelies D10% above the upper error bar of the observations.However, the Hb broad-to-narrow ratio does agree withobservations, favoring fromfeqB 0.08È1.0 (T

e/T

p\ 0.7È1.0

eq. [10]). Excluding the uncertainty in broad FWHM, theimplied range of shock velocities is 300È365 km s~1.

The disagreement between the observed and predictedHa broad-to-narrow ratios may be due to the excitationcross sections utilized in the models. In the range of shockvelocities implied by the observations, the thermal energy ofthe postshock electrons is ryd (1 ryd\ 13.6 eV). At[15these energies, the collisional excitation cross sections arenot as well determined as the cross sections at higher ener-gies. This uncertainty is translated directly into the calcu-lated broad-to-narrow ratios.

The Cygnus P7 shock lies D25@ NW of the Ðlament pre-viously studied by RBFG83 and HRB94. In their analysis,HRB94 concluded that their observed shock has recentlyslowed due to an encounter with a density enhancement inthe ISM. The shock observed by RBFG83 and HRB94exhibits an Ha broad component width D130 km s~1 andan Ha broad-to-narrow ratio D1.6. In addition, faint[O III], [N II], and [S II] emission can be seen in this Ðla-ment. These features are markedly di†erent from those ofCygnus P7. In a pure Balmer line Ðlament in the westernCygnus Loop, Tre†ers (1981) also measured an Ha broadcomponent width D130 km s~1, with broad-to-narrowratio D1. Since the broad-to-narrow ratio is proportionalto the ratio of the charge exchange rate to the ionizationrate (CKR80; Smith et al. 1991), the di†erence between theCygnus P7 broad-to-narrow ratio and those of other Ðla-ments may be due to the fact that the collisional ionizationrate of hydrogen rises strongly with shock velocity for

km s~1. At a given equilibration, this would100 [ vS[ 300

result in a decreasing broad-to-narrow ratio for this rangeof shock velocities.

Overall, our derived shock velocity for Cygnus P7 lieswell above values obtained in most other optical studies ofthe Cygnus Loop. Previous observations using spectropho-tometry (Raymond et al. 1980 ; RBFG83; Fesen, Blair, &Kirshner 1982 ; Fesen & Itoh 1985 ; Raymond et al. 1988 ;HRB94), Fabry Perot spectrometry (Kirshner & Taylor1976 ; Tre†ers 1981 ; Shull & Hippelein 1991), and narrow-band imagery (Fesen, Kwitter, & Downes 1992 ; Levensonet al. 1998) have detected shock waves in various stages ofevolution, from nonradiative to fully radiative. These obser-vations have yielded shock velocities D100 km s~1 for thebright, radiative Ðlaments and D 150È200 km s~1 for thefaint, partially radiative/nonradiative Ðlaments. However,Kirshner & Taylor (1976) and Shull & Hippelein (1991)detected face-on Ha emission near the center of the CygnusLoop, blueshifted to velocities D 350È400 km s~1. Theseauthors suggested that the face-on emission is produced bythe supernova blast wave propagating into the low-densityISM. Soft X-ray observations of the Cygnus Loop (Ku et al.1984 ; Levenson, Graham, & Snowden 1999) conÐrm thepresence of fast (D400 km s~1) shocks in this remnant,including locations where there is no optical emission (thepreshock gas is fully ionized). The Cygnus P7 shock issimilar in speed to both the X-rayÈdetermined blast-wavevelocity and the blueshifts seen by Kirshner & Taylor (1976)and Shull & Hippelein (1991). Therefore, we conclude thatthe Balmer-dominated Cygnus P7 Ðlament marks the


current location of the supernova blast wave in the north-eastern Cygnus Loop.

