Francesca Fragkoudi*

*[email protected]
2005MNRAS.358.1477A
Effect of Boxy/Peanut
Bulges
on Galaxy
LAM (Laboratoire
d'Astrophysique
de Marseille) Potentials
The research leading to these results has received funding
from the People Programme (Marie Curie Actions) of the
European Union's Seventh Framework Programme
WHY THE POTENTIAL?
E. Athanassoula, A. Bosma, F. Iannuzzi
HOW DO YOU OBTAIN IT?
Follow us on our (DAGAL) outreach
pages!
3
To
We
To obtain
obtain potential
potential we
we need
need the
the density:
density:
We need
need the
the potential,
potential, the
the
facebook.com/DAGALaxies
1
forces
forces and
and the
the derivatives
derivatives of
of
the
the forces
forces of
of aa galaxy
galaxy in
in order
order
Surface
brightness
Context
Surface
brightness
to
model:
4 Fragkoudi et al.
to model:
Educated
guess
at
the
thickness
of
the
disk
Educated
guess
at
the
thickness
of
the
disk
●●
from
image:
from
image:
The
gas
flows
in
barred
The gas flows in barred
and
the
height
function
and
the
height
functionand
galaxies
Boxy/Peanut (B/P) bulges are formed due to vertical instabilities in thespiral
bar and
they “built” from the same orbits
To create a dynamical model of a galaxy, we need its potential
spiral
galaxies
●●
hence the 3D density distribution of the galaxy: !
which constitute the bar (Fig. 1), i.e. they are the bar seen edge-on (Combes
Sanders
1981,
Athanassoula
Test
manifold
theory
of
Test&the
the
manifold
theory
of
!
2005). Therefore, once a bar is formed, a B/P bulge is likely to follow and
hence
B/P
bulges
are
a
common
ring
and
spiral
arm
formation
ring and spiral arm formation
1. The 2D surface density
is obtained
from images of
face
on (or
feature of barred disc galaxies (Lutticke et al 2000).
Derived
from
simulations
and
observations
Derived from simulations and observations
((E.
E. Athanassoula,
Athanassoula,Romero-Gomez,
Romero-Gomez,
moderately inclined) disc galaxies (left panel Fig. 2) !
of
edge-on
galaxies
Bosma,
of
edge-on
galaxiesa height
Bosma, Masdemont,
Masdemont,2010,
2010,and
and
2. The three-dimensional density is obtained by assigning
HEIGHT FUNCTION STUDY
Potential, Forces
and
Derivatives
of
Forces,
will
be
Forces
and
Derivatives
of
Forces,
will
be
*[email protected]
affected
affected by:
by:
●●
The
The height
height function
function we
we chose
chose (how
(how the
the mass
mass is
is
distributed
distributed in
in the
the zz direction)
direction)
●●
The
The thickness
thickness z0
z0 of
of the
the disk
disk
2
We
want
to
study
how
much
these
choices
We want to study how much these choices will
will
affect
affect our
our results.
results.
1.5
Potential,
@DAGAL_EU
Creating a Model with a Boxy/Peanut Bulge
The aim of this work is to study the effect that B/P Bulges will have on galaxy
models.
Specifically,
we study
references
within
))
references
within
the effect they will have on the potential and forces of the galaxy and by●●extension
on
the
periodic
orbits and
To
probe
the
dark
matter
To probe the dark matter
on bar strength. This ultimately suggests what the effects of B/P bulges will be on the stellar and gaseous
distribution
of
galaxies
from
distribution
of
galaxies
from
kinematics of the galaxy.
dynamical
dynamical tests
tests
Results
0.0023
0.0023
0.0016
0.016
0.071
0.071
z0 (kpc)
0.29
0.29
1.2
1.2
0.0023
0.0023
0.0016
0.016
0.071
0.071
0.29
0.29
1.2
1.2
-0.00035 -0.00016
-0.00016 -0.00003
-0.00003
0.00021
-0.00035
0.00021
0.0033
0.1
0.42
0.0023 0.023
0.016
0.071
0.29
0.00042
0.00042
1.7
1.2
F
Side-on
End-on
observed galaxies directly from images. However the height function is
fit of the two two-dimensional gaussians for the side-on projec
2 assume, for simplicity,
order to decrease the noise2 of the image we require
respect to position we
thatFitthe dennot known for face-on or mildly inclined galaxies. Therefore for In
the
2 z
Fit
z0 sity distribution can
z0
F
(
z)=sech
)
a snapshot with a large number of particles. We create
a
be
written as, (
Sech-squared
Sech-squared
purpose of our study we use a simulated Nbody + SPH isolated
galaxy
z 0 previous section.
snapshot
with 40 times the particles
1.5 of the original snapshot,
1.5 y, z)
⇢(x,
= ⌃(x, y)Fin
(z),the
(7)
simulation
introduced
We recover
following
the
procedure
described
in
Athanassoula
(2005).
