The Physics of Supernova 1987A

Richard McCray

Abstract

We describe multiwavelength observations of the evolving spectra and images ofSupernova (SN) 1987A, and we review the principles used to infer the physicalconditions in the explosion debris. We interpret the early optical and gamma-ray light curves with a simple diffusion model. We review the evidence for dustformation in the debris. We show X-ray and optical observations that enable us tocharacterize and map the shock fronts caused by of the interaction of the debriswith circumstellar matter. We describe how observations of millimeter emissionlines due to rotational transitions of CO and SiO enable us to map the distribution,masses, and temperatures of these molecules in the debris.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Supernova Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 The Light Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 X-Rays and Gamma Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Spectral Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Dust Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Circumstellar Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 The Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

8.1 Plane-Parallel Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108.2 Blast Wave and Reverse Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

9 Radiation from Shocked Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149.1 X-Ray Emission from Non-radiative Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149.2 Radiative Shocks: The Hotspots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.3 Balmer-Dominated Shocks: The Reverse Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . 209.4 Doppler Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

10 Interior Debris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.1 Molecular Emission from Inner Debris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2310.2 Rotational Transitions of CO and SiO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

R. McCray (�)Department of Astronomy, University of California, Berkeley, CA, USAe-mail: mccrayr@me.com

© Springer International Publishing AG 2017A.W. Alsabti, P. Murdin (eds.), Handbook of Supernovae,DOI 10.1007/978-3-319-20794-0_96-1

1

2 R. McCray

10.3 Modeling the Spectral Line Energy Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1 Introduction

Supernova 1987A in the Large Magellanic Cloud was first observed on February23, 1987. It is the brightest supernova since Kepler’s supernova of 1604. By virtueof its proximity, SN is the first supernova to be observed at every band of theelectromagnetic spectrum and the first to be observed through its initial flash ofneutrinos.

SN1987A was classified as a Type II supernova, i.e., its spectrum was dominatedby hydrogen lines. But it had an unusual light curve, which continued to brightenfor about three months after its initial outburst before it began to fade. We nowunderstand that this behavior was a consequence of the fact that its progenitor was ablue giant rather than a red giant star. Modern supernova surveys show that roughly1–2% of Type II supernovae display light curves and spectral evolution similar toSN1987A.

SN1987A is surrounded by a system of three circumstellar rings (Fig. 1), whichevidently were ejected by the progenitor some 20,000 years before the supernovaoutburst. The cylindrical symmetry of the ring system strongly indicates that theprogenitor of SN1987A was a binary star system. The absence of any evidence fora surviving companion star suggests that the two stars merged before the explosion,

Fig. 1 SN1987A as seenwith the Hubble SpaceTelescope in 2010 (Courtesyof Peter Challis)

The Physics of Supernova 1987A 3

and this scenario may account for the facts that the progenitor was a blue giant andthat it ejected the triple ring system.

Today, almost 30 years after its discovery, SN1987A has made the transition tothe supernova remnant phase, in which its luminosity is dominated by emission fromshocks formed where the supernova blast wave encounters the inner circumstellarring. Although the supernova debris has faded by a factor �10�7, it is stillobservable at wavelengths ranging from radio to gamma rays, and it continues tobe the most intensively observed supernova in history.

In this chapter, I aim to provide a simple framework for understanding the physicsof this evolution. To do this, I will rely on rough order-of-magnitude estimates.The reader can find many references to the observations and more details of theirinterpretation in review articles by Arnett et al. (1989), McCray (1993) and McCrayand Fransson (2016).

2 Supernova Energetics

SN1987A is a core-collapse supernova, which means that the explosion is the resultof the gravitational collapse of the iron core of a massive star. In the final stagesof evolution, thermonuclear reactions in the core convert H to He, He to C, O,Ne, Si, and ultimately to Fe. Lacking any source of thermonuclear energy, the ironcore cools and shrinks until it is supported by degeneracy pressure of electrons. Atthis point the core resembles a white dwarf star, having mass comparable to Mˇ,the mass of the Sun, and radius RC � 1000 km. The mass of the core continuesto increase as a result of thermonuclear reactions in shells surrounding the core.When the core mass exceeds MC D 1:4 Mˇ (the Chandrasekhar limit), electrondegeneracy pressure can no longer withstand the pull of gravity. The core collapseson the free-fall timescale, tff � R

3=2C .GMC /

1=2 � 2ms. The collapse is halted at aradius Rcore � 12 km, at which point the density of the core, �core � 1015 g cm�3,is comparable to that of an atomic nucleus and the nuclear force becomes stronglyrepulsive. The kinetic energy of infall, EG � .3=5/GM2

C =Rcore � 3 � 1053 ergs, is

converted to heat, which is divided equally among photons, electrons, positrons, andthree species of neutrinos and antineutrinos. The temperature of the resulting fireballcan be estimated from the relationship EG D .43=8/aT 4Œ4�R3core=3�, which yieldskT � 100MeVŒRcore=10 km��3=4. At such a temperature and density, the ironnuclei dissolve into neutrons and protons. Most of the protons are converted toneutrons by inverse beta decay. The result is a nascent neutron star.

During the first few seconds after the collapse is halted, the hot neutron staris opaque, even to neutrinos. The neutrinos carry the internal heat by convectionto the neutrinosphere, the surface above which the neutrinos can freely streamoutward. The convection is violently unstable, and this instability reverses the infallof matter into the neutron star and deposits ESN � 1% ofEG as thermal and kineticenergy of outflowing matter. This outflowing matter acts as a piston, driving a shockwave through the envelope of the star. Thermonuclear reactions in the shocked gassynthesize heavy elements, including the radioisotopes 56Ni, 57Ni, and 44Ti.

4 R. McCray

Hydrodynamic instabilities during the first few days after the explosion cause theheavy elements to be mixed with the O, C, Ne, Si, etc., that were synthesized duringthe late stages of evolution of the progenitor star and with the hydrogen/helium ofthe stellar envelope. This mixing is macroscopic, not microscopic, in the sense thatthe fragments of different chemical composition do not interact chemically exceptat their boundaries.

Most of the binding energy of the newly formed neutron star emerges as a burstof neutrinos lasting several seconds. In the case of SN1987A, this neutrino flashwas detected through flashes of Cerenkov light seen in deep underground tanks ofwater in Japan and Ohio. The total energy of the neutrino flash was E� � 5 �

1052 ergs, as expected for the formation of a neutron star. (E� D EG=6, because thegravitational collapse energy was divided equally among six types of neutrinos, onlyone of which was detected). The duration, �10 s, and the characteristic temperatureof the neutrinos, kT� � 4MeV, were also just as predicted for the formation of aneutron star.

As mentioned, a small fraction of the energy of the neutrinos is depositedas thermal and kinetic energy in a hot bubble of gas that reverses the flow ofmatter falling toward the nascent neutron star. This bubble, which resides withinthe neutrinosphere at �100 km, drives a shock wave through the envelope of thestar. The passage of the shock deposits half the energy of the bubble as kineticenergy of the exploding star and half as thermal energy. The thermal energy residesmostly as photons. One can estimate the temperature of the radiation by equating.4�R3�=3/aT

40 D 1:5 � 1051 ergs, which gives T0 � 107 K ŒE51=R

312��1=4, where

E51 D ESN=1051 ergs and R12 is the radius of the progenitor in units 1012 cm.

