Relativistic jets
Accretion energy sets limits on the mass of the central SMBH, by
connecting the luminosity L to the gravitational field.
•
αβγδεζηθκλμνξοπρςστυφχψω
ΓΔΘΛΞΡΣΦΩ
ṁ
→≃∾∝∞≤≥⊙●≾≿∅∑±∓∊∈∄∀≫≪≡∗Δ∂∫∮∇Å
ÅÅ
Eddington accretion limits
Accretion energy sets limits on the mass of the central SMBH, by
connecting the luminosity L to the gravitational field.
Hypotheses:
a. L is mostly produced near the SMBH → the QSO's is a pointlike, isotropic source;
Eddington accretion limits
Accretion energy sets limits on the mass of the central SMBH, by
connecting the luminosity L to the gravitational field.
Hypotheses:
a. L is mostly produced near the SMBH → the QSO's is a pointlike, isotropic source;
b. The accretion region around the QSO is fully ionised.
•
Eddington accretion limits
Accretion energy sets limits on the mass of the central SMBH, by
connecting the luminosity L to the gravitational field.
Hypotheses:
a. L is mostly produced near the SMBH → the QSO's is a pointlike, isotropic source;
b. The accretion region around the QSO is fully ionised.
Then the radiation force at distance r from the QSO is:
•
Eddington accretion limits
Accretion energy sets limits on the mass of the central SMBH, by
connecting the luminosity L to the gravitational field.
Hypotheses:
a. L is mostly produced near the SMBH → the QSO's is a pointlike, isotropic source;
b. The accretion region around the QSO is fully ionised.
Then the radiation force at distance r from the QSO is:
•
a.
4
2 4
where σT = (8π/3)(e /me c ) is the Thomson scattering cross section, σT
-25
2
= 6.652458∗10 cm .
Eddington accretion limits
Accretion energy sets limits on the mass of the central SMBH, by
connecting the luminosity L to the gravitational field.
Hypotheses:
a. L is mostly produced near the SMBH → the QSO's is a pointlike, isotropic source;
b. The accretion region around the QSO is fully ionised.
Then the radiation force at distance r from the QSO is:
•
a. p
4
2 4
where σT = (8π/3)(e /me c ) is the Thomson scattering cross section, σT
-25
2
= 6.652458∗10 cm .
NOTE: this expression is obtained under 2 further hypotheses:
2
1.The energy if the scattered photons Eγ ≪ mec ;
Eddington accretion limits
Accretion energy sets limits on the mass of the central SMBH, by
connecting the luminosity L to the gravitational field.
Hypotheses:
a. L is mostly produced near the SMBH → the QSO's is a pointlike, isotropic source;
b. The accretion region around the QSO is fully ionised.
Then the radiation force at distance r from the QSO is:
•
a.
4
2 4
where σT = (8π/3)(e /me c ) is the Thomson scattering cross section, σT
-25
2
= 6.652458∗10 cm .
NOTE: this expression is obtained under 2 further hyotheses:
2
1.The energy if the scattered photons Eγ ≪ mec ;
2.Electrons are at rest in the laboratory rest frame
Eddington accretion limits (contd.)
Intuitive derivation:
Eddington accretion limits (contd.)
Intuitive derivation:
[L] = E ∗T
a.
-1
Eddington accretion limits (contd.)
Intuitive derivation:
-1
2
[L] = E ∗T → S = L/(4πr c) is the global momentum per unit area
2 -2
-1
-1
-2
carried by the photons ([L/c] = (ML T ∗T )∗(L/T) = MLT ≡
[MV/T] thus it is a momentum).
a.
Eddington accretion limits (contd.)
Intuitive derivation:
-1
2
[L] = E ∗T → S = L/(4πr c) is the global momentum per unit area
2 -2
-1
-1
-2
carried by the photons ([L/c] = (ML T ∗T )∗(L/T) = MLT ≡
[MV/T] thus it is a momentum). By multiplying this by the electron
cross section we get the force acting on the electrons, i.e. the
momentum transferred per unit time.
a.
Eddington accretion limits (contd.)
