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Lecture 14
Supernova remnants
• Supernova remnants
• A simple model of the supernova explosion
• Energy conserving phase
• Momentum conserving phase
• Efficiency of energy conservation
• Effects of supernovae
• Motions of interstellar clouds
• The production of coronal gas in the interstellar
medium
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Supernovae remnants
• Supernovae provide the most obvious examples of shocks in the ISM
• SNe are ‘instantaneous’ releases of large amounts of energy into the ISM
• The local surrounding gas is heated to high temperatures. The
pressure increases and the gas expands supersonically with
respect to the ambient medium.
• A shock wave is immediately set up, moving out and
sweeping up the ISM
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Supernovae remnants
• Stars with M > 8M⊙ end their lives with an explosive ejection (core-collapse SNe of >1/2
stellar mass.
• The total kinetic energy released is ~1043 - 1044J and the ejection velocity is v ~ 0.02×c
• The kinetic energy is deposited instantaneously in the ISM of density n0. The expansion
velocity drops as the bubble expands.
• The surrounding ISM is typically low density (n ~ 1cm-3). This means the shocked post-shock
gas has very high initial T, since the gas cannot cool effectively by radiation processes (shock is
adiabatic)
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Supernovae blast wave
• The shock wave which moves into the surrounding gas and sweeps up gas is driven by the
pressure (i.e. the thermal energy) of the hot expanding gas bubble interior to the shock
• The shock wave S has a radius R and a velocity dR/dt. The latter is very high but drops as
the bubble expands
• It is important to consider the phase when radiative cooling in the shocked interstellar gas
(which now fills the bubble) is unimportant
• …this occurs because of the very low density
of the surrounding medium (n0 ~ 1cm-3) which
results in a long cooling time
S
dR/dt
n0
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Energy conserving phase
• We assume minimal radiative loss (i.e. adiabatic shock)
• Conservation of energy (RH3 condition) then requires that the total energy (ETOT) of the gas
behind the shock (= kinetic + thermal) is equal to the explosion energy (E*)
• We assume that dR/dt = VS since the gas ahead of the shock is taken to be at rest. We
remember that for a strong adiabatic shock we have:
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Energy conserving phase
• We assume also assume that the gas everywhere in the bubble has the specific energies
(thermal and kinetic) as given by the above equations.
• We see that the thermal and kinetically energy contribute equally to the total energy during
this energy conserving phase.
• The total energy (ETOT) of the gas in the bubble of density ρ0 is then:
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Energy conserving phase
• Since E* = ETOT, we can derive the following equation of motion for the boundary of the
bubble:
Remember VS = dR/dt
• Since the bubble starts of at a very small radius, we take R → 0 as t → 0 as the boundary
condition for the above differential equation. Adopt the Ansatz R = A×tα, then:
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Energy conserving phase
• Since E* and ρ0 are independent of time, we must have
• And thus:
• And so the solution to our differential equation for the bubble radius at time t is:
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Energy conserving phase
• And the expansion velocity of the bubble at time t is:
Note that dR/dt ↓ as t ↑
This means that TS ↓ since
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Momentum conserving phase
• We have seen that the shock velocity decreases with time and that this implies that the
immediate post-shock temperature also decreases with time
• The radiative cooling rate of the gas will actually increase as the temperature decreases as
more ions which are able to cool the gas via collisional excitation of lines become available
• The radiative cooling of the gas immediately behind the shock therefore becomes more
important as times goes on
• At some point: rate of decrease of temperature due to radiative cooling of gas > rate of
decrease of temperature due to expansion
• The gas tries to loose pressure but is pushed
up against the shock by the very hot interior
gas (which is so hot and tenuous that it does
not cool appreciably over the time-scales
relevant here)
• The compression → increase in gas density
→ increase in the cooling rate →
‘catastrophic’ cooling → formation of a thin
shell of cool material behind the shock
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Momentum conserving phase
• We shall make a simple analysis of the motion of the shell
• We ignore the internal pressure from the interior hot gas and assume shell moves outwards,
sweeping up matter and conserving momentum (but not energy since it radiates). This is the
snowplough model: high pressure, thin shell
• Since the gas cools so efficiently the shell will be thin and so a single radius R and velocity
dR/dt define its position and dynamics. Momentum conservation demands that:
• Suppose the thin shell is formed
instantaneously at a time t0 when R=R0
and dR/dt = (dR/dt)0. Then:
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Momentum conserving phase
• And so:
• If we integrate between t0 and t, and between r0 and r we get:
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Momentum conserving phase
• And so:
• At large enough times (i.e., t >> R0/(dR/dt)0), we have that R ∝ t1/4 and dR/dt ∝ t-3/4
• This is characteristic for the momentum
conservation phase: R ∝ t1/4 and dR/dt ∝ t-3/4
• We see that R increases more slowly, and dR/dt
decreases more rapidly, with time in the
momentum conservation phase than in the
energy conservation phase
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Momentum conserving phase
• The transition (energy conserving to momentum conserving) occurs at dR/dt ~ few 100 km
s-1. During energy conserving (adiabatic) phase, T is high (> 106 K) and the gas emits Xrays, i.e. young SNRs emit X-rays (Cas A, Tycho, Kepler, Crab)
• During momentum conserving phase T is lower (< 105 K) and gas strongly emits
collisionally excited visible line radiation, i.e. old SNRs e.g. Vela, Cygnus Loop.