We may estimate the characteristic expansion velocity, V , of the supernovadebris by equating the explosion energy, ESN D MV 2=2, which yields V D3200ŒE51=M10�

1=2 km s�1, whereM10 is the debris mass in units of 10 solar masses.The time for the supernova blast to propagate through the debris is given byt0 D R=V � 1 h R12M

1=210 E

�1=251 .

3 The Light Curve

Figure 2 shows light curves of various components of SN1987A.The optical display of the supernova will commence when the blast arrives at

the photosphere, an event called shock breakout. According to the (very rough)estimate above, this event should occur about 1 h after the neutrino flash. At shockbreakout, the temperature of the photosphere will suddenly rise to TS � 106 Kand subsequently decrease rapidly (within a few hours) as the debris expands.The shock breakout yields a flash of ionizing radiation having initial luminosityL0 � 1045 ergs=s and total fluence of ionizing (>13:6 eV) radiation Fi D 2 �

1057 photons. The net energy of ionizing radiation released at shock breakout,Ei�10

47 ergs, is negligible compared to the total energy of the supernova explosion,ESN � 3 � 10

51 ergs.

The Physics of Supernova 1987A 5

Fig. 2 SN1987A Light Curves (McCray and Fransson 2016). Solid curves are debris: green –radioactive deposition; violet – far infrared; cyan – optical. Dashed curves are equatorial ring: pink– radio (3–20 cm); green –X-rays (0.5–3 keV); red – UV/optical; gold – near infrared (5–30�m)

For the first few months, the supernova envelope is opaque to the radiationdeposited in the debris by the blast wave, so that most of the radiation producedby the supernova blast cannot escape freely. The opacity is dominated by Comptonscattering, and one may estimate roughly the optical depth by assuming that thesupernova envelope has uniform electron density, ne D M=ŒmH4�R

3=3�. Theoptical depth is given by � D ne�T R, where �T D 0:67 � 10�24 cm�2 is theThomson scattering cross section. Accordingly, we estimate � � 2 � 109M10R

�212 .

The characteristic time for radiation to escape from such an envelope is given bytesc � R�=c � 0:7� 10

11 sM10R�112 . The radiation will remain trapped until a time

t D tesc , or t � 1:5 � 107 s M3=410 E

�1=451 , or about 4.6 months for M10 D 1 and

E51 D 3. This estimate agrees fairly well with the observed time, t D 3 months, ofmaximum light for SN1987A.

By maximum light, the characteristic radius of the debris will have increased toR12 � 4:3�10

3M1=410 E

1=451 (about 380 AU forE51 D 3). The trapped radiation cools

according to the adiabatic law, Prad / � , where the adiabatic index is D 4=3 fora radiation-dominated fluid. Since � / R�3, Prad / R�4. Given that Prad / T 4,we find that T / R�1 and the energy of trapped radiation decrease as Erad / R�1.By the time that the trapped radiation can escape, its energy will have diminishedby a factor �4:3 � 103. As the fireball expands, the radiation pressure causesthe expanding debris to accelerate, so that the characteristic expansion velocityincreases by a factor

p2.

If the explosion were the only source of energy in the supernova debris, theoptical display would be very faint, with total energy <1048 ergs, far less than

6 R. McCray

the observed value. But the debris contains another source of energy: radioactivityof newly synthesized isotopes, principally 56Ni and its daughter 56Co. The fastpositrons and gamma rays from the decay of these isotopes deposit their energyin the debris in timescales comparable to their mean lifetimes, tN i D 8:76 d for56Ni and tC o D 111:3 d for 56Co. Assuming that the gamma rays are absorbed,the radioactive energy deposited by 56Ni is 1.75 MeV per decay and, by 56Co,3.83 MeV per decay. The total energy deposited in the supernova envelope by 56NiisENi D 0:6�1050MNi ergs and that by 56Co isECo D 1:3�1050MNi ergs, whereMNi is the initial mass (in Solar units) of 56Ni produced by the supernova.

The energy deposition rate by these isotopes is, therefore, D 0:8 � 1044 ergs/sMNi exp.�t=tN i / C 1:36 � 1043 ergs/s MNi Œ1 � exp.�t=tN i /� exp.�t=tCo/�. By7 months after the explosion, tesc � t and the bolometric luminosity of thesupernova will be equal to the nuclear energy deposition, now dominated by 56Codecay. A fit to the observed bolometric luminosity for 7 months < t < 14 months,which actually decays exponentially with mean lifetime tCo D 111:3 days, enablesus to infer that MNi D 0:071 solar masses.

Although the net energy deposited by 56Co, ECo D 0:9 � 1049 ergs, is muchless than the thermal energy, 1:5 � 1051 ergs, produced by the blast, it dominatesthe light from the supernova. This is so because the 56Co energy is deposited on atimescale �tCo D 111:3 days, by which time the thermal energy from the blast hasdiminished to �4� 1047 ergs. Likewise, the energy from 56Ni decay, most of whichis deposited in a timescale �tN i , makes a minor contribution to the light curve.

The same kind of analysis explains why most core-collapse supernovae areinitially much brighter than SN1987A. The main reason is that their progenitorsare red supergiants, which have initial radii �400Rˇ or about 30 times greater thanthe progenitor of SN1987A. Accordingly, the internal radiation from the initial blastis not diminished nearly so much by adiabatic expansion when t � tesc .

4 X-Rays and Gamma Rays

The decay of 56Co produces gamma ray emission lines at 847 and 1238 keV, andthese were observed as early as 190 days Sunyaev et al. (1987). As McCray (1993)describes, these observations imply that a significant amount of 56Co resides atThomson optical depths as low as �T � 1, whereas the Thomson optical depthfrom the center of the debris has the value � � 25 at 190 days according toour simple uniform density model with E51 D 3 and M10 D 1. The fact thatsome of the newly synthesized radioactive elements lie in the outer portion of thedebris is consistent with three-dimensional hydrodynamic simulations (Fig. 3) of theexplosion that show plumes of 56Ni extending beyond the inner debris, where mostof the nucleosynthesis products reside.

For t > 2 years, the supernova envelope became transparent to gamma rays.Thereafter, most of the gamma rays emerged without depositing their energy inthe supernova debris. In that case, the energy deposition by radioactive elementsis dominated by stopping of the fast positrons that accompany their decay. (It is

The Physics of Supernova 1987A 7

Fig. 3 Isocontours of the 56Ni distribution for a 20 Mˇ (left) and 15 Mˇ (right) progenitor�16 h after explosion (Wongwathanarat et al. 2015). The color bars give the expansion velocityof the surface. Note the higher velocities in the strongly fragmented distribution for the 15 Mˇmodel. The heating from the 56Ni decay has not been included

generally assumed that the positrons do not propagate far from the place where theyare emitted because a very weak magnetic field is sufficient to trap them.)

By t D 4:25 years, the abundance of 56Co had decreased by a factor �10�6, andthe radioactive energy deposition was dominated by less abundant isotopes withlonger mean lifetimes, notably 57Co (t57 D 271 days). Today, the radioactive energydeposition is dominated by the decay (electron capture) of 44Ti (t44 D 86:7 years),which is accompanied by the emission of gamma ray lines at 67.9 and 78.3 keV. The44Ti decay is followed by the prompt .tSc44 D 5:7 h) ˇC decay of its daughter 44Sc,which deposits 0.73 MeV of positron kinetic energy. Observations of the gamma raylines by the INTEGRAL (Grebenev et al. 2012) and NUSTAR (Boggs et al. 2015)observatories enable us to infer the mass, M44 D .1:5 ˙ 0:5/ � 10�4Mˇ, of 44Tiand the net energy deposition due to its decay.