5.2. SW RCW 86The Ha broad component width of the nonradiative

shock SW RCW 86 (Fig. 6) is 562 ^ 18 km s~1, placing thisshock at a Mach number intermediate between that ofCygnus P7 and Tycho knot g. The narrowband Ha image(Fig. 5) suggests that the shock geometry here is more com-plicated than for the Cygnus P7 shock, with multiple Ðla-ments surrounding the bright, clumpy emission covered bythe slit. The forbidden line emission in the two-dimensionalspectrum (Fig. 5) indicates that part of the clumpy regionhas become radiative. Narrowband Ha and [S II] images ofRCW 86 indicate that most of the bright clumpy emissionseen in Figure 5 is produced by radiative shock waves(Smith 1997). From the one-dimensional spectrum in Figure6, we estimate an Ha surface brightness of 4.2 ] 10~5 ergscm~2 s~1 sr~1 for the nonradiative shocks in southwesternRCW 86. This surface brightness is nearly twice that of knotg (KWC87; Ghavamian et al. 2000) and a factor of 2 ormore brighter than the Ðlament observed by HRB94. Thisproperty, along with the presence of radiative emissionnearby, suggests an enhanced preshock neutral density inthis part of the SNR. The velocity shift between the broadand narrow component centers in Figure 6 is smaller thanthe uncertainty in broad component width, indicating thatthe shock is viewed nearly edge-on.

Broad-to-narrow ratios were previously measured inRCW 86 by Long & Blair (1990), who obtained opticalspectra of Balmer-dominated shocks in the north andsouthwest (near our slit position). In southwestern RCW 86,they obtained an Ha broad component width of 600È780km s~1 and a broad-to-narrow ratio of 0.65È0.94 for thatline. These parameters are noticeably di†erent from ours ;the discrepancy is likely due to the fact that Long & Blair(1990) observed a partially radiative shock close to our slitposition. This is evidenced by the faint [N II] and [S II]emission lines in their southwestern RCW 86 spectrum. Inthis case, slight contamination of the Ha narrow componentby radiative Ha has likely lowered the broad-to-narrowratio. In addition, the S/N in the Ha proÐle of southwesternRCW 86 is much lower in the Long & Blair (1990) data thanours, making their broad component line Ðt more uncer-tain.

We assembled a grid of shock models for equilibrations0.03, 0.05, 0.2, 0.5, 0.8, and 1.0. These equilibrationsfeq \ 0,

yield the observed FWHM for shock velocities of 565, 570,573, 575, 634, 710, and 775 km s~1, respectively. The pre-dicted broad-to-narrow ratios appear in Figure 11, com-puted for the case cm~3. As with Cygnus P7, modelsn0\ 1with higher equilibration match the observations somewhatbetter than the low-equilibration models. From Figure 11,

simultaneously matchesfeq B 0.4È0.5 (Te/T

pB 0.25È0.33)

the Ha and Hb broad-to-narrow ratios. The implied shockvelocity is 600È635 km s~1.

The ability to simultaneously determine both the velocityand equilibration of Balmer-dominated shocks is a poten-tially useful aid in modeling the X-ray emission from SNRs.The ASCA study of RCW 86 by Borkowski et al. (2000) isone particular example. The Balmer-dominated shocks insouthwestern RCW 86 lie close to one of the ASCA spectralextraction windows used by Borkowski et al. (2000). In thissection of the remnant, the X-ray emission is dominated by

FIG. 11.ÈPredicted broad-to-narrow ratios for southwestern RCW 86section 2 (broad FWHM \ 562 ^ 18 km s~1), shown vs. equilibra-tion fraction for three di†erent preshock neutral fractions cm~3).feq (n0\ 1Each equilibration combines with a unique shock velocity to yield theobserved broad FWHM. Horizontal dashed lines mark the range ofbroad-to-narrow ratios derived from observations.

nonthermal continuum from fast km s~1)(vSD 600È1000

shocks, with thermal emission lines barely detected. Bor-kowski et al. (2000) noted that the portion of southwesternRCW 86 lying in the ASCA window features a variety ofshock speeds and that characterizing the thermal and non-thermal emission requires a large number of free param-eters. This makes the X-ray modeling of southwestern RCW86 quite difficult. We note that in the future, velocities andequilibrations predicted by our Balmer-dominated shockmodels may be used to reduce the number of free parame-ters needed to model the X-ray data of RCW 86 (and otherSNRs).

5.3. Tycho Knot gWe compared data from knot g1 with shock model pre-

dictions for equilibrations of 0, 0.03, 0.05, 0.2, 0.5, 0.8, and1.0, corresponding to shock velocities of 2050, 2075, 2090,2175, 2375, 2635, and 2900 km s~1. We allowed the pre-shock neutral fraction to vary from 0.99 to 0.1. We presentthe predicted broad-to-narrow ratios for knot g1 in Figure12. The models are unable to match the observed Ha broad-to-narrow ratio, although the Hb broad-to-narrow ratio isconsistent with This is similar to thefeq[ 0.2 (T

e/T

p[ 0.1).

