1. Alignment of Magnetized Accretion Disks and Relativistic Jets with
Spinning Black Holes
Jonathan C. McKinney et al.
Science 339, 49 (2013);
DOI: 10.1126/science.1230811
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2. REPORTS
viscosity aligns a very thin disk out to some dis-
Alignment of Magnetized Accretion tance [estimated to be out to r ~ 10rg to 105 rg,
where rg is a gravitational radius (19), depending
Disks and Relativistic Jets with on assumptions] from the BH. Whereas the vis-
cosity has been thought to result from turbulence
Spinning Black Holes driven by the magneto-rotational instability (MRI)
that amplifies weak small-scale (<~H, the disk
height) magnetic fields (20), magnetohydrody-
Jonathan C. McKinney,1,2* Alexander Tchekhovskoy,3 Roger D. Blandford1 namical (MHD) simulations of weakly magnet-
ized disks have not yet produced any BP alignment
Accreting black holes (BHs) produce intense radiation and powerful relativistic jets, which are effect (21). The BP effect and LT precession re-
affected by the BH’s spin magnitude and direction. Although thin disks might align with the main commonly invoked mechanisms to under-
BH spin axis via the Bardeen-Petterson effect, this does not apply to jet systems with thick stand how tilt affects the evolution of BH mass
disks. We used fully three-dimensional general relativistic magnetohydrodynamical simulations and spin (22, 23), how merging BHs are affected
to study accreting BHs with various spin vectors and disk thicknesses and with magnetic flux by any nearby plasma (24), and how disks and
reaching saturation. Our simulations reveal a “magneto-spin alignment” mechanism that causes their jets are oriented (25, 26).
magnetized disks and jets to align with the BH spin near BHs and to reorient with the outer Large-scale electromagnetic (EM) fields might
disk farther away. This mechanism has implications for the evolution of BH mass and spin, BH also affect the jet’s and disk’s orientations via ex-
ternal confinement forces (27, 28). Estimates based
Downloaded from www.sciencemag.org on January 4, 2013
feedback on host galaxies, and resolved BH images for the accreting BHs in SgrA* and M87.
on the presumption that large-scale magnetic fields
strophysical black holes (BHs) operate the angle between the BH spin axis and the disk’s are weaker than turbulent disk fields suggested
A as engines that convert the gravitational
binding energy of accreting plasmas into
intense radiation (1) and release BH spin energy
angular momentum axis at large distances, influ-
ences the intensity of the radiation via changes in
the gravitational potential felt by the plasma; it
that EM forces are insufficient to align the disk
with the BH (27) or the BH with the disk (29, 30).
Simulations without disks have given ambiguous
(2–4) into powerful relativistic jets (5, 6). Rel- also influences what an observer at different results for the EM jet direction. For a uniform ver-
ativistic jets from accreting BHs are commonly viewing angles sees as a result of disk warping tical magnetic field and no disk, the jet is directed
observed to emerge from active galactic nuclei and jet bending. along the magnetic field direction rather than along
(AGN) or quasars, x-ray binaries that behave as One mechanism known to possibly affect the the BH’s tilted spin axis (31), whereas isolated mag-
microquasars, and gamma-ray burst (GRB) orientation of a disk or jet is the Bardeen-Petterson netic threads tend to align with the BH spin axis
events. GRB jets allow one to probe the earliest (BP) effect (15–18), where Lense-Thirring (LT) when there is no disk to restrict their motion (32).
epochs of star formation, whereas radiation and forces induced by the BH frame-dragging cause a We have used fully three-dimensional (3D)
jets from AGN play a direct dynamical role via misaligned disk to precess and warp until a local general relativistic (GR) MHD simulations (33)
feedback that suppresses star formation in their
host galaxies (7).