The green curve in Fig. 2 shows the energy deposition in the supernova debrisdue to 56Ni, 56Co, 57Co, and 44Ti.

5 Spectral Evolution

For the first few months after the explosion, the supernova debris was opaque.Accordingly, its optical spectrum was dominated by a continuum. Blackbody fits tothe continuum give a temperature Tph � 5500K, which remained fairly constantin time. During this photospheric phase of spectral evolution, the radius of thephotosphere can be inferred from the observed luminosity according to the equationLph D 4�R2ph�T

4ph. A fit to the evolution of Lph shows that Rph increased as

8 R. McCray

Rph / t1=2 to a maximum value Rmax � 100 AU by tmax � 3 months. Note that the

velocity, Vph D dRph=dt , decreases with time, indicating that the photosphere wasmoving inward with respect to the debris. At tmax, the radial velocity of the matterat the photosphere was Vmat ter D Rmax=tmax � 1800 km/s.

After this time, the photosphere shrank rapidly as the optical continuum faded.This event marked the transition from the photospheric phase to the nebular phase,in which the radiation from the supernova debris was dominated by emission lines.

Even during the photospheric phase, the optical continuum of SN1987A waspunctuated by emission lines, notably H˛, [O I]��6300; 6364 [Ca II] 7291, 7324,and Ca II��8600. The H˛ line had a P-Cygni profile, consisting of bright redshiftedemission and a blueshifted absorption feature, evidently caused by resonant scat-tering of continuum photons by excited hydrogen atoms in the rapidly expandingdebris above the photosphere.

As the supernova continued to expand, the emission lines became narrower,indicating that the fast-moving gas was becoming less dense and fading, while thephotosphere moved inward to reveal emission from more slowly expanding matterdeeper within the supernova debris. The widths of the emission lines during thenebular phase implied that most of the line emission was confined to a sphericalvolume expanding with radial velocity�1800 km/s.

McCray (1993) gives a detailed discussion of the evolution of the nebularspectrum. Here, we only summarize the most important conclusions from theanalysis of this spectrum. First, the supernova debris cooled rapidly, owing toadiabatic expansion and radiative cooling. For example, infrared emission bandsfrom vibrationally excited CO molecules indicated that the gas emitting these bandscooled from T � 4000K at 192 days to T � 1800 K at 377 days. Recent(Kamenetzky et al. 2013) observations of rotational transitions of CO made with theALMA observatory show that the emitting gas has continued to cool to T < 100K.

Second, different temperature evolution histories were inferred from emissionlines of different elements, clear evidence that the debris was not mixed at themicroscopic level but instead remained fragmented into chemically distinct regions.

6 Dust Formation

A dramatic change in the light curve and spectrum of SN1987A occurred in theinterval 250 days � t � 550 days. First, the optical light curve began to fademore rapidly than the 111.3 day exponential decay. Second, a far-infrared (FIR)(� > 8�m) continuum appeared. Third, at the same time, the red sides of the opticaland near-infrared (NIR) emission lines from the interior debris began to vanish.

All these observations could be explained by the formation of dust grains inthe inner debris. The dust was evidently black, in the sense that there was nopreferential extinction of the optical lines compared to the NIR lines. Moreover, thedust continuum could be fit better with a Planck spectrum rather than the emissionspectrum of typical interstellar dust grains. The fact that the dust blocked almost the

The Physics of Supernova 1987A 9

entire red sides of the emission line profiles from the interior debris but roughly halfof the blue sides indicated that the dust was confined to opaque clouds that blockedalmost all the light from the far hemisphere of the debris and � half of the lightfrom the near hemisphere.

By �600 days, the FIR continuum from dust was equal to �30% of thebolometric luminosity of the supernova debris. The temperature of the dust wasT � 600K at 400 days and decreased to T � 140K at 1316 days. At theseearly times, the minimum mass of dust required to account for the optical and NIRextinction was �3 � 10�4 Mˇ.

Twenty-eight years after the explosion, an observation with the Herschel obser-vatory (Matsuura et al. 2011, 2015) discovered a strong FIR continuum, whichdominated the bolometric luminosity of SN1987A. A model fit to the continuumspectrum gave a dust temperature Td � 17�23K and a minimum mass of�0:4Mˇof silicate dust (Dwek and Arendt 2015) An image of the continuum source takenwith the ALMA (Indebetouw et al. 2014) confirmed that most of the dust is confinedto the inner debris.

The fact that the extinction by dust obscures most of the optical and NIR emissionfrom the inner debris makes it difficult to infer anything about nucleosynthesis yieldsor the morphology of the inner debris from optical or NIR spectra taken after thedust began to form. The only way to see the entire debris is to observe it at sub-mmwavelengths, at which the dust becomes transparent.

7 Circumstellar Matter

At t � 80 days, narrow (FWHM � 30 km s�1) emission lines of N V��1238,1242, N IV]�1486, N III]�1750, C III]�1909, He II�1640, [O III]�4363,[O III]�5007, and a few other ions appeared in the spectrum of SN1987A. Theseemission lines brightened to a maximum at t � 400 days and then began to fade.

Fransson et al. (1989) recognized that these lines came from nearly stationarygas surrounding the supernova that was photoionized by the initial flash of EUVradiation from the supernova. From the time to reach maximum, they inferred thatthe circumstellar matter was located at a distance �0:6 lt-years from the supernova.From the rate of fading of the emission lines they could infer that most of theemitting gas had electron density ne � .2 � 4/ � 104 cm�3, but the persistenceof [O III]�5007 required an additional component of circumstellar gas with lowerdensity, ne � 103 cm�3.

The structure of the circumstellar matter became clear in 1990, when imagesobtained with the ESO NTT and the HST showed that the line emission was comingfrom a circular ring of diameter dr D .1:27˙0:07/�1018 cm that was inclined withi D 43ı. Assuming that the ring was thin and uniform, Dwek and Felten (1992)constructed a model for the light curves of the narrow emission lines. The modelconsists of a geometrical model for the illumination of the ring by the supernovaflash convolved with physical models for the decay of emission lines from flash-ionized gas in the ring (Lundqvist and Fransson 1991, 1996; Mattila et al. 2010;see also Sect. 9). This model fits the data qualitatively but not quantitatively (Gould

10 R. McCray

1995). By comparing the angular diameter of the ring with its physical diameter asinferred from the light curves of the narrow lines, one can infer that the distance tothe supernova is D D 46:8˙ 0:8 kpc.

This distance estimate relies on an assumption that we now have reason to doubt.That is, that the narrow emission lines come from the same gas as is observed inthe image seen by the HST. We now know, from observations of the impact of theblast wave with the ring, that the ring is not uniform but consists of some thirtyhotspots of high-density gas protruding inward from a substrate of lower-densitygas. When the ring is illuminated by the ionizing flash, most of the narrow lineemission comes from the hotspots. But, since the timescale for fading of the lineemission is inversely proportional to the gas density, the hotspots vanish first. By thetime that the HST images became available, the image was dominated by the lower-density gas, which has angular diameter about 5% greater than the hotspots. To beconsistent, one should compare the image of the hotspots with the light curves fromthe narrow emission lines. One can estimate that correction will yield a distanceabout 5% greater than the estimate above.