limit on derived by Laming et al. (1996) for a shock ofTe/T

psimilar speed in SN 1006. Ignoring the measurement uncer-tainty in broad component width, the corresponding shockvelocity of knot g1 is 2050È2175 km s~1. The minimum in

occurs for an equilibration of 3%È5%; this corre-Ib/I

nsponds to the optimum electron temperature for Balmerline excitation in the narrow component (the opposite ofwhat occurs in the slower shock models). Models with high-preshock neutral fractions match the Hb broad-to-(Z0.8)narrow ratio best ; this result agrees with predictions of ourphotoionization precursor models (Ghavamian et al. 2000).The high neutral fraction may be explained by the H I 21 cm


FIG. 12.ÈPredicted broad-to-narrow ratios for Tycho knot g1 (broadFWHM\ 1765 ^ 118 km s~1), shown vs. equilibration fraction forfeqthree di†erent preshock neutral fractions cm~3). Each equili-(n0\ 1bration combines with a unique shock velocity to yield the observed broadFWHM. Horizontal dashed lines mark the range of broad-to-narrowratios derived from observations.

observations of Reynoso et al. (1999), which suggest that theeastern edge of Tycho is encountering a warm H I cloud. Onthe other hand, X-ray observations (Hwang, Hughes, &Petre 1998 ; Seward, Gorenstein, & Tucker 1983) and evolu-tionary models (Dwarkadas & Chevalier 1997) of TychoÏsSNR indicate a total preshock density cm~3. Taken[1.1together, the optical, X-ray, and radio data suggest thatknot g is probably encountering the extreme (low density)edge of the H I cloud.

6. DISCUSSION

One obvious problem is the failure of the models toreproduce the observed Ha broad-to-narrow ratio forTycho knot g1. The predicted Hb broad-to-narrow ratiosdo agree with the observations, although this is due in partto the larger error bars. Taking the predictedA

V\ 1.6,

narrow-component Balmer decrement falls below theobserved value, regardless of the equilibration or Lymanline trapping efficiency. The disagreement between theobserved and predicted broad-to-narrow ratios may bepartly due to the proton excitation cross sections utilized bythe shock code. The proton cross sections are most uncer-tain at low energy, D10 keV. Since the proton thermalenergy in the knot g1 shock lies close to this value, the Haand Hb excitation rates computed from these cross sections(including charge transfer into excited states) will be corre-spondingly uncertain. This situation is similar to theproblem mentioned earlier with the Cygnus P7 models,where the uncertain quantity was the electron excitationcross section near threshold.

An alternate explanation of the discrepant Ha broad-to-narrow ratio is collisional excitation of H in a spatiallyunresolved precursor. If the upstream H is collisionallyexcited before entering the knot g shock, the resulting Haand Hb emission will add to the overall narrow-component

Ñux from the shock. In an earlier paper (Ghavamian et al.2000) we presented a high-resolution Ha spectrum whichindicated that the neutrals are heated to a temperatureD40,000 K just before entering the shock. At temperaturesD104 K, collisional excitation produces an optical spec-trum with a steep Balmer decrement ; this could explainwhy the narrow-component Ha Ñux is so enhanced relativeto that of Hb. The Balmer-dominated spectrum ofTycho would then result from the superposition ofprecursor ] postshock neutral excitation.

It would be appropriate here to speculate on the relation-ship between and the magnetosonic Mach numberfeq where is the sound speed and is[M

S\ v

S/(c

S2] vA2)1@2, c

SvSthe speed in the preshock gas]. There is a clear trendAlfve� n

of decreasing equilibration with higher (see Table 5).MSCargill & Papadapoulos (1988) have argued that for low

Mach number shocks propagating perpendicular to theinterstellar magnetic Ðeld, a series of plasma instabilities areinitiated by the protons just behind the shock. As theprotons gyrate about the Ðeld lines, some of them re-enterthe upstream region, counterstreaming into the preshockgas. Laming (1998, 2001) and Bingham et al. (1997) havepointed out that if the reÑected protons follow a beamlike(monoenergetic) distribution, a two-stream instability willdevelop. Lower hybrid waves generated by the instabilitycan be in resonance with protons and electrons simulta-neously, allowing energy transfer between the two particlespecies. A key di†erence between perpendicular shocks with

(like Cygnus P7 and southwestern RCW 86)MSD 20È50

and those with like those in Tycho) is that theMSD 200

former are expected to be laminar (i.e., steady) while thelatter are expected to be highly turbulent (Tidman & Krall1971). Due to the unsteady nature of high Mach numbershocks, reÑected protons are likely to exhibit an angularspread in velocities, inhibiting the growth of the two-streaminstability and reducing the efficiency of electron heating.Electron heating by lower hybrid waves has been used toexplain X-ray emission from comet C/Hyakutake (Binghamet al. 1997), the nonthermal X-ray tail observed in Cas Aspectra (Laming 2001), and the injection of thermal particlesinto the cosmic ray acceleration process (McClements et al.1997).