• Ex: For E* = 1044J, n0 = 106m-3, we have
(R in pc)
And
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Momentum conserving phase
• For (dR/dt)0 = 250km/s then R0 = 24pc after t0 = 4×104yrs. And the total mass swept up after
this time is ~1400M⊙ (>> ejected mass)
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Efficiency of energy conservation
• We have shown that in the adiabatic phase the explosion energy stored is approximately
equally between kinetic and thermal energy of the gas.
• In the ‘snowplough' phase, the fraction of the explosion energy (E*) stored in kinetic energy
is:
• At later times (momentum conserving phase), we saw that:
Note dR/dt = R
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Efficiency of energy conservation
• Inserting into f, we get:
• Inserting the values from the previous examples: (dR/dt)0 = 250km/s, R0 = 24pc, n0 = 106m-3,
t0 = 4×104yrs:
Note f decreases with
time as expected
• While if (dR/dt)0 = 10km/s, then t/t0 = 44, f ~ 0.04, i.e. rather low.
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Effects of Supernovae - Motions of interstellar clouds
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Effects of Supernovae - Motions of interstellar clouds
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Effects of Supernovae - Motions of interstellar clouds
• When two diffuse clouds collide, there is a significant dissipation of kinetic energy through
shocks.
• If there is a steady-state in the ISM, we need to supply energy: could supernovae supply
enough energy?
• The mean free path for cloud-cloud collisions = λC = (nσ)-1 where n = # clouds per unit
volume and σ is a cloud cross-section
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Effects of Supernovae - Motions of interstellar clouds
• Let:
• Note that:
⇒
• We then have the cloud free mean path:
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Effects of Supernovae - Motions of interstellar clouds
• The time-scale between cloud-cloud collisions is:
where uC is the typical cloud velocity and λC is the inter cloud separation = mean free path
• The rate at which collisions occur an in the Galaxy is:
• And in each collision an energy EC = 1/2 MC uC2 (×2 for clouds)
is dissipated, where nC is the gas number density in a cloud
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Effects of Supernovae - Motions of interstellar clouds
• Thus the kinetic energy dissipated at a rate per unit volume in the Galaxy is:
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Effects of Supernovae - Motions of interstellar clouds
• Ex.: Assume nC = 3×107m-3, NT = 3×107 (=clouds in Galaxy), VT = 2×1060m3 = volume of
disk (R=10kpc, thickness=250pc). Also, let VC = volume of clouds of diameter 5pc. Then we
get:
• If each SN liberates E* Joules of which a fraction g is converted to kinetic energy injection,
then the energy injection rate per unit volume is:
where tex is the time between SN events in the galaxy
• If tex ~ 30yr and g=5%, then:
• Hence it seems plausible that the energy balance is maintained by supernovae!
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The production of coronal gas in the ISM
• Coronal gas in this context refers to very hot gas in the ISM - named by analogy with the
solar corona. It is detected through emission of X-rays. The X-ray spectrum extends from
0.1-2keV, indicating T ~ 106K
• The density of the emitting gas is deduced to be ~104m-3 and it must be collisional excited
and ionised since radiative ionisation produces ionised gas with a temperature of only
~104K, which would not emit at X-ray wavelengths
• Old expanding supernovae remnants are believed to be the origin of the coronal gas. The
coronal gas is often referred to as the ‘hot interstellar medium (HIM)’ phase
• The far-UV spectra of hot stars often show interstellar absorption lines of OVI (five-times
ionised oxygen). Temperatures of ~few 105K are needed to produce this ion by collisional
ionisation. This material is cooler than the coronal gas and is believed to represent an
interface between the hot X-ray emitting ‘HIM’ phase and the cooler (5000 - 10000K) ‘warm
interstellar medium (WIM)’ phase
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