The cylindrical geometry of the triple ring system suggests that rotation playsa key role in its formation, but the actual mechanism remains uncertain. Proposedscenarios range from mass loss from a rapidly rotating single star (Chita 2008) tothe merger of a binary star system (Morris and Podsiadlowski 2007). An importantclue comes from the radial velocity of the ring, vr D 10:3 km s�1. Assuming thatthe equatorial ring has been expanding at constant velocity, we can estimate the timewhen it was expelled, t D dr=2vr � 20;000 years.

8 The Impact

As discussed above, most of the energy of the supernova explosion resides askinetic energy of the expanding debris. This energy becomes manifest when thedebris strikes circumstellar matter. The collision creates a system of shocks thatsuddenly slow down the debris while accelerating the circumstellar matter. Theshocks suddenly heat the gas, causing it to radiate. The radiating system resultingfrom this impact comprises the supernova remnant.

The circumstellar matter surrounding SN1987A has a complex structure (Fig. 4).Consequently, the system of shocks resulting from the impact of the debris ofSN1987A with this matter is complex, and the radiation from such shocks hasa complex spectrum. To interpret the observations, we review the basic physicsof plane-parallel shocks, following Ryden (http://www.astronomy.ohio-state.edu/~dhw/A825/notes7.pdf).

8.1 Plane-Parallel Shocks

In a frame of reference in which the shock is stationary, the gas entering theshock from upstream is characterized by its density, �1, velocity, u1, pressure,P1 D nkT1, and internal energy density, �1. The corresponding parameters

The Physics of Supernova 1987A 11

Fig. 4 Schematic diagram of SN1987A and its inner circumstellar ring, viewed normal to theequatorial plane. The nucleosynthesis products in the interior debris are confined mostly within acomoving sphere expanding with velocity �1800 km s�1. The blue represents the outer envelopeof the supernova, which is composed mostly of hydrogen and helium. The blue-yellow interfacerepresents the reverse shock, while the yellow annulus represents the X-ray-emitting gas, boundedon the outside by the blast wave. The white fingers represent protrusions of relatively dense gas.As the blast wave overtakes these fingers, they light up as hotspots

describing the downstream flow are �2, u2, P2, and �2. The internal energy densityof the gas is related to its density and pressure by � D P=Œ. � 1/��. The adiabaticindex is D 5=3 for a monatomic nonrelativistic gas and D 4=3 for a relativisticgas (such as a radiation field).

The Rankine-Hugoniot equations representing conservation of mass, momentum,and energy across a plane-parallel shock are:

�1u1 D �2u2; (1)

�1u21 C P1 D �2u

22 C P2; (2)

and

u21=2C �1 C P1=�1 D u22=2C �2 C P2=�2 (3)

We define the shock Mach number as the ratio of the velocity of the upstreamflow divided by its sound speed:

12 R. McCray

M1 D

��1u21P1

�1=2(4)

In terms of the Mach number, the Rankine-Hugoniot equations yield

�2=�1 D u1=u2 D C 1

. � 1/C 2=M21

(5)

and

P2 D2�1u21 C 1

�1 �

� 1

M21

�(6)

A strong shock is the limit M � 1, and hence P1 is negligible. In that case wefind (for D 5=3)

�2 D 4�1; (7)

u2 D u1=4 (8)

and

kT2 D3

16 u21 (9)

where is the mean molecular weight per particle.

8.2 Blast Wave and Reverse Shocks

The simplest model for the propagation of the blast from a supernova explosioninto uniform interstellar gas is the famous Sedov solution. In this solution, theradius, R, of the blast wave depends on only three parameters: E, the energy ofthe explosion; �, the density of interstellar gas; and t , the time since the explosion.The only combination of these three parameters that has the dimensions of length isR D AŒE=��1=5t2=5, where A is a dimensionless parameter. A detailed solution ofthe hydrodynamic equations gives A D 1:17.

The Sedov solution is based on the assumption that the energy of the explosionresides entirely in the interstellar gas overtaken by the blast wave. It is a goodapproximation for the propagation of a thermonuclear explosion in the Earth’satmosphere but not for young supernova remnants. In the latter, a substantial fractionof the hydrodynamic energy resides in the expanding debris of the explosion.

In the simplest case, in which both the supernova debris and the circumstellarmatter are spherically symmetric and have power-law radial distributions, the impactresults in two shocks, as illustrated in Fig. 5. The shock that propagates forward into

The Physics of Supernova 1987A 13

Fig. 5 Simulation of the hydrodynamics of a supernova striking circumstellar gas. Gray scalerepresents gas density. The expanding outer atmosphere has a power-law density distribution � /v�9t�3. Left panel: in the polar direction the structure obeys the similarity solution describedby Chevalier (1982). In the equatorial direction the blast wave is decelerated suddenly when itencounters a toroidal distribution of relatively dense gas and a reflected shock merges with thereverse shock. Right panel: the blast wave has encountered the dense equatorial ring, sending atransmitted shock into the ring and another reflected shock backward, where it merges with thereverse shock

the circumstellar matter is called the blast wave, while the shock that propagatesback into the debris is called the reverse shock. Sandwiched between these twoshocks is a double layer consisting of shocked supernova debris on the inside andshocked circumstellar gas on the outside. The boundary between these two layers iscalled the contact discontinuity.

We may characterize the density distribution of the circumstellar medium as�c / r

�s , where s D 0 for uniform density and s D 2 (for a steady wind.) Modelsfor the hydrodynamics of a supernova explosion show that the outer debris will havea power-law density distribution, �d .r; t/ / t�3.r=t/�n, where the index n rangesfrom n D 7 (Chevalier 1982) to n D 9:6 (Ensman and Burrows 1992). In thiscase, it is possible to construct a self-similar solution for the propagation of shocks(Chevalier 1982). Self-similarity implies that the ratio of the radius of the blast waveto that of the reverse shock is constant and that the density of the circumstellar matterat the blast wave is a constant fraction of the density of the supernova debris at thereverse shock. Assuming so, we find from dimensional analysis that the radius ofthe blast wave evolves as RB / t .n�3/=.nCs/ (e.g., RB / t 2=3 for n D 9 and s D 0).

Chevalier presents detailed solutions of the structure of the similarity solutionsfor n ranging from n D 6 to n D 14 and both s D 0 (uniform density circumstellarmatter) and s D 2 (steady stellar wind). For these solutions the ratio of the radiusof the blast wave to that of the reverse shock is typically �1:2. Moreover, thedensity of the shocked debris at the contact discontinuity is greater than that ofthe shocked circumstellar gas, so that the contact discontinuity will be unstable inthe decelerating flow.

14 R. McCray

If, as is the case with SN1987A, the supernova debris strikes a more complexdistribution of circumstellar gas, a more complex distribution of shocks will ensue.In particular, if the blast wave strikes a density discontinuity in the circumstellar gas,it will split into two shocks: a transmitted shock that propagates into the denser gasand a reflected shock that propagates backwards into the gas behind the blast wave.If the density contrast is great, the dense gas acts as a rigid obstacle. In that limit,one may show from Eqs. (1), (2), (3), (4), (5), (6), (7), (8), and (9) that the reflectedshock propagates backward with twice the velocity of the blast wave, compressingthe gas by a further factor 2:5 and raising the temperature by a factor 2:4.