There is strong evidence of an inverse relation between feqand from spacecraft observations of solar wind shocks.MSSchwartz et al. (1988) presented ISEE data from 14 inter-

planetary shocks, as well as 66 crossings of the EarthÏs bowshock. According to the ISEE data, electrons are heatedrelative to the protons by an amount which scales as 1/M

S.

Interestingly, the fractional equilibrations we derive fromour analysis follow roughly the same trend. The shocksstudied by Schwartz et al. only reach while ourM

SD 20,

observations cover The equilibration frac-25 [ MS[ 200.

tion determined by Laming et al. (1996) for a nonradiative

TABLE 5

SHOCK PARAMETERS PREDICTED BY NUMERICAL MODELS

vS

Shock feq Te/T

p(km s~1)a

Cygnus P7 . . . . . . . . . . . . . . . . . . . . 0.8È1.0 0.67È1.0 300È400Southwestern RCW 86 . . . . . . 0.4È0.5 0.25È0.33 580È660Tycho knot g1 . . . . . . . . . . . . . . . ¹0.2 ¹0.1 1940È2300

a Includes measurement uncertainty of broad component width.


shock in SN 1006 agrees with the value found simply byextrapolating from the lower Mach numberfeqP 1/M

Sshocks of Schwartz et al. (1988) to the values found in SN1006 (M

SD 250).

It is interesting to note that according to the results ofCargill & Papadapoulos (1988), collisionless heating pro-ceeds more efficiently for shocks propagating perpendicularto the interstellar magnetic Ðeld. There may be a connectionbetween the more efficient equilibration predicted for per-pendicular shocks and the limb brightening seen in radioobservations of barrel shaped remnants. In these cases, thelimb brightening occurs in parts of the remnant which pro-pagate perpendicular to the magnetic Ðeld (Gaensler 1998).The radio (and sometimes X-ray) emission from these rem-nants is synchrotron radiation from electrons energized byÐrst-order Fermi acceleration, a process where electrons areboosted to relativistic energies by scattering back and forthbetween upstream and downstream turbulence (Jones &Ellison 1991 ; Reynolds & Gilmore 1986 ; Blandford &Eichler 1987). Fulbright & Reynolds (1990) proposed thatthe enhanced radio emission occurs because the Fermiacceleration process is more efficient for perpendicularshocks. From the work of McClements et al. (1997) andLaming (1998), it appears that the same plasma instabilitieswhich heat the electrons in strong shocks can also injectparticles into the cosmic ray acceleration process. Clearly,the relationship between magnetic Ðeld orientation, equili-bration, and cosmic ray acceleration deserves a moredetailed investigation.

7. CONCLUSIONS

To investigate the properties of nonradiative shocks, wehave obtained high S/N spectra of the Cygnus Loop, RCW86, and Tycho covering a factor of 10 in Mach number. Ineach remnant, we have measured broad-to-narrow ratios ofboth Ha and Hb. We Ðnd that the broad-to-narrow ratiosshow considerable variation from one remnant to the next,with the Hb ratio systematically larger than that of Ha. Thedi†erence between the two ratios is evidence of Lyman linetrapping in the narrow component.

We have devised a numerical code which predicts theionization structure behind a nonradiative shock and uses a

Monte Carlo simulation to calculate the inÑuence of Lyband Lyc trapping on the Ha and Hb emission lines. Wehave modeled the Ha and Hb broad-to-narrow ratios inthree of the observed shocks : Cygnus P7 km(v

S\ 300È400

s~1), southwestern RCW 86 km s~1), and(vS\ 580È660

Tycho knot g km s~1). Overall, the shock(vS\ 1940È2300

code matches the observations in the Ðrst two remnants butyields only marginal agreement for Tycho knot g. Themodels indicate nearly complete equilibration for CygnusP7 and half equilibration for southwestern RCW 86, evi-dence for substantial collisionless heating in these non-radiative shocks. In knot g, the predicted Ha broad-to-narrow ratios are systematically larger than the observedvalues for all equilibrations. The difficulty in modeling theknot g broad-to-narrow ratio may be due to the largeuncertainty in proton excitation cross sections near thresh-old. However, the models do successfully reproduce theobserved Hb broad-to-narrow ratios. From the Hb results,we infer a low equilibration for Tycho, This value is[20%.consistent with the Ðndings of Laming et al. (1996), whoused ultraviolet observations with HUT to determine theequilibration of shocks of similar strength in SN 1006.