Intuitive derivation:
-1
2
[L] = E ∗T → S = L/(4πr c) is the global momentum per unit area
2 -2
-1
-1
-2
carried by the photons ([L/c] = (ML T ∗T )∗(L/T) = MLT ≡
[MV/T] thus it is a momentum). By multiplying this by the electron
cross section we get the force acting on the electrons, i.e. the
momentum transferred per unit time.
-2
NOTE: Frad ∝ r
Eddington accretion limits (contd.)
Intuitive derivation:
-1
2
[L] = E ∗T → S = L/(4πr c) is the global momentum per unit area
2 -2
-1
-1
-2
carried by the photons ([L/c] = (ML T ∗T )∗(L/T) = MLT ≡
[MV/T] thus it is a momentum). By multiplying this by the electron
cross section we get the force acting on the electrons, i.e. the
momentum transferred per unit time.
-2
NOTE: Frad ∝ r
a.
The radiation force is contrasted by the gravitational force:
Eddington accretion limits (contd.)
Intuitive derivation:
-1
2
[L] = E ∗T → S = L/(4πr c) is the global momentum per unit area
2 -2
-1
-1
-2
carried by the photons ([L/c] = (ML T ∗T )∗(L/T) = MLT ≡
[MV/T] thus it is a momentum). By multiplying this by the electron
cross section we get the force acting on the electrons, i.e. the
momentum transferred per unit time.
-2
NOTE: Frad ∝ r
a.
The radiation force is contrasted by the gravitational force:
a.
where M● is the SMBH mass and we have considered only protons
because me ≪ mp (in a fully ionised plasma protons and electrons are
bound together by Compton scattering, see Spitzer 1962)
Eddington accretion limits (contd.)
The condition that gravity dominates ver radiation pressure sets an
upper limit on the luminosity L:
Eddington accretion limits (contd.)
The condition that gravity dominates ver radiation pressure sets an
upper limit on the luminosity L:
a.
which translates into the condition:
Eddington accretion limits (contd.)
The condition that gravity dominates ver radiation pressure sets an
upper limit on the luminosity L:
a.
which translates into the condition:
a.
and in the Eddington luminosity:
Eddington accretion limits (contd.)
The condition that gravity dominates ver radiation pressure sets an
upper limit on the luminosity L:
a.
which translates into the condition:
a.
and in the Eddington luminosity:
1. LEdd is independent of r because both radiation force and gravity
-2
scale with the same power (r );
Eddington accretion limits (contd.)
The condition that gravity dominates ver radiation pressure sets an
upper limit on the luminosity L:
a.
which translates into the condition:
a.
and in the Eddington luminosity:
1. LEdd is independent of r because both radiation force and gravity
-2
scale with the same power (r );
2.Being σT independent of radiation frequency, LEdd is a bolometric
luminosity.
Eddington accretion limits (contd.)
To estimate the maximum BH mass we impose the condition that
accretion is only possible if radiation pressure does not exceed in any
place gravity.
Eddington accretion limits (contd.)
To estimate the maximum BH mass we impose the condition that
accretion is only possible if radiation pressure does not exceed in any
place gravity. Thus the necessary condition to have a QSO is that its
luminosity is always sub-Eddington: L > LEdd, which translates into:
Eddington accretion limits (contd.)
To estimate the maximum BH mass we impose the condition that
accretion is only possible if radiation pressure does not exceed in any
place gravity. Thus the necessary condition to have a QSO is that its
luminosity is always sub-Eddington: L > LEdd, which translates into:
The lower limit is reached by the nucley of Seyfert I (type I AGN), while
8
for bright, high-z QSOs we have M● ≥ 10 M⊙.
Eddington accretion limits (contd.)
To estimate the maximum BH mass we impose the condition that
accretion is only possible if radiation pressure does not exceed in any
place gravity. Thus the necessary condition to have a QSO is that its
luminosity is always sub-Eddington: L > LEdd, which translates into:
The lower limit is reached by the nucley of Seyfert I (type I AGN), while
8
for bright, high-z QSOs we have M● ≥ 10 M⊙.
Eddington accretion rate
Eddington accretion limits (contd.)