BHs are also intrinsically interesting because Table 1. Tilted black hole disk-jet systems. The simulation models are listed by model name, which
they act as laboratories for probing Einstein’s identifies the approximate BH spin of j (values following “A”), something about the magnetic field choices
general relativity theory and for testing theories (values following “B” and “N”) (28), and the initial relative tilt (qtilt,0 in radians) between the BH spin axis
about accreting BHs and jets. Astrophysical BHs and the disk rotation axis (values following “T”; if T is absent, qtilt,0 = 0). Successive columns give the
are characterized primarily by their mass (M) and dimensionless BH spin ( j); the evolved quasi–steady-state value of the disk height/radius ratio H/R
dimensionless spin angular momentum ( j). BHs between r ~ 20rg and r ~ 30rg; the initial relative tilt (qtilt,0) between the disk + jet (having the same tilt
have been measured to have masses of tens to initially) and the BH spin axis; the evolved relative tilt between the BH spin axis and the disk and jet,
respectively, at r = 4rg; and the same measurements at r = 30rg. If the relative tilt is 0.0, the disk or jet
billions of solar masses M◉; the mass of the BH
remained aligned with the BH spin axis; if the tilt is equal to the initial tilt, the disk or jet was unaffected.
in M87 is ~6 × 109 M◉ (8). Spins have also been
measured and span over many of the possible Disk tilt Jet tilt Disk tilt Jet tilt
values, including near the maximal value of j ~ 1 Model name BH j Disk H/R Initial tilt
r = 4rg r = 4rg r = 30rg r = 30rg
in the BH x-ray binary GRS1915+105 with
A0.94BfN40 0.9375 0.6 0 0 0 0 0
M ~ 14 M◉ and in the AGN MCG-6-30-15 with
A0.94BfN40T0.35 0.9375 0.6 0.35 0.0 0.0 0.2 0.2
M ~ 3 × 106 M◉ (9, 10). Structures on a few BH
A0.94BfN40T0.7 0.9375 0.6 0.70 0.0 0.0 0.4 0.3
event horizon length scales have recently been
A0.94BfN40T1.5708 0.9375 0.6 1.5708 0.0 0.1 0.5 0.7
resolved by Earth-sized radio telescope interfer-
A-0.9N100 –0.9 0.3 0 0 0 0 0
ometry for SgrA* (11, 12) and M87 (13, 14).
A-0.9N100T0.15 –0.9 0.3 0.15 0.1 0.1 0.1 0.2
Although the BH’s present angular momen-
A-0.9N100T0.3 –0.9 0.3 0.30 0.2 0.2 0.2 0.2
tum axis is set by the history of plasma accretion
A-0.9N100T0.6 –0.9 0.3 0.60 0.2 0.3 0.4 0.3
and mergers with other BHs, the gas being cur-
: A-0.9N100T1.5708 –0.9 0.3 1.5708 0.2 0.4 0.9 0.8
rently supplied (at mass accretion rate M ) to the
A0.9N100 0.9 0.3 0 0 0 0 0
BH can have an arbitrarily different angular mo-
A0.9N100T0.15 0.9 0.3 0.15 0.0 0.0 0.1 0.1
mentum axis. This relative tilt, given by qtilt for
A0.9N100T0.3 0.9 0.3 0.30 0.1 0.1 0.2 0.2
A0.9N100T0.6 0.9 0.3 0.60 0.1 0.1 0.3 0.3
1
Kavli Institute for Particle Astrophysics and Cosmology, Stanford A0.9N100T1.5708 0.9 0.3 1.5708 0.2 0.3 0.7 0.6
University, Stanford, CA 94309, USA. 2Department of Physics A0.9N100 0.99 0.3 0 0 0 0 0
and Joint Space-Science Institute, University of Maryland, Col- A0.99N100T0.15 0.99 0.3 0.15 0.0 0.1 0.1 0.1
lege Park, MD 20742, USA. 3Center for Theoretical Science, A0.99N100T0.3 0.99 0.3 0.30 0.1 0.1 0.2 0.2
Princeton University, Princeton, NJ 08544, USA.