9 Radiation from Shocked Gas

We may define three categories of shocks in supernova remnants: non-radiativeshocks, radiative shocks, and Balmer-dominated shocks.

9.1 X-Ray Emission from Non-radiative Shocks

In non-radiative shocks, the timescale for energy loss by radiation exceeds thecharacteristic timescale for the system, so that the hydrodynamics is unaffected bythe radiation. Actually, the term non-radiative shock is a misnomer. Radiation atX-ray wavelengths is commonly observed from non-radiative shocks in supernovaremnants.

According to Eq. (9), a shock entering stationary gas with velocity u1 D1000 km s�1 V1000 will heat the gas to a temperature

kT D .1:95 keV/ V 21000; (10)

where we have set D 1 for a composition of 50% hydrogen and 50% heliumby number, appropriate for the circumstellar matter around SN1987A. Accordingly,gas behind shocks moving with velocity 300 < V < 3000 km s�1 will be heated totemperature 0:176 keV < kT < 17:6 keV. Such gas will radiate primarily in the softX-ray band.

The radiation from such a gas will be dominated by electron impact excitation ofemission lines. The luminosity of each emission line may be calculated from

Li;j;k.T / D EMAifjCi;j;kEi;j;k; (11)

where the volume integral

EM D

ZdV nen (12)

The Physics of Supernova 1987A 15

is called the emission measure. To calculate the complete spectrum, one should adda similar expression for continuum emission due to bremsstrahlung, recombination,and excitation of metastable levels that decay by 2-photon emission.

TheEi;j;ks are the energies of photons emitted following excitation of ions ni;j Dnifi;j having fraction fi;j in ionization state j , and Ai is the fractional abundanceof element i , ni D Ain. The functions Ci;j;k.T / are the rate coefficients for electronimpact excitation of states Ei;j;k , i.e.,

Ci;j;k D .2�kT =me/�3=2

Z4�v2dv expŒ�mev

2=kT �v�.v/ (13)

where the �i;j;ks are the electron impact excitation cross sections. References tovalues of �i;j;k and Ci;j;k for hundreds of ions can be found in Sutherland and Dopita(1993).

To complete the task of calculating the emission spectrum of hot gas, it’snecessary to calculate the distribution of ionization states for each element. Thisis accomplished by solving coupled rate equations for ionization and recombinationof each element:

d

dtfj D �nefj

Xk

Ij;k.T /C neXk

fkIk;j .T /; (14)

where the quantities Ij;k.T / are rate coefficients for the reactions .njCe ! nkCe,which include impact ionization to states k > j plus radiative and dielectronicrecombination to states k < j .

If the temperature of the gas remains constant, the various atomic processeswill drive the ionization rate equations to a stationary state such that dfj =dt D 0

for all j . In that case Eq. (14) becomes a set of algebraic equations, the solutionsof which are called coronal ionization equilibrium or CIE models. According tothe CIE approximation, the solutions, fj .T /, are functions of temperature alone,independent of density (because all reaction rates are linear in ne). Figure 6 showsa typical example, the case of neon.

Given the solutions fj .T /, one can calculate the emission spectrum, L� , for ahot gas from Eq. (11) and the total luminosity

L.T / DXi;j;k

Li;j;k D nen�.T /; (15)

where the second equality shows explicitly the fact that L.T / is proportional tothe square of the gas density. The function �.T /, called the cooling function, isindependent of density and is illustrated in Fig. 7.

16 R. McCray

Fig. 6 Population of ionization states of neon in coronal ionization equilibrium (Landini andMonsignori Fossi 1990)

Fig. 7 Radiative cooling function (Sutherland and Dopita 1993). The curves are labeled bylog10 [Fe]/[H], where [Fe]/[H]D 1 corresponds to solar system abundance ratio nFe=nH D3�10�5. The dashed curve labeled CEI represents the coronal ionization equilibrium model (with[Fe]/[H]D 1), while the other curves represent NEI models

The Physics of Supernova 1987A 17

To account for departures from CIE in modeling the X-ray emission fromshocked gas, one can solve the coupled differential equations (Eq. 10), assuming thatthe gas entering the shock has relatively low ionization. Therefore, accurate modelsfor the emission of radiation from shocked gas require the simultaneous solution ofthe coupled differential equations (Eq. 10). Assuming that the gas enters the shockat relatively low ionization and that its temperature remains constant downstreamfrom the shock, the solutions will depend on two parameters, temperature, T, andionization age, net . One may then calculate the spectrum of the shocked gas byintegrating the line emissivity (Eq. 11) along the downstream flow from the timethat the gas first entered the shock to the present. This procedure yields a set ofnon-ionization equilibrium or NIE, models for fi .T; nt/ and Li;j;k.T; nt/.

Although the CIE approximation provides much physical insight into the radi-ation from a hot gas, it can be off the mark when calculating the radiation fromshocked gas. The error arises from the fact that electron impact ionization rates andradiative recombination rates are typically slow compared to excitation rates. Asa result, the ionization balance tends to lag changes in temperature. Gas suddenlyheated to a given temperature on passage through a fast shock will be less ionized atthat temperature than dictated by the CIE approximation. Typically, lower ionizationstates have more bound levels that can be excited by thermal electron collisions.Consequently, a suddenly heated gas will be under-ionized compared to the CIEmodel, and its actual line emission can be substantially greater than that given bythe CIE model.

Conversely, if a gas at a given temperature is allowed to relax by radiativecooling from an initial state of CIE, it will be over-ionized, so that the emissionof radiation will be less than given by the CIE model. Therefore, accurate modelsfor the emission of radiation from shocked gas require the simultaneous solution ofthe differential equations (Eq. 14) coupled with the appropriate fluid equations forthe evolution of density and temperature.

For 105 K < T < 107 K, the radiative cooling function shown in Fig. 7 isdominated by emission lines of O, Ne, and Fe in the range 100–912 Å, which cannotbe observed directly owing to interstellar absorption (Landini and Monsignori Fossi1990).

Figure 8 shows an X-ray spectrum of SN1987A. For wavelengths 6 Å < � <

12 Å (energies 2 keV > E > 1 keV), it is dominated by emission lines of hydrogen-and helium-like Ne, Mg, and Si, while for 12 Å < � < 20 Å (energies 1 keV >

E > 0.6 keV), it is dominated by emission lines of Fe XVII and Fe XX. Fits ofNIE models to such spectra enable one to determine the parameters T; nt; EM , andelemental abundances Ai and their confidence bounds.

It is not possible to find a satisfactory fit to the observed X-ray spectrum witha single shock temperature. This is not surprising, given that the young remnantof SN1987A has complex hydrodynamics, with shocks entering gas having a dis-tribution of densities. Accordingly, we generalize the emission measure, EM.T /,to a distribution function over temperature, d.EM/=dT . The additional degrees offreedom introduced by this generalization enable a good fit to the observed X-rayspectrum. We find that the fitting procedure yields a bimodal distribution functionwith peaks at�0:5 keV and�2 keV.