To estimate the maximum BH mass we impose the condition that
accretion is only possible if radiation pressure does not exceed in any
place gravity. Thus the necessary condition to have a QSO is that its
luminosity is always sub-Eddington: L > LEdd, which translates into:
The lower limit is reached by the nucley of Seyfert I (type I AGN), while
8
for bright, high-z QSOs we have M● ≥ 10 M⊙.
Eddington accretion rate
For a given efficiency ε of rest mass to radiation conversion the
accretion rate is given by:
Eddington accretion rate
But we have seen that a necessary condtion for accretion is LEdd > L
thus, after having substituted in the equation for ṁ, we get an Eddington
accretion rate and we can express the accretion rate in units of it:
The "Soltan argument"
The "Soltan argument"
A simple, global, quantitative argument put forward by Soltan (1982)
allows one to set lower limits to the local density of SMBHs in our
present (local) Universe, and to prove that accretion can be the engine
of QSO and AGNs' activity.
The "Soltan argument"
A simple, global, quantitative argument put forward by Soltan (1982)
allows one to set lower limits to the local density of SMBHs in our
present (local) Universe, and to prove that accretion can be the engine
of QSO and AGNs' activity.
Assume that all radiation from QSOs originates from accretion. Then
the luminosity of a given QSO is given by
The "Soltan argument"
A simple, global, quantitative argument put forward by Soltan (1982)
allows one to set lower limits to the local density of SMBHs in our
present (local) Universe, and to prove that accretion can be the engine
of QSO and AGNs' activity.
Assume that all radiation from QSOs originates from accretion. Then
the luminosity of a given QSO is given by
The actual accreted mass-energy by a QSO of luminosity Lbol will then
be given by: Macc = (1-ε)Lbol,
The "Soltan argument"
A simple, global, quantitative argument put forward by Soltan (1982)
allows one to set lower limits to the local density of SMBHs in our
present (local) Universe, and to prove that accretion can be the engine
of QSO and AGNs' activity.
Assume that all radiation from QSOs originates from accretion. Then
the luminosity of a given QSO is given by
The actual accreted mass-energy by a QSO of luminosity Lbol will then
be given by: Macc = (1-ε)Lbol, and the total mass-energy density at any
given redshift z will be obtained by convolving with the QSOs
luminosity function Φ(L,z)
The "Soltan argument"
A simple, global, quantitative argument put forward by Soltan (1982)
allows one to set lower limits to the local density of SMBHs in our
present (local) Universe, and to prove that accretion can be the engine
of QSO and AGNs' activity.
Assume that all radiation from QSOs originates from accretion. Then
the luminosity of a given QSO is given by
The actual accreted mass-energy by a QSO of luminosity Lbol will then
be given by: Macc = (1-ε)Lbol, and the total mass-energy density at any
given redshift z will be obtained by convolving with the QSOs
luminosity function Φ(L,z):
The "Soltan argument"
The average SMBH density at z=0 will be given by integrating over all
epochs:
The "Soltan argument"
The average SMBH density at z=0 will be given by integrating over all
epochs:
The "Soltan argument"
The average SMBH density at z=0 will be given by integrating over all
epochs:
The calculation is not easy, because
Φ(L,z) evolves with z
The "Soltan argument"
The average SMBH density at z=0 will be given by integrating over all
epochs:
The calculation is not easy, because
Φ(L,z) evolves with z
The final value is: ρBH,0 =
5
-3
2.1∗10 [0.1(1-ε)/ε] M⊙ Mpc
The "Soltan argument"
The average SMBH density at z=0 will be given by integrating over all
epochs:
The calculation is not easy, because
Φ(L,z) evolves with z
The final value is: ρBH,0 =
5
-3
2.1∗10 [0.1(1-ε)/ε] M⊙ Mpc
Now, the local SMBH density
estimated from direct measurements
5
(M●-σ relation) is: ρM-σ = 2.5∗10
-3
M⊙ Mpc
Structure of accretion disc
We can now derive the radial structure of an optically thick, geometrically
thin (height ≪ radius) accretion disc.
Structure of accretion disc
We can now derive the radial structure of an optically thick, geometrically
thin (height ≪ radius) accretion disc.