A0.99N100T0.6 0.99 0.3 0.60 0.1 0.1 0.3 0.4
*To whom correspondence should be addressed. E-mail:
A0.99N100T1.5708 0.99 0.3 1.5708 0.1 0.1 0.6 0.6
jcm@umd.edu
www.sciencemag.org SCIENCE VOL 339 4 JANUARY 2013 49

3. REPORTS
of accreting BHs to show that near the BH, both Over the horizon and in the jet, U e 10 for Our self-consistent fully 3D general rela-
the disk angular momentum and jet direction our thinner disk models and U e 17 for our tivistic magnetohydrodynamic (GRMHD) simu-
reorient and align with the BH’s spin axis. Our thick disk models (28). Also, for both thicknesses, lations started with a disk around an untilted BH
simulations were designed so that the magnetic (r/rg)aeff ~ 15 and roughly constant with radius, where the BH spin axis, disk rotational axis, and
field built up to a natural saturation strength with, and vr /vf ~ 1 near the horizon (28). So, at all dis- emergent jet’s direction all pointed in the vertical
roughly, the disk’s thermal + ram + gravitational tances, the jet’s EM forces lead to tEM/tLT ~ 2 (z) direction. As the simulation proceeded, the
forces balancing the disk’s and jet’s magnetic for our thinner disk models and tEM/tLT > ~5 for mass and magnetic flux readily advected from
forces such that the trapped large-scale magnetic our thick disk models. Hence, we expect EM forces large distances onto the BH. The magnetic flux
field threading the BH and disk became strong to dominate LT forces for both our thinner and versus radius saturated on the BH and within the
relative to the disk’s turbulent field. The saturated thick disk models [including for small spins (33)]. disk near the BH after magnetic forces balanced
field strength has been shown to be (i) indepen- EM alignment forces are effective when they the disk’s thermal + ram + gravitational forces.
dent of the strength of the initial magnetic field are larger than forces associated with the newly ac- Magnetic braking causes such disks to become
when the surrounding medium has a sufficient creted rotating plasma with torque per unit area of
: even more sub-Keplerian than weakly magnetized
supply of magnetic flux, (ii) weakly dependent on tacc e Mvf =ð2prÞ. Therefore, tEM =tacc e U2 WF = thick disks (28), which means that the classical
BH spin, and (iii) proportionally dependent on disk ð4rvf Þ, and when these torques are equal one thin disk innermost stable circular orbit position
thickness (4, 28, 34). We considered various BH obtains an implicit equation for a “magneto-spin is even less applicable than for weakly magne-
spins (35), BH tilts, and disks with a quasi– alignment” radius of tized thick disks. The simulations were evolved
steady-state height/radius ratio (H/R) of ~0.6 WF r2 U2
g
for a long time period so that the disk reached a
rmsa e ð5Þ
Downloaded from www.sciencemag.org on January 4, 2013
for thick disks and ~0.3 for thinner disks. Nu- quasi-stationary magnetically saturated state out
4vf
merical convergence of our results was determined to about r ~ 40rg (28, 33).