18 R. McCray

Fig

.8

X-r

aysp

ectr

umof

SN19

87A

(Dew

eyet

al.2

008)

The Physics of Supernova 1987A 19

We can derive an important clue to the shock hydrodynamics from the profiles ofthe X-ray emission lines, which have FWHM� 500 km/s. According to Eq. (9), toheat gas to temperature �2 keV requires a shock having velocity �1000 km/s. Butthe gas is not moving this fast. This apparent paradox is resolved when we recognizethat the hottest gas has been shocked a second time by a reflected shock from a denseobstacle. The reflected shock simultaneously slows the gas down as it elevates thetemperature and density.

The low temperature peak in the bimodal distribution function probably comesfrom shocks that have been transmitted into clumps of intermediate-density gas asthey are overtaken by the blast wave. As Fig. 7 shows, the cooler gas will cool morerapidly than the hotter gas. In this case, the radiative cooling may be rapid enoughto modify the hydrodynamics of the shocked gas, as described below.

9.2 Radiative Shocks: The Hotspots

In radiative shocks, energy loss due to radiation modifies the hydrodynamics ofthe shocked gas. Unlike non-radiative shocks, radiative shocks convert a substantialfraction of the thermal energy of the shocked gas into optical radiation. The opticalhotspots in SN1987A are due to radiative shocks.

After passage of the shock, the hot gas begins to cool and compress as a resultof emission of radiation. After sufficient time, this cooling will rob the hot gas of itsinternal energy. As the temperature falls in the downstream flow, the density will risein order to maintain approximate pressure equilibrium. Since the rate of radiativecooling (Fig. 7) increases with falling temperature and rising density, the coolingincreases rapidly. One may estimate the timescale for cooling from the equation:

d

dt

5

2nkT D �nen�.T / (16)

where we have taken coefficient 5/2 rather than 3/2 because the cooling takesplace at constant pressure rather than at constant density. Equation (16) yields acharacteristic cooling time:

tC �5kT

Œ2n0�.T /�(17)

where we have set ne D n, assuming pure hydrogen.Figure 9 illustrates the structure of a radiative shock, calculated by integrating the

hydrodynamic equations downstream from the shock front including NIE radiativecooling. The passage of the shock front heats the ions to a temperature T � 2� 106

K. Initially, the electrons are cooler than the ions, but in a relatively short timeCoulomb collisions equilibrate the ion and electron temperatures. In the ionizationzone the gas is under-ionized, but by t � 103=n0 years the system has relaxedto CIE. Then, by t � 104=n0 years, radiative cooling causes the temperature to

20 R. McCray

Fig. 9 Temperature (thin line) and density (thick line) structure for electrons (dashed line) andprotons (solid line) for a radiative shock of velocity 250 km s�1 (Pun et al. 2002)

decrease by a factor�10�2, while the density increases by a factor�102. The densegas in the photoionized zone would continue to cool, but the radiative cooling isbalanced by heating due to photoionization by EUV and soft X-rays produced inthe cooling region. In this way, roughly half of the radiation produced in the coolingzone is trapped in the photoionization zone and converted to optical radiation, whilethe other half propagates upstream into the unshocked gas. The photoionization zoneradiates optical radiation with a spectrum similar to that of a planetary nebula.

Actually, radiative shocks are violently unstable (Pun et al. 2002). With aconstant driving pressure, the shock front does not have constant velocity. Asradiative cooling sets in, the shock front slows down, causing a drop in temperatureof the shocked gas. With decreasing temperature, the shock front slows down more,resulting in a runaway collapse of the cooling layer. The collapse is halted when thephotoionization zone catches up with the shock and causes it to speed up again. Thisinstability manifests itself in three dimensions, causing the optical line emission tohave a velocity dispersion comparable to the shock velocity.

9.3 Balmer-Dominated Shocks: The Reverse Shock

As the name implies, Balmer-dominated shocks are characterized by the emissionof strong Balmer lines of hydrogen and little else. Balmer-dominated shocks werefirst recognized in the spectra of supernova remnants such as Tycho and wereinterpreted by Chevalier et al. (1980) as the result of the passage of the supernovablast wave into interstellar gas containing neutral hydrogen atoms. Downstream

The Physics of Supernova 1987A 21

from the shock, the atoms can be excited and/or ionized by collisions with hotions. If they are excited to levels n 3, the atoms will radiate Balmer lines. Theexcited hydrogen atoms retain the velocity distribution that the neutral atoms hadbefore shock passage. Once ionized, the atoms will cease emitting line radiation.On average, 0.2 H˛ photons are emitted for every HI atom that passes through a fastshock.

Note that the mechanism described above is entirely different from the mech-anism for Balmer line emission from gaseous nebulae. The former mechanism,electron-ion impact excitation by fast electrons and ions, excites all ions withcomparable cross sections, so that the strengths of emission lines from differentelements are roughly proportional to the element abundances. Thus, for shocksin gas with Magellanic Cloud abundances, hydrogen emission lines are somefour orders of magnitude brighter than emission lines from other elements. Incontrast, emission lines from gaseous nebulae are produced by collisions of ionswith thermal electrons, which have insufficient energies to excite hydrogen atomsto states n 2. In photoionized nebulae, Balmer lines are produced by radiativerecombination, which proceeds at a rate much slower than impact excitation.Moreover, forbidden lines such as [O I]��6300; 6364 and [N II]��6548; 6583 canhave strengths comparable to H˛, despite the fact that oxygen and nitrogen havemuch lower abundance than hydrogen.

The Balmer emission from the reverse shock in SN1987A takes place in adifferent frame of reference from the Balmer emission seen in supernova remnants.In the latter, the blast wave overtakes nearly stationary interstellar gas containingHI atoms, so the Balmer lines have narrow profiles (FWHM � 20 km s�1. Inaddition to this narrow line emission, line profiles from Balmer-dominated shocksin supernova remnants have components with linewidths comparable to the thermalvelocity of protons in the shocked gas, caused by charge transfer collisions ofhydrogen atoms with hot protons in the shocked plasma. But the cross sectionsfor such charge transfer collisions are unimportant in the high velocity flow throughthe reverse shock of SN1987A.) In the case of SN1987A, the HI atoms have thevelocity of the freely expanding supernova debris, so the Balmer line emission willhave Doppler shifts ranging up to several thousand km/s.

9.4 Doppler Tomography

As described in Sect. 2, the supernova blast wave propagated through the envelopeof the progenitor in a few hours. After that, the debris is further accelerated by thepressure of trapped radiation resulting from the initial explosion and from the depo-sition of energy by the decay of 56Ni (Li et al. 1993). Within a time �t�10 days,this acceleration becomes negligible, and the supernova debris expands freely.

If we can image the supernova debris at a given Doppler shift within an emissionline, we are seeing a slice of the debris at a depth z D vDt . This relation betweendepth and Doppler shift for supernova debris is a kind of Hubble’s law, vD D H0z,where the Hubble constant H0 D 1=t . In fact, it is more accurate than the Hubble’s

22 R. McCray

law that describes the expansion of the universe, which has significant departuresdue to cosmic evolution and gravitational interactions among galaxies and clusters.

The slices in SN1987A are nearly planar: the departures from flatness are ız=z ��t=t � 10 days=104 days � 10�3. The slices have finite thickness owing to themolecular velocity dispersion. The fractional thickness is given by�z=z � cs=vD �10�3 for thermal velocity cs � 1 km s�1 (assuming temperature 100 K) and Dopplervelocity vD � 1000 km s�1.