We have previously seen the expression for the emissivity of a ring of
width Δr from a thin accretion disc:
Structure of accretion disc
We can now derive the radial structure of an optically thick, geometrically
thin (height ≪ radius) accretion disc.
We have previously seen the expression for the emissivity of a ring of
width Δr from a thin accretion disc:
If this ring is optically thick it will re-emit as a blackbody of (to be
determined) temperature T(r):
Structure of accretion disc
We can now derive the radial structure of an optically thick, geometrically
thin (height ≪ radius) accretion disc.
We have previously seen the expression for the emissivity of a ring of
width Δr from a thin accretion disc:
If this ring is optically thick it will re-emit as a blackbody of (to be
determined) temperature T(r):
-5
-2
-4
where σSB = 5.67∗10 erg cm K is the Stefan-Boltzmann constant.
Structure of accretion disc
We can now derive the radial structure of an optically thick, geometrically
thin (height ≪ radius) accretion disc.
We have previously seen the expression for the emissivity of a ring of
width Δr from a thin accretion disc:
If this ring is optically thick it will re-emit as a blackbody of (to be
determined) temperature T(r):
-5
-2
-4
where σSB = 5.67∗10 erg cm K is the Stefan-Boltzmann constant.
Equating these two expressions we get an expression for T(r):
Structure of accretion disc (contd.)
A more lenghty derivation would start from the energy conservation
equation and includes viscous dissipation (a loss term) and thermal
heating (a gain term).
Structure of accretion disc (contd.)
A more lenghty derivation would start from the energy conservation
equation and includes viscous dissipation (a loss term) and thermal
heating (a gain term). The final (correct) result differs only by a factor of
order unity from the previous approximation:
Structure of accretion disc (contd.)
A more lenghty derivation would start from the energy conservation
equation and includes viscous dissipation (a loss term) and thermal
heating (a gain term). The final (correct) result differs only by a factor of
order unity from the previous approximation:
Correct
Structure of accretion disc (contd.)
A more lenghty derivation would start from the energy conservation
equation and includes viscous dissipation (a loss term) and thermal
heating (a gain term). The final (correct) result differs only by a factor of
order unity from the previous approximation:
Previous
Correct
Structure of accretion disc (contd.)
A more lenghty derivation would start from the energy conservation
equation and includes viscous dissipation (a loss term) and thermal
heating (a gain term). The final (correct) result differs only by a factor of
order unity from the previous approximation:
Previous
Correct
It is customary to express accretion disc profiles in units of the
Schwarzschild radius (approx. size of the BHs' event horizon):
Structure of accretion disc (contd.)
A more lenghty derivation would start from the energy conservation
equation and includes viscous dissipation (a loss term) and thermal
heating (a gain term). The final (correct) result differs only by a factor of
order unity from the previous approximation:
Previous
Correct
It is customary to express accretion disc profiles in units of the
Schwarzschild radius (approx. size of the BHs' event horizon):
Thus we finally obtain:
Structure of accretion disc (contd.)
A more lenghty derivation would start from the energy conservation
equation and includes viscous dissipation (a loss term) and thermal
heating (a gain term). The final (correct) result differs only by a factor of
order unity from the previous approximation:
Previous
Correct
It is customary to express accretion disc profiles in units of the
Schwarzschild radius (approx. size of the BHs' event horizon):
Thus we finally obtain:
Structure of accretion disc (contd.)
A few consequences:
Structure of accretion disc (contd.)
A few consequences:
-3/4
• T ∝ r at fixed M● and accretion rate ṁ
Structure of accretion disc (contd.)
A few consequences:
-3/4
• T ∝ r at fixed M● and accretion rate ṁ
1/4
4
• T ∝ ṁ because the total radiated energy ∝ T and the total
emission ∝ ṁ, thus one expects this scaling
Structure of accretion disc (contd.)
A few consequences:
-3/4
• T ∝ r at fixed M● and accretion rate ṁ
1/4
4
• T ∝ ṁ because the total radiated energy ∝ T and the total
emission ∝ ṁ, thus one expects this scaling
-1/2
• T ∝ M● ! Colder discs are associated to more massive BHs !!