on the basis of convergence quality measures for (with rg reintroduced), within which EM forces Then, the BH spin axis was instantly tilted by
how well the MRI and turbulence were resolved can torque the accreting dense material. For suf- an angle of qtilt,0 (see Table 1 for tilts used for
(table S2) as well as by explicit convergence test- ficiently small values of U or j, no alignment can different spins and disk thicknesses). The tilted
ing (33). occur. We obtain rmsa > ~30rg for our thinner and disk-jet system underwent a violent rearrangement
Let us motivate these MHD simulations by thick disk models that are sub-Keplerian by a for the larger tilts. The frame-dragging forces
estimating whether EM forces are expected to factor of 0.5 to 0.1, respectively (28), although caused the nearly split-monopole BH magneto-
dominate LT forces on the rotating heavy disk. accurate estimates require performing more sim- sphere to align with the BH spin axis, as expected
Imagine a toy model with a flat heavy disk tilted ulations or accounting for more physics that could because the misaligned angular momentum was
and pushed up against the magnetized jet gen- lead to much different rmsa (33). radiated away as part of the electromagnetic
erated directly by the rotating BH. For a magnetic
field B bending on scale r, the EM torque per unit
area is tEM ~ rBrBf/4 (33) for a jet magnetic field
that has both radial (Br) and toroidal (Bf ~ rBr ΩF)
components and rotates with an angular frequen-
cy of ΩF [where rgΩF/c ~ j/8 for j ~ 1 (28)]. The
radial field is written in terms of a dimensionless
magnetic flux given by
0:7F
U ≈ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
: ð1Þ
4pr2 Mcg
for magnetic flux F ~ 4pr2Br for B in Gaussian
units (rg and c reintroduced for dimensional
clarity) that is consistent with measurements in
our previous work (28). This gives
r2 :
g
tEM e MWF U2 ð2Þ
8pr2
Meanwhile, the LT torque per unit area is tLT ~ ΩLTL
with LT precession rate ΩLT ~ 2j/r3, disk angular
momentum per unit area L ~ Srvf , and disk sur-
:
face density S e M=ð2prvr Þ. This gives
:
jcrg Mvf
2
tLT ð3Þ Fig. 1. 3D snapshot for an evolved model with j = 0.99, initial relative tilt qtilt,0 ≈ 90°, and disk thickness
e pr3 v
r H/R ~ 0.3. The rotating BH sits at the center of the box of size r = −40rg to r = +40rg in each dimension.
The snapshot shows the disk near the BH (yellow isosurface, which is mostly flat in the figure plane), the
The ratio of the EM to LT torques for j ~ 1 is then highly magnetized jet region (blue isosurface, with magnetic energy per unit rest-mass energy equal to
tEM =tLT e U2 rvr =ð64rg vf Þ, with rg reintroduced about 70), the rotational axis of the disk both initially and at large distances (orange cylinder), outer disk
for dimensional clarity. Far beyond the horizon, (green-yellow volume rendering, more aligned with disk rotational axis at large distances), magnetic field
vectors (like iron filings on that surface) for a cross section of the jet (cyan vectors), and jet magnetic field
tEM 1 2 raeff H 2 lines (white lines) that trace from the BH out to large distances. The disk and jet near the BH are aligned
tLT e 64 U r R
ð4Þ
with the BH spin axis and point mostly in and out of the figure plane, whereas at larger distances the jet
g
points roughly halfway between the BH spin axis and the disk’s rotational axis (pointing along the orange
for an effective viscosity aeff ≡ vr/[(H/R)2vf]. cylinder).
50 4 JANUARY 2013 VOL 339 SCIENCE www.sciencemag.org

4. REPORTS
outflow on Alfvén time scales. The magnetic torque S1), where more tilt led to reduced efficiency due to accretion at either very low (37) or very high
then caused the heavy disk to lose its misaligned to more spatially and temporally irregular mass rates when H/R ~0.5] are expected. For ex-
component of angular momentum and so reorient inflow. ample, for SgrA* and M87, if the BHs rotate
with the BH’s rotating magnetosphere. The time Our most extreme case of a tilted BH accre- sufficiently rapidly (33), then we expect the photon
scale for alignment seems to be roughly the tion disk and jet system is the j = 0.99 model with spectra, temporal behaviors, and resolved images
Alfvén crossing time near the heavy disk and a full tilt of qtilt,0 = 1.5708 ≈ 90° and disk thick- of their jets and disks to be affected by nonzero
BH. This “magneto-spin alignment” occurs be- ness H/R ~ 0.3. Even in this extremely tilted case, relative tilts as a result of disk warping and jet
cause the magnetic field built up to a natural the evolved disk and jet near the BH aligned with bending near the BH.