Doppler tomography gives us a unique opportunity to map the three-dimensionalstructure of the reverse shock in SN1987A with the Space Telescope ImagingSpectrograph (STIS). Figure 10 shows an image of a STIS spectrum with the slitplaced along the minor axis of the equatorial ring.

Each point in the STIS spectra is located by two parameters: � , the angle alongthe slit, and ��, the Doppler shift. The physical height of the point is given byh D D� , where D � 50 kpc is the distance to the supernova, and the projecteddistance from the midplane of the supernova is given by d D �ct��=�0. Givenh, d , and i D 45ı, the inclination of the equatorial ring, we can remap the STISimage of Fig. 10 into cylindrical coordinates z; �, where z is the height above theplane defined by the equatorial ring and � is the cylinder radius, as illustrated inFig. 11.

In this way, we transform the STIS image shown in Fig. 10 into a map of thereverse shock shown in Fig. 12. Note that the reverse shock has moved into theequatorial ring, as illustrated in Fig. 5.

Fig. 10 STIS spectrum of the reverse shock of SN1987A. (France et al. 2011). Left panel:location of the STIS slit on image of SN1987A. Right panel: STIS spectrum. RS denotes emissionfrom reverse shock. D denotes emission from interior debris. The vertical bar is due to H˛ and[NII]��6548, 6563 emission from nearly stationary gas in the ring, while the pair of spots near theleft border is due to emission by [OI]�6364 by the ring

Fig. 11 Transformation ofcoordinates inferred fromSTIS spectra (d, h) tocylindrical coordinates (�, z)

h

z

iα

d

ρ

r

R

The Physics of Supernova 1987A 23

Fig. 12 Map of reverseshock and supernova debrisinferred from STIS spectrum(France et al. 2011). The solidred dot denotes the center ofthe supernova. The red circlesdenote the equatorial ring.The part of the STIS imagealong the slit, whichrepresents stationary gas, hasbeen masked out

10 Interior Debris

The faint irregular feature near the center of SN1987A seen in Fig. 1 represents H˛emission from the freely expanding supernova debris. This emission is also evidentin the STIS spectrum (the feature marked D in Figs. 10 and 12). Like the reverseshock, this emission can be mapped by Doppler tomography. Figure 12 shows that itis concentrated in the equatorial plane and is predominately blueshifted, with radialvelocity extending to �6000 km/s, well beyond the expansion velocity �1800 km/sthat was inferred from the profiles of the emission lines during the nebular phase.

Note that there is little or no redshifted counterpart to the feature marked D inFig. 12. Its absence is probably due to extinction by internal dust.

Figure 2 shows that the optical luminosity of SN1987A began to increase after�5000 days. Before that time, the optical emission from the debris was morecentrally concentrated than the source shown in Fig. 2, and it was fading. Thebrightening can be attributed to illumination of the debris by the annular sourceof X-rays caused by the impact of the outer debris with the circumstellar ring(Larsson et al. 2011). The X-rays cannot penetrate into the inner debris where thenucleosynthesis products reside, because the photoabsorption cross sections of theseelements are much greater than those of hydrogen and helium.

10.1 Molecular Emission from Inner Debris

As the simulations of Fig. 3 show, the structure of the inner debris is sensitive to themass of the progenitor envelope. Fortunately, it is now becoming possible to mapthis structure through Doppler tomography of molecular rotational emission lines

24 R. McCray

70

60

50

40

30

20

10

0220 240 260 280 300 320 340 360

1.0

0.8CO3-2

SiO8-7

SiO7-6

SiO6-5

3-2HCO+ HCO+4-3

SiO5-4

CO2-1

0.6

0.4

0.20.35−

0.35−0.3−

0.25−

0.22− 0.18−0.15−

0.0

frequency [GHz]

flux

dens

ity [m

jy]

atm

osph

eric

tran

smis

sion

Fig. 13 Spectrum of SN1987A as observed by ALMA (Matsuura et al. 2017). The black curveindicates atmospheric transmission at the ALMA site

with the Atacama Large Millimeter Array (ALMA). Unlike optical radiation, whichis strongly absorbed by the dust in the debris, millimeter radiation can pass throughthe dust with negligible absorption.

Figure 13 shows the spectrum of SN1987A in the mm/sub-mm band as observedwith the ALMA. Emission lines from the rotational transitions .j; j � 1/ D .2; 1/

and .3; 2/ of CO and (5,4), (6,5), and (7,6) of SiO are clearly evident. (The (8,7) lineof SiO is blended with the (3,2) line of CO.)

To interpret these results, we need a model for how the observed luminosities ofthe emission lines depend on the physical conditions in the debris. To characterizethe simplest such model we assume that the CO (or SiO) resides in zones all havinguniform temperature and density. If so, the luminosity in each emission line from athin (thickness d z) section displaced by a line-of-sight distance z from the midplaneof the debris is given by

dLul D h�ulAulnuA.z/d zPesc (18)

where Aul is the Einstein A coefficient for the radiative transition u ! l and nu

is the number density of molecules in state u. A.z/ is the projected area of a thinsection displaced by a line-of-sight distance z from the midplane of the debris, andd z is the thickness of the section. For the freely expanding debris d z D ctd��=�ul .The mean escape probability, Pesc , accounts for the possibility that a spectral linephoton may undergo many resonant scatterings before it escapes the emitting region.In the optically thin limit, Pesc D 1, we have

The Physics of Supernova 1987A 25

Lul D h�ulAulnj

ZA.z/d z (19)

whereRA.z/d z D Vem, the net volume occupied by the emitting gas.

Typically, however, a line photon will be reabsorbed and reemitted many timesbefore it escapes. To account for this probability, we calculate the optical depth:

�ul D

Zd znl�lu

�1 �

glnu

gunl

�(20)

where �lu D .�e2=mec/flu�.��/ is the absorption cross section, flu is theoscillator strength for the transition l ! u, and �.��/ is the line profile function,R�.��/d�� D 1. The term in square brackets accounts for induced emission.

Then, taking d z D ctd.��/=�j;j�1, Eq. (20) becomes

�lu D nl.�e2=mec/flu.ct=�ul /

�1 �

glnu

gunl

�D�30tgunlAul

8�gl

�1 �

glnu

gunl

�(21)

The mean escape probability is given by

Pesc D

Z 1

0

dx exp.�x�lu/ D1 � exp.��lu/

�lu(22)

In the limit �lu � 1 Eqs. (19), (20), (21), and (22) yield

Lul D4��ul

ctB�.Tul /Vem; (23)

where B�.T / D .2h�3=c2/Œ1 � exp.h�=kTul /� is the Planck function and Tul is theexcitation temperature, defined by nu=nl D .gu=gl / exp.�h�ul =kTul /.

Note the difference between the expressions for the line luminosity in theoptically thin limit (Eq. 19) and the optically thick limit (Eq. 23). In the formercase, the luminosity is proportional to nuVem, the total number of molecules in theupper state, while in the latter case, the luminosity is proportional to the net emittingvolume, Vem, and is independent of the density of molecules.