Thus we expect soft X-ray emission from SMBH and very hot discs
from stellar-size BHs (→ X-ray galactic binaries)
References
• Spitzer, L., Physics of Fully Ionized Gases, 1962
•
• 't Hooft, G., introduction to the Theory of Back Holes
(http://www.staff.science.uu.nl/~hooft101/lectures/blackholes/
BH_lecturenotes.pdf)
• Soltan, A., MNRAS 200, 115 (1982)
αβγδεζηθκλμνξοπρςστυφχψω
ΓΔΘΛΞΡΣΦΩ
ṁ
→≃∾∝∞≤≥⊙●≾≿∅∑±∓∊∈∄∀≫≪≡∗Δ∂∫∮∇Å
ÅÅ
References
• Black holes before Einstein:
http://www.narit.or.th/en/files/2009JAHHvol12/2009JAHH...
12...90M.pdf
•
• 't Hooft, G., introduction to the Theory of Back Holes
(http://www.staff.science.uu.nl/~hooft101/lectures/blackholes/
BH_lecturenotes.pdf)
• Soltan, A., MNRAS 200, 115 (1982)
Black Holes and AGNs
How much energy can be extracted during accretion? Eddington
luminosity
• Zel’dovich & Novikov (1964), E. Salpeter (1964), D. LyndenBell (1969) and M. Rees (1984): SMBHs at the centers of
galaxies create thin accretion disks from which gravitational
energy is extracted and converted into radiaton (QSOs) and jets
(AGNs, radiogalaxies)
Schwarzschild Black Hole (ST, chap 5)
Einstein's equations:
where the Einstein Tensor is given by:
finds:
where:
and
αβ
T is
Here
αβ
G
=
αβ
R
-
αβ
(1/2)Rg
and one
αβ
g
and
are the coefficients of the
metrics
the stress-energy tensor, specifying the matter properties.
αβ
u is
the matter's 4-velocity, subject to the normalization:
E.g. in the fluid rest frame one has:
General Relativity (cont.)
Note: ∇βG = 0, because ∇βT = 0 (matter conservation laws → Euler
equation, continuity equation)
αβ
•
αβ
αβ
G :
αβ
g
only depends on metrics
→ space-time geometry
(intrinsic curvature of ST)
αβ
αβ
αβ
• T : depends on ρ, p(ρ,T), u and also on g → matter and
geometry
•
•
Geometrized units
Adopt units of length, time and mass (L, T, M) s.t.: G = c = 1.
• 3 units and 2 constraints → 1 unit can be freely chosen
c = 1 → L = T, i.e. length measured in units of time.
6
6
7
13
E.g.: T = 10 yr ≃ 10 ∗ (π∗10 ) sec →L ≃ c* π∗10 cm ≃ 0.31 Mpc.
3 -2 -1
3 -2
44
[G] = L T M , G = 1→ M = L T . In our case: M ≃ 8.48∗10 g ≃
11
4.26∗10 M⊙ , i.e. the mass of a typical Milky Way galaxy
Schwarzschild solution
In GR gravity arises from the postulate that space-time has an intrinsic
geometry
Gravity is then not a force, but a property (intrinsic curvature) of
space-time.
λ
(x )
The metric gλμ
is a solution of the Einstein's equations for any
λ
coordinate system in space-time. {x }, λ=0,1,2,3 denotes the position
in space-time of a generic event for a particular coordinate system.
The Einstein (strong) equivalence principle states that all physical
law have the same mathematical form in any inertial coordinate frame
→ physical laws are covariant.
λ
x (τ)
Inertial coordinate frame: Associated to space-time trajectories
of those observers in free fall in the space-time, i.e. whose motion only
depends on the local space-time Riemannian curvature Rαβγδ.
Schwarzschild solution (cont.)
λ
{x }(τ)
Inertial frames define a unique orthonormal coordinate system
whose axis define a tetrad {et}, i.e. 4 unit tangent vectors defined by
the orthonormality requirements:
The coordinate displacements are then the 4-vectors:
In spherical coordinates these become:
so that the orthogonality requirement above results into:
because from the orthonormality condition and fom the definition of
metric: