HIGH-ENERGY ASTROPHYSICS

26.02.2018

Accretion power in astrophysics

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Textbook

• Accretion power in astrophysics - Frank, King and Raine http://qxyang.lamost.org/uploads/books/Accretion_Power_in_Astrophysics.pdf

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Goals

• Eddington limit with derivation

• Bondi accretion (Spherical symmetry):– Accretion radius

– Supersonic accretion

• Wind accretion vs Roche Lobe overflow

• Accretion disk properties: temperature and typical spectrum

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Introduction• Accretion luminosity and

role of compactness.

• Eddington luminosity

– Balance radiation with gravitation

– Assume ionized hydrogen

– Discuss deviations

• Typical temperatures:

– Temperature if radiation were a black body

– Temperature if gravitational energy were all converted into heat

• Perform these calculations as in book

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Bondi accretion• Spherical accretion

• Solution with sonic point

• Bernoulli eq.

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Accretion radius• Accretion radius:

– Radius at which gravitational pull equals the sonic speed of medium (thermal motion)

• Efficiency of spherical accretion is low.

• For X-ray binaries, we need to have others mechanisms.

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Inefficiently radiating flow?• Density of the flow is

𝜌 =ሶ𝑀

4𝜋𝑟2𝑣= 𝜌𝑎𝑐𝑐

𝑟

𝑟𝑎𝑐𝑐

−3/2

• Density near a stellar black hole reach 1022𝑐𝑚−3 like the Earth atmosphere, near a supermassive (106 − 109) 𝑀𝑠𝑢𝑛 black-hole ?

• Matter falls on a free-fall time scale

𝑡𝑓𝑓 =𝑅𝑎𝑐𝑐

32

𝐺𝑀 1/2~𝐺𝑀

𝑟𝑎𝑐𝑐𝑟𝐵𝐻

3/2

~10−5𝑀

𝑀𝑠𝑜𝑙

𝑟𝑎𝑐𝑐

𝑟𝐵𝐻

3/2s

• Accretion flow could be heated at the temperature in the GeV range

𝑇~𝑚𝑣𝑓𝑓2 ~

𝐺𝑀𝑚

𝑟~mc2

• However, emission can be produced oly if the time scale of energy loss is lower than the free-fall time scale

𝑡𝐵𝑟~ 10−5𝑛

1020 𝑐𝑚−3

−1

𝑠

• For very low accretion rate, particles cannot interact and there is no emission.

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Galactic center• Density of medium is around

102𝑐𝑚−3

• Temperature from X-rays around 0.1-1 keV

• From motion of stars, we derive a mass of 4 × 106𝑀𝑠𝑢𝑛 and the closest star orbits at 1000 gravitational radii

• Variability occurs on time scales of 10 min, comparable to a Keplerian orbit around the BH

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Sgr A*• We can use the accretion radius and the

free-fall time scale (Exercise) to derive an accretion rate corresponding to a luminosity of 1041 erg/s (efficiency 0.1), at 8 kpc, this would be way too high for observations (10−10erg/s/cm2)

• We think of a very inefficient way of producing X-rays from accreting material: radio, flares in IR, in X-ray and TeV gamma-rays -> particles do not interact while falling!

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Observations• Quiescent and flaring

periods

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F. Yusef-Zadeh M. Et al.ApJ 144, 1 (2012)

Galactic center as an emitter• Let’s focus on IR and X-ray flare, which are

correlated• Electrons are heated up at the energy of

protons, which fall onto the central object at ~GeV energy giving rise to IR-range photons

𝜖𝑠𝑦𝑛𝑐ℎ = 0.05𝐵

1𝐺

𝐸

1𝐺𝑒𝑉

2eV

• These photons can Inverse Compton on the same population of electrons in the hard X-ray range

𝜖𝐼𝐶 =𝐸𝑒𝑚𝑒

2

𝜖𝑠𝑦𝑛𝑐ℎ

= 100𝜖𝑠𝑢𝑛𝑐ℎ0.01𝑒𝑉

𝐸𝑒1𝐺𝑒𝑉

2

𝑘𝑒𝑉

• We know the peak energy, but what about the intensities? We need to know the magnetic field and radiation densities

• Let’s estimate the radiation field as the observed luminosity in a dimension close to the Schwarschild radius

𝑈𝑟𝑎𝑑 =𝐿

4𝜋𝑟2= 1012.5𝐿36

𝑅

10𝐺𝑀

2 𝑒𝑉

𝑐𝑚3

• X-ray is comparable or smaller than IR, then the energy density of magnetic field should be similar

𝑈𝐵 =𝐵2

8𝜋= 1012.5

𝐵

10 𝐺

2 𝑒𝑉

𝑐𝑚3

Value of B-field? (~10 G)

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TeV and radio emissions• TeV and radio emissions cannot

be straightforwardly interpreted and are probably due to ifferent populations of electron and or radiation processes

• They are not variable, therefore they could be roduced at larger distances

• Very active research of Sgr A* and its surroundings

• For instance echo of past activity on molecular clouds

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Wind accretion• Early-type stars have a strong

wind, powered by absorption of lines in the UV. Mass loss rate is 10−10 − 10−6𝑀𝑠𝑢𝑛/𝑦𝑟

• if a compact object is embedded, it can accrete. Bondi-Hoyle accretion

• Wind speeds is 𝑣𝑤𝑖𝑛𝑑 𝑟 =

𝑣∞ 1 −𝑅∗

𝑟

−𝛽with terminal

velocity of ~1000 km/s• A strong tidal tail is formed, but

when accretion s possible?

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Manousakis et al. (2014)

Wind accretion• In stedy state, accretion radius is

𝑟𝑎𝑐𝑐 =2𝐺𝑀

𝑐𝑠2

• With a mass accretion rateሶ𝑀 = 𝜋𝑟𝑎𝑐𝑐

2 𝜌𝑚𝑐𝑠• In a binary system, the medium

velocity is not the sound speed, but the wind speed plus the orbital motion

• 𝑣𝑂𝑟𝑏 = 2𝐺𝑀/𝑎 with a the binary separation

• We must substitute to 𝑐𝑠2 the

value 𝑣𝑤𝑖𝑛𝑑2 + 𝑣𝑜𝑟𝑏

2 ~𝐺𝑀

𝑎+

𝑣𝑤𝑖𝑛𝑑,∞2

• Considering the density 𝜌𝑤𝑖𝑛𝑑 =ሶ𝑀𝑤𝑖𝑛𝑑

4𝜋𝑎2𝑣𝑤𝑖𝑛𝑑2 , we can derive

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Beyond a simple theory

• Acrretion from close binaries: Vela X-1, Cyg X-1

• We need to consider several other aspects, such as the photionization of the wind, which decellerates wind

• Possible to form a small and possibly intermittent accreiton disk.

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Useful parametrization• We use a frame rotating with

the system and account for Coriol force to obtain the effective potential

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• The solution of this equation is not straightforward, but we obtain the potentials of the previous figure. A good parametrization (𝑞 =

𝑀2

𝑀1)

Exercises• Derive the free-fall time scale.• Derive the energy gained by a free falling

proton on a body of 1 Solar mass and radius R. Compute it it for the Schwartschild radius.

• Compute the temperature of a fluid of protons fallen onto the surface of a star for Solar mass, Solar radius, Earth radius, neutron star radius (10 km), and Scharschild radius.

• Get an estimate of the accretion radius for a one solar mass star in the ISM (T~10000 K).

• Estimate the mass accretion rate and accretion luminosity for one solar mass with the above radii

• Compute the temperature of a black-body emitting all the gravitational energy as a black-body for the previous cases.

• Estimate the sonic point radius and speed as function of the value at infinity (use Bernoulli equation). Can we use the adiabatic approximation ? (gamma=5/3)

• Get the temperature radial dependency at any radius from a central object for a mass accretion rate ሶ𝑚 . Is temperature higher far or close to the central object? Waht is the luminosity?

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Roche Lobe overflow

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• Matter can inflow from inner lagrangian point

• It posseses a sizeable angular momentum

Formation of an accretion disk• When streaming from the inner

Lagrangian point, particles have an angular momentum.

• Gas speed is dominated by the orbital motion (~1000 km/s) while the sound speed is ~10 km/s

• The stream will intersect itself in different ellipses and create shocks. It will settle at the first stable orbit which conserves initial angular momentum, the circularization radius, obtained from equating the angular momentum at the Lagrangian point to a circular orbit

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𝑀 = 𝑀1 +𝑀2, 4𝜋𝑎3 = 𝐺𝑀𝑃2

Thin accretion disk

• H<<R, radiation pressure negligeable, strictly valid only for 𝐿 ≪ 𝐿𝐸𝑑𝑑

• Gas pressure must support disk vertically against gravitation

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Thin disk• Remember that sound

speed is 𝑐𝑠2 ≈

𝑃

𝜌

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• Is the flow supersonic or subsonic?

• 𝑣 = 1.2 × 1010𝑀/

𝑀𝑠𝑜𝑙 𝑅6cm/s

• Like in Bondi accretion, but matter remains very long in orbit, so efficient emission !!

Thin disk/2

• Radial acceleration due to pressure, negligeable as compared to gravitation

• Radial speed is very low, almost Keplerian motion

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• ሶ𝑀 = −2𝜋 𝜎 𝑣𝑟 with sigma the surface density

• Vertical density scales exponentially due to assumption that the pressure balances gravity

• 𝐹𝑧 ∝𝐺𝑀

𝑅2𝑧

𝑅∝ 𝑧

• 𝑛 𝑧 ∝ exp(−𝑧

𝐻)

Thin disks/3• Since angular speed is

𝑟 =𝐺𝑀

𝑅3

1

2, there is

differential motion and thus, possibly shear visccosity.

• Angular momentun decreases with decreasing radius (write equation)

• Accretion disks are pumps of angular momentum

• Force due to viscosity is

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Torque

• This torque need to balance the loss of angular momentum

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Temperature profile• We need to equate the

dissipation torque to thermal emission at each radius

• 2𝜎𝑆𝐵𝑇4 = 𝐷(𝑅)

• Why the 2?

• 𝑇 𝑅 ∝ 𝑅−3

4

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Accretion disk spectrum• Disk emits as a suprposion of

black bodies• In reality other factors also

need to included, such as metallicity, viscous parameter, preseence of an atmosphere, properties of the compact object etc.

• Pivot temperature is the inner disk temperature

𝑇~5 𝑀9−1

4eV

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• Spectrum of AGN in the UV, of X-ray binaries in X-rays

Dissipation• Disk dissipate

half of potentl energy of the accretion flow.

• From virial theorem

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