(Athanassoula
et
al
2013),
for
which
we
can
obtain
both
the
twoOnce we have the 3-dimensional density distribution we need to use a method to obtain the
a rather
complex behaviour,
one that cannot
be grasped e
where ⇢ is the
three-dimensional
density distribution,
⌃ is
To further reduce the noise in the image we apply some
dimensional
density,
and the height function.
has a
tirely by
a simple
analytic
we found th
1
1 surface
the two-dimensional
density,
andzfunction.
F is the However,
height
smoothing, by Fourier decomposing
and recomposing the
potential. There exist various methods, and we
use the surface
so dubbed,
3D-Integration
method.The galaxy
Fgeneral
( z)=sech(
) canfunction
twomore
two-dimensional
gaussians
provide a reasonable a
function. In the
case where
the height
fairly strong bar and B/P (the bar strength of the galaxy is in image.
the range of
Sech
Sech depends on position
z 0 be for
galaxy,
as would
proximation
the B/P
shape.
As example
can be seen in Fig. 3 a
The Fourier components are0.5
calculated as follows:
0.5in the to
an SB galaxy).
the height function describing a B/P bulge, the scaleheight
∞
l
0.5
In
our
study
we
use
a
disk
with:
In our study we use a disk with:
function to the galaxy (right panel of Figs. 2 and Fig. 3)!
An
with
aa
0
An exponential
exponential disk,
disk, which
which is
is superimposed
superimposed
with
3. We then perform a three-dimensional integration on the density
-10
-5
0
2-dimensional
ferrer’s
such
that
itit
0.0033 0.023 ellipsoid,
0.1
0.42
1.7
x (kp
2-dimensional
ferrer’s
ellipsoid,
such
that
distribution to obtain the potential.
Figure 2:(b)
Our
simulated galaxy seen
face-on
(a) Original Image
Model
(c) Residual
(a)
gtr116:
Side-on
(b) Edge-on: S
resembles
a
disk
with
a
bar.
resembles
a
disk
with
a
bar.
(left) and edge-on (right)
Figure 2. Left: We show the surface density of the original image of the stellar component from the gtr116 simulation. Middle: The
ρ(x , y , z)=Σ( x , y) F (z)
Figure
3. Left:
side-on
of the surface
model from the Fourier recomposition, using up to n =26 even Fourier components.
Right: The
residual
imageimage
after subtracting
the stellar density
B/P
on Galaxy
5 projectio
3-dimensional
density
distribution
error between E↵ects
the originalof
image
and Bulges
the
model.
simulation
(red
crosses)
along the Models
y=0 axis (side-on
The above
process
is particularly suited for obtaining modelsmodel
of from the original image. The units give the relativeFor
3-dimensional
density
distribution
we
are
using:
For the
the height
height functions
functions
are
using:
simulation (red we
crosses)
along
x=3 which is where the maximu
z0 (kpc)
2
AAstudy
study on
onhow
howthe
theheight
heightfunction
functioncan
canaffect
affectthe
thebar
bar
1
strength
can
be
found
in
Laurikainen
&
Salo
(2002)
strength can be found in Laurikainen & Salo (2002)
∞
ρ(x , y , z)dx dy dz
∞
Φ(x p , y p , z p )=∫−∞ ∫−∞ ∫−∞
√( x−x
2
p
z0 (kpc)
Francesca Fragkoudi*
as commented below, this choice may fail at certain poin
Z
changes as abut
function
of position.
In this
case,
the density of the structu
provides
an
overall
fair
representation
1 ⇡
0
would be0-10
given by,-5 1
d✓,
2
The B/P2 height function
was assigned by examining the scaleheight ofan (r) = ⇡ ⌃(r, ✓)cos(n✓)
-10
-5
0 (4)
5 distribution
10
0
5
10
The resulting B/P height
function
is a non-separab
z⩽z
⇡
0
0.0033
0.023
0.1
0.42
1.7
0.0033 0.023
p
p
xUniform
(kpc)
y y,
(kpc)
z)=
⇢(x,F
y,(z)
=position
⌃(x,2z
y)F0and
(x,→
Slab
the galaxy
for side-on
and end-on projections (Fig. 3). We
use an 0.1
an 0.42 1.7
function
of
isz),given by: (8)
Uniform
Slab
(a) gtr116: Side-on
(b)3:
Edge-on:
Side-on peanut height function (c)
Edge-on:
End-on
✓
◆
z>
z
Z ⇡
Figure
Our
fiducial
is
constructed
by
0
where
the
normalisation
of
the
height
function
is,
analytic function for the B/P height function (two two-dimensional
1
z
0 = 1 sech2
Z 1 F (x, y, z)
bn (r) =
⌃(r, ✓)sin(n✓)
d✓,
(5)
.