For a given shock velocity, we Ðnd that the thickness ofthe ionization layer depends on the electron-proton tem-perature equilibration. At shock velocities km s~1,[1000the collisional ionization of hydrogen is dominated by theelectrons ; the greater the equilibration, the thinner the ion-ization layer. This trend is reversed at high shock velocities.Above 2500 km s~1, proton ionization dominates electronionization ; therefore, the more equilibrated the shock, thethicker the ionization layer.

P. G. would like to the thank J. Weisheit, J. M. Laming,and R. Bandiera for several helpful discussions. We wouldalso like to thank the anonymous referee for valuable sug-gestions in improving the presentation of this paper. Thework of P. G. was supported by grant D70832 from RiceUniversity, STScI grant GO 07515-02.96A, NSF atomicphysics theory grant PHY 97-72634, and student travelsupport from CTIO. P. G. also acknowledges the hospital-ity of the Harvard-Smithsonian Center for Astrophysics,where some of this paper was completed. The work of J. R.was supported by NASA grant NAG 5-2845.

REFERENCESAldcroft, T. L., Romani, R. W., & Cordes, J. M. 1992, ApJ, 400, 638Bingham, R., Dawson, J. M., Shapiro, V. D., Mendis, D. A., & Kellett, B. J.

1997, Science, 275, 49Blair, W. P., Long, K. S., & Vancura, O. 1991, ApJ, 366, 484Blandford, R. D., & Eichler, D. 1987, Phys. Rep., 154, 1Borkowski, K. J., Rho, J., Reynolds, S. P., & Dyer, K. K. 2000, ApJ, in pressCargill, P. J., & Papadapoulos, K. 1988, ApJ, 329, L29Chevalier, R. A., Kirshner, R. P., & Raymond, J. C. 1980, ApJ, 235, 186

(CKR80)Chevalier, R. A., & Raymond, J. C. 1978, ApJ, 225, L27Detle†sen, D., Anton, M., Weiner, A., & Schartner, K.-H. 1994, J. Phys. B.,

27, 4195Draine, B. T., & McKee, C. F. 1993, ARA&A, 31, 373Dwarkadas, V. V., & Chevalier, R. A. 1998, ApJ, 497, 807Fesen, R. A., Blair, W. P., & Kirshner, R. P. 1982, ApJ, 262, 171Fesen, R. A., & Itoh, H. 1985, ApJ, 295, 43Fesen, R. A., Kwitter, K. B., & Downes, R. A. 1992, AJ, 104, 719Freeman, E. L., & Jones, E. M. 1974, Atomic Collision Processes in Plasma

Physics Experiments I (UKAEA Rep. CLM-R137 ; Abingdon : CulhamLab.)

Fulbright, M. S., & Reynolds, S. P. 1990, ApJ, 357, 591Gaensler, B. M. 1998, ApJ, 493, 781Ghavamian, P. 1999, Ph.D. thesis, Rice Univ.Ghavamian, P., Raymond, J., Blair, W. P., & Hartigan, P. 2000, ApJ, 535,

266Giovanardi, C., Natta, A., & Palla, F. 1987, A&AS, 70, 269Hesser, J. E., & van den Bergh, S. 1981, ApJ, 251, 549

Hester, J. J., Raymond, J. C., & Blair, W. P. 1994, ApJ, 420, 721 (HRB94)Hwang, U., Hughes. J. P., & Petre, R. 1998, ApJ, 497, 833Janev, R. K., Langer, W. D., Evans, K., Jr., & Post, D. E., Jr. 1987, Elemen-

tary Processes in Hydrogen-Helium Plasmas (New York : Springer)Jones, F. C., & Ellison, D. C. 1991, Space Sci. Rev., 58, 259Kaastra, J. S., Asaoka, I., Koyama, K., & Yamauchi, S. 1992, A&A, 264,