saturation strength on the horizon where U e 10 the BH spin axis (Fig. 1, fig. S1, and movie S1). Tidal disruption flare events such as Swift
(depending on the thickness and tilt), which led The jet’s magnetic field wound around the per- J164449.3+573451 are thought to be produced
to the BH magnetosphere’s forces dominating the sistent relativistic jet, and the magnetic field was by very high accretion rates onto BHs, which
disk dynamics and LT forces. well ordered even for this highly tilted case. The launch fairly persistent jets that dissipate and
All simulations (including with zero tilt) were jet was not symmetric around the jet axis, and give the observed emission (38). Our results sug-
then further evolved in time until all the tilted instead there was a broad wing (with opening gest that the inner disk and inner jet are both
simulations reached a new quasi-stationary state half-angle of about 25° by r = 40rg) and a narrow aligned with the BH spin axis, but the observed
out to r ~ 40rg. This ensured that any differences wing (with opening half-angle of about 5° by r = jet dissipating at large distances need not point
at later times in the disk and jet between tilted and 40rg). At large distances from the BH, the jet along the BH spin axis. EM forces do not direct-
untilted simulations were due to the BH tilt. This drilled its way through the disk material and grad- ly cause any precession (27), so the lack of LT
Downloaded from www.sciencemag.org on January 4, 2013
also ensured that each of the tilted and untilted ually got pushed away from the disk (Fig. 2 and precession-induced variability does not alone im-
simulations reached their own quasi–steady state movie S2). The magnetic field wound around with ply that the jet is necessarily driven by the BH
values for magnetic flux near the BH, magnetic a pitch angle of about 45° near the BH and a spin power (26). Further, quasi-periodic oscilla-
flux in the disk, mass accretion rate, etc. We then smaller pitch angle at larger distances. By r ~ tions (39) and long-term dips seen in this sys-
measured the evolved relative tilt between the 300rg, the jet had become parallel with (but offset tem’s light curve might be explained by oscillations
BH spin axis and the disk and jet at r = 4rg and from) the outer disk rotational axis, and so the jet in the disk-jet magnetospheric interface (28) or
r = 30rg (Table 1). For all our models, the disk and counterjet were also offset. by periods of magnetic flux accumulation and re-
and jet aligned with the BH spin axis near the Thus, our simulations have revealed a “magneto- jection by the BH (4) (both occurring for untilted
BH, whereas the disk axis and jet direction de- spin alignment” mechanism that aligns the disk systems) rather than by LT precession.
viated at larger distances. Such deviations are ex- and jet axes with the BH spin axis near the BH Jet dissipation and emission (e.g., in blazars)
pected because the jet interacted with circulation once the magnetic field has saturated on the BH might be due to the jet ramming into the disk
with stronger mass inflows at larger distances and within the disk (36). Unlike the BP effect, the until the jet aligns with the disk rotational axis at
(33). Despite the tilts and jet deviations, the BH’s mechanism actually works best for thick disks, large distances. Measurements of BH spin in
efficiency (defined as the ratio of energy out to and so the magneto-spin alignment mechanism AGN and x-ray binaries might be affected by
energy in) was roughly 100% for j ~0.9 (table should control jet systems where thick disks [due assumptions about the alignment between the
disk, BH spin, and jet (9, 10). The cosmological
evolution of BH mass and spin and AGN feed-
back for accretion at high rates might be affected
by the higher BH spin-down rates and jet effi-
ciencies relative to standard thin disk spin-down
rates and radiative efficiencies (28) and also by
how the jet aligns the disk material before LT
torques can be effective, thus possibly leading to
less change in the BH spin direction relative to
the BP effect.