10.2 Rotational Transitions of CO and SiO

We briefly summarize the physics of rotational transitions of CO and SiO, asdescribed in the NRAO website Molecular Line Spectra (http://www.cv.nrao.edu/course/astr534/MolecularSpectra.html). The energy levels of the excited rotationalstates are given byEj D j .jC1/E0, and the frequency of the transition j ! j �1

is given by

26 R. McCray

�j;j�1 D 2j �B; (24)

where �B D 57:65GHz for CO and �B D 21:71GHz for SiO. Accordingly, weidentify the strong emission lines seen in Fig. 13 as the (2,1) transition of CO andthe (5,4), (6,5), and (7,6) transitions of SiO. The strong feature at �345GHz is ablend of the 345.9 GHz (3,2) transition of CO and the 347,4 GHz (8,7) transition ofSiO.

The Einstein A coefficients for radiative transitions of diatomic molecules aregiven by

Aj;j�1 D64�4

3hc3 2j

2j C 1�3j;j�1; (25)

where is the electric dipole moment of the molecule. For CO, D 0:11 DebyeD 0:11 � 10�18 cgs, and for SiO, D 3:15 Debye. It follows that for CO

Aj;j�1 D 21:6 � 10�8 j 4

2j C 1s�1; (26)

and for SiO, by

Aj;j�1 D 9:15 � 10�6 j 4

2j C 1s�1: (27)

Note that, for a given frequency, the Einstein A coefficient for SiO is 820 timesgreater than that for CO because the dipole moment of SiO is 28 times greater thanthat of CO.

10.3 Modeling the Spectral Line Energy Distribution

If we have a model for the number densities nj D nfj of molecules in exci-tation states j , we can calculate the luminosities, Lj , of emission lines fromEqs. (19), (20), (21), and (22):

Lj;j�1 D h�j;j�1Aj;j�1nfj VemPesc.j; j � 1/: (28)

For example, if the energy levels are populated in local thermodynamic equilibrium(LTE), we have

fj .Te/ Dgj

G.Te/exp.�Ej=kTe/; (29)

where gj D 2j C 1, Ej D j .j C 1/h�B , and the partition function is, to a goodapproximation, given by

The Physics of Supernova 1987A 27

G.T / �kTe

E0C1

3C

E0

15kTe: (30)

If the LTE model is valid, one can calculate the luminosity of any given line fromEqs. (21), (22), (23), (24), (25), (26), (27), and (28) by specifying the local density,n, of molecules, the excitation temperature, Te , and the total emitting volume, V .

The LTE model will give a good approximation to the level populations, fj ,provided that the rate of excitations/deexcitations by collisions exceeds the rate ofradiative transitions, i.e.,

ncol lCj;j�1 � Aj;j�1Pesc.j; j � 1/; (31)

where ncol l is the density of collision partners and Cj;j�1.Tk/ D˝v�j;j�1

˛is the

rate coefficient for transitions driven by collisions with partners having kinetictemperature Tk .

There is considerable uncertainty in the value of ncol lCj;j�1. Rate coefficientsfor excitations of rotational transitions of CO by H2 and He have typical valuesC2;1 � 3:4 � 10�11 cm3 s�1 (Schöler et al. 2005: http://home.strw.leidenuniv.nl/~moldata/). So, for example, the population of CO in level j D 2 will be in LTE ifthe CO is embedded in molecular hydrogen with n.H2/� 2�104 cm3. However, itis unlikely that the CO-emitting gas is predominately H2. If the gas is predominatelyCO, the relaxation rates for CO C CO collisions may be substantially greater thanthose for COCH2 collisions, so the critical density of CO molecules may be muchless than that of H2 molecules.

To take into account departures from LTE, one can calculate the populations,fj .Tk/, by solving the coupled rate equations:

dfj

dtD 0 D �fjRj C fjC1RjC1;j C fj�1Rj�1;j (32)

where

Rj D Aj;j�1Pesc.j; j � 1/C ncol l .Cj;j�1 C Cj;jC1/; (33)

RjC1;j D AjC1;j Pesc.j C 1; j /C ncol lCjC1;j (34)

Rj�1;j D ncol lCj�1;j : (35)

Note that the solutions, fj .Tk/, are functions of the kinetic temperature, and Tk , ofthe colliding partners, in contrast to the rotational excitation temperature, Te , of themolecules in the LTE approximation.

The code RADEX (http://var.sron.nl/radex/radex_manual.pdf, van der Tak et al.2007) solves the set of Eq. (32). One can see that the solutions are functions of twolocal parameters, ncol l and Tk . They also depend on the density, n, of moleculesthrough the escape probabilities Pesc.j; j 0/, which depend on nj (Eqs. 20 and 22).

28 R. McCray

RADEX was developed to interpret observations of molecular line emissionfrom clouds of interstellar gas, characterized by ncol l , Tk , column density, N , andlinewidth, �v. One can find the solutions by specifying these parameters in theRADEX online site http://var.sron.nl/radex/radex.php. Since RADEX online asks usto specify bothN and�v, we choose an arbitrary value of�v, say,�v D 100 km/s,and, for given local density of molecules, n, specify the column density from therelationship N D n�vt , where t is the time since explosion. RADEX onlinereturns, for each transition, the excitation temperature defined by

fj

fj�1D.2j C 1/

.2j � 1exp

��h�j

kTR

�(36)

and the brightness temperature of the emitting surface, TR.j /, defined by

I� D B.�j ; TR/ D2h�3j

c2

�exp

�h�j

kTR

�� 1

��1; (37)

where I�.TR/ is the specific intensity radiated by the surface. The luminosity of anemission line is given by

Lj D 4�

Zd��A.z/I�.TR/ D

4��j

ctB.�j ; TR/Vem (38)

where we have used the relation d��=�j D d z=ct .To infer the physical conditions (n; ncol l ; TK; and Vem) of the emitting gas, we

search the space defined by these four parameters and map the surface for which theline luminosities predicted by Eq. (38) agree with the observed luminosities. Notethat Vem, the volume occupied by the emitting molecules, can be much less than thevolume, V , containing the emitting debris. We define a filling factor, f D Vem=V ,where V may be estimated from the observed widths and images of the emissionlines. If, as is likely, the molecules are confined within unresolved clumps, we expectf < 1.

The task of inferring the physical conditions (Vem, Tkin, ncol l , and n) from theobserved lines is still incomplete. Kamenetzky et al. (2013) fitted the luminosities ofthe CO 2� 1 and 1� 0 lines observed with ALMA and the CO 7� 6 and 6� 5 linesobserved with the Herschel-SPIRE spectrometer with an LTE model and inferredthat the CO-emitting gas had MCO > 0:2Mˇ, Te.CO/ > 100K. Matsuura et al.(2017) fitted the CO observations with a NLTE RADEX model (assuming ncol l D105 cm�3) and find MCO D 0:01 � 0:03Mˇ and Tkin D 30 � 50K. Matsuura et al.also fitted the SiO 6�5 and 5�4 lines and foundMSiO D 2�10

�5�1�10�3Mˇ,Tkin D 20 � 170K.

These estimates of the properties of the emitting regions are based on the ques-tionable assumption that they have uniform densities and temperatures. Indebetouwet al (2017) have employed ALMA to obtain images and spectra of SN1987A at

The Physics of Supernova 1987A 29

angular resolution �0:1” When these data become available, we will be able toobtain more refined measurements of the properties of the emitting regions.

Cross-References

�Lightcurves of Type II Supernovae� Spectra of Supernovae in the Nebular Phase�The Multi-dimensional Character of Nucleosynthesis in Core Collapse Super-

novae�X-Ray Emission Properties of Supernova Remnants

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