654Kennel, C. F., Edmiston, J. P., & Hada, T. 1985, in Collisionless Shocks in

the Heliosphere : A Tutorial Review, ed. R. G. Stone & B. T. Tsurutani(Washington, DC: Am. Geophys. Union)

Kirshner, R. P., & Taylor, K. 1977, ApJ, 208, L83Kirshner, R. P., Winkler, P. F., & Chevalier, R. A. 1987, ApJ, 315, L135

(KWC80)Ku, W. H.-M., Kahn, S. M., Pisarski, R., & Long, K. S. 1984, ApJ, 278, 615Laming, J. M. 1998, ApJ, 499, 309ÈÈÈ. 2001, ApJ, in pressLaming, J. M., Raymond, J. C., McLaughlin, B. M., & Blair, W. P. 1996,

ApJ, 472, 267Leibowitz, E. M., & Danziger, J. J. 1983, MNRAS, 204, 273Levenson, N. A., Graham, J. R., Keller, L. D., & Richter, M. 1998, ApJS,

118, 541Levenson, N. A., Graham, J. R., & Snowden, S. L. 1999, ApJ, 526, 874Long, K. S., & Blair, W. P. 1990, ApJ, 358, L13Long, K. S., Blair, W. P., & van den Bergh, S. 1988, ApJ, 333, 749

F. 1999, J. Phys. B, 32, 501Mart•� n,McClements, K. G., Dendy, R. O., Bingham, R., Kirk, J. G., & Drury,

L. OÏC. 1997, MNRAS, 291, 241


McLaughlin, B. M., Winter, T. G., & McCann, J. F. 1998, J. Phys. B, 30,1043

Petruk, O. 1999, A&A, 346, 961Raymond, J. C., Blair, W. P., Fesen, R. A., & Gull, T. R. 1983, ApJ, 275, 636

(RBFG83)Raymond, J. C., Davis, M., Gull, T. R., & Parker, R. A. R. 1980, ApJ, 238,

L21Raymond, J. C., Hester, J. J., Cox, D. P., Blair, W. P., Fesen, R. A., & Gull,

T. R. 1988, ApJ, 324, 869Reynolds, S. P., & Gilmore, S. P. 1986, AJ, 92, 1138Reynoso, E. M., Velazquez, P., Dubner, G., & Goss, W. M. 1999, AJ, 117,

1827Rosado, M., Ambrocio-Cruz, P., Le Coarer, E., & Marcelin, P. F. 1991,

Rev. Mexicana Astron. AstroÐs., 315, 243Schwartz, S. J., Thomsen, M. F., Bame, S. J., & Stansberry, J. 1988, J.

Geophys. Res., 93, 12923Seward, F., Gorenstein, P., & Tucker, W. 1983, ApJ, 266, 287Shah, M. B., Elliott, D. S., & Gilbody, H. B. 1987, J. Phys. B, 20, 2481

Shah, M. B., Geddes, J., McLaughlin, B. M., & Gilbody, H. B. 1998, J.Phys. B, 31, L757

Shah, M. B., & Gilbody, H. B. 1981, J. Phys. B, 14, 2361Shull, P., & Hippelein, H. 1991, ApJ, 383, 714Smith, R. C. 1997, AJ, 114, 2664Smith, R. C., Kirshner, R. P., Blair, W. P., & Winkler, P. F. 1991, ApJ, 375,

652Smith, R. C., Raymond, J. C., & Laming, J. M. 1994, ApJ, 420, 286Tidman, D. A., & Krall, N. A. 1971, Shock Waves in Collisionless Plasmas

(New York : Wiley)Tre†ers, R. R. 1981, ApJ, 250, 213Truelove, J. K., & McKee, C. F. 1999, ApJS, 120, 299Tuohy, I. R., Dopita, M. A., Mathewson, D. S., Long, K. S., & Helfand,

D. J. 1982, ApJ, 261, 473Vink, J., Kaastra, J. S., & Bleeker, A. M. 1997, A&A, 328, 628Whelan, C. T. 1986, J. Phys. B, 19, 2355Winkler, P. F., & Long, K. S. 1997, ApJ, 491, 829

Documents

THE ASTROPHYSICAL JOURNAL 2001. The …srk/Ay126/Lectures/Lecture... · No. 2, 2001 BALMER-DOMINATED SPECTRA OF NONRADIATIVE SHOCKS 997 uncertainty in broad component width, indicating