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Negative Absolute Temperature for In Fig. 1A, we schematically show the rela-
tion between entropy S and energy E for a ther-
mal system possessing both lower and upper
Motional Degrees of Freedom energy bounds. Starting at minimum energy, where
only the ground state is populated, an increase in
energy leads to an occupation of a larger number
S. Braun,1,2 J. P. Ronzheimer,1,2 M. Schreiber,1,2 S. S. Hodgman,1,2 T. Rom,1,2
of states and therefore an increase in entropy. As
I. Bloch,1,2 U. Schneider1,2*
the temperature approaches infinity, all states be-
come equally populated and the entropy reaches
Absolute temperature is usually bound to be positive. Under special conditions, however, its maximum possible value Smax. However,
negative temperatures—in which high-energy states are more occupied than low-energy the energy can be increased even further if high-
states—are also possible. Such states have been demonstrated in localized systems with finite, energy states are more populated than low-energy
discrete spectra. Here, we prepared a negative temperature state for motional degrees of ones. In this regime, the entropy decreases with
freedom. By tailoring the Bose-Hubbard Hamiltonian, we created an attractively interacting energy, which, according to the thermodynamic
ensemble of ultracold bosons at negative temperature that is stable against collapse for definition of temperature (8) (1/T = ∂S/∂E), re-
arbitrary atom numbers. The quasimomentum distribution develops sharp peaks at the upper sults in negative temperatures. The temperature is
band edge, revealing thermal equilibrium and bosonic coherence over several lattice sites. discontinuous at maximum entropy, jumping from
Negative temperatures imply negative pressures and open up new parameter regimes for positive to negative infinity. This is a consequence
cold atoms, enabling fundamentally new many-body states. of the historic definition of temperature. A con-
tinuous and monotonically increasing tempera-
bsolute temperature T is one of the cen- with energy. If we were to extend this formula to ture scale would be given by −b = −1/kBT, also
A tral concepts of statistical mechanics and
is a measure of, for example, the amount
of disordered motion in a classical ideal gas. There-
negative absolute temperatures, exponentially in-
creasing distributions would result. Because the
distribution needs to be normalizable, at positive
emphasizing that negative temperature states are
hotter than positive temperature states, i.e., in
thermal contact, heat would flow from a negative
fore, nothing can be colder than T = 0, where temperatures a lower bound in energy is re- to a positive temperature system.
classical particles would be at rest. In a thermal quired, as the probabilities Pi would diverge for Because negative temperature systems can ab-
state of such an ideal gas, the probability Pi for a Ei → –∞. Negative temperatures, on the other sorb entropy while releasing energy, they give
particle to occupy a state i with kinetic energy Ei hand, demand an upper bound in energy (1, 2). In rise to several counterintuitive effects, such as
is proportional to the Boltzmann factor daily life, negative temperatures are absent, be- Carnot engines with an efficiency greater than
cause kinetic energy in most systems, including unity (4). Through a stability analysis for thermo-
Pi º e−Ei =kB T ð1Þ particles in free space, only provides a lower en- dynamic equilibrium, we showed that negative
ergy bound. Even in lattice systems, where kinet- temperature states of motional degrees of free-
where kB is Boltzmann’s constant. An ensemble ic energy is split into distinct bands, implementing dom necessarily possess negative pressure (9) and
at positive temperature is described by an occu- an upper energy bound for motional degrees of are thus of fundamental interest to the description
pation distribution that decreases exponentially freedom is challenging, because potential and in- of dark energy in cosmology, where negative pres-
teraction energy need to be limited as well (3, 4). sure is required to account for the accelerating
1
So far, negative temperatures have been realized expansion of the universe (10).
Fakultät für Physik, Ludwig-Maximilians-Universität München, in localized spin systems (5–7), where the finite, Cold atoms in optical lattices are an ideal
Schellingstraße 4, 80799 Munich, Germany. 2Max-Planck-Institut
für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, discrete spectrum naturally provides both lower system to create negative temperature states be-
Germany. and upper energy bounds. Here, we were able to cause of the isolation from the environment and
*To whom correspondence should be addressed. E-mail: realize a negative temperature state for motional independent control of all relevant parameters
ulrich.schneider@lmu.de. degrees of freedom. (11). Bosonic atoms in the lowest band of a
52 4 JANUARY 2013 VOL 339 SCIENCE www.sciencemag.org