(1
fitting two two-dimensional gaussians to the
scaleheight
of
the
⇡ of
Left: side-on image
Middle:
Plot
of
the
scaleheight
of
the
2z
(x,
y)
z
(x,
y)
⇡ the surface stellar density of the simulated galaxy gtr116. F
gaussians), which we fit to the values of the scaleheightFigure
of the3.stellar
0
(x, y, z)dz = 1.0
(9)
{ }{
) +( y− y ) +(z−z )
}
Figure 1: The Boxy/Peanut bulge, which can be seen when in edgeon galaxies (top), is composed by the same orbits
which make
the Miyamoto-Nagai
We
with
analytic
the
potential,
in
order
Wecompare
compare
withup
analytic
Miyamoto-Nagaisolution
solutionfor
for
the
potential,
in
order
simulation (red crosses) along the y=0 axis
(side-on
projection). The
solidthe
black
line gives
the
Right:
Plot
of the scaleheight of the
1 fit.on
stellar
component
along
x-axis
(shown
the
left).
where
an and bn are the even and odd Fourier components,
component
of
our
simulation.
We
call
this
our
fiducial
peanut
model.
The
scaleheight
z
(x,y)
varies
like
two the
two-dimension
0
simulation
(red
crosses)
along
x=3
which
is
where
the
maximum
of
the
peanut
occurs
(end-on
projection).
The
solid
black
line
gives
bar seen in face-on galaxies (bottom). (Schematic
representation)
to
toget
getan
anestimate
estimateof
ofthe
theerrors
errorsof
of the
themethod.
method.In
Ingeneral
generalthe
themethod
methodcan
canbe
be
✓ is the azimuthal angle, and r the radius. We thenPeanut
reduce
F ( z , r)
Peanut
gaussians:
fit of the two two-dimensional gaussians for the side-on projection.
3.1
Flat
Height
Functions
the high frequency noise by recomposing the image as,
slow
slowbut
butitit isisrobust
robusti.e.:
i.e.:
✓ ✓
◆◆
●
2 or “flat”
2
Up to now in the literature, position-independent
●
ItItgives
errors
within
the
range
of
the
precision
we
ask
for
(x x0 )
(y y0 )
r
gives errors within the range of the precision we ask for
m=n
z
(x,
y)
=A
exp
+
+
X
height functions
used whenthe
modelling
barred
disc 2
0have been
simulation
introduced
in
the
previous
section.
We
recovered
centre
and
the
peanut
maximum,
gaussian
approximaa
2
●
0
2
2
● Although for forces and derivatives of forces we might need to introduce a
⌃(r, ✓) =
+
(am (r)cos(m✓) + bm (r)sin(m✓)) , (6)
Although for forces and derivatives of forces we might need to introduce a
galaxies.
We
therefore
also
use
two
flat
height
functions
inwith ◆◆
a rather complex2 behaviour, one that cannot be grasped ention fails to represent the behaviour
of the scaleheight
✓
✓
2
2
m=2
8 Fragkoudi et al.
(x
x
)
(y
y
)
this
paper,
to
check
the
discrepancy
which
will
be
created
1
1
softening
due
to
the
big
power
in
the
denominator
(a
lot
of
power
in
very
disc
by a simple
function. However, we found
that scaleheights:
y. However, as it turns A
out,
the
fitted function
shown
in
softening due to the big power in the denominator (a lot of power in very
h
E↵ects of B/P Bulges tirely
on Galaxy
Modelsanalytic
9
The
exp
+
+
z
h
0
r2
The
scaleheights:
2
r a flat2 height
in the model of the galaxy when using
function
using
only
a
limited
(n
)
number
of
even
Fourier
compo2
F
two
two-dimensional
gaussians
can
provide
a
reasonable
apFig.
3
underestimates
the
value
of
z
at
these
points.
This
Effect on Bar Strength
=5
0
narrow
Effect on Forces
=10
narrowpeaks)
peaks)Effect on Orbits
and
a
B/P
height
function.
We
use
two
common
functions
We
know
that
bars
are
the
main
drivers
of
secular
nents. We show in Fig. 2 the surface density of the original
h =3 kpc
(1
z
Conclusions:
F
proximation to the B/P shape. As can be seen in Fig. 3 and
directly
translates into an underestimation
of the e↵ect
of
r
z
0
0
from
the
literature,
i.e.
the
isothermal-sheet
model
(van
der
no B/P
no B/P
image of the
simulation
(in arbitrary
units),
the surface
densech
2
as
commented
below,
this
choice
may
fail
at
certain
points,
the
peanut
in
those
regions.
In
summary,
our fiducial
model
B/P
B/P
sech
where
A
is
the
maximum
scaleheight
of
Kruit
&
Searle
1981),
and
the
sech-law
model,
since
it
is the peanut abo
z
=0.6
kpc
sity
of
the
model
from
the
Fourier
recomposition,
and
the
z
=0.3
kpc
0
peanut
0
but provides an overall fair representation of the structure.
for
the peanut
function
shown
inThe
Fig.variance
3 will
result
4
0.6
4
Thin
disk
Thick
disk
possible
for discs
to deviate
from the
isothermal
sheet
near
the height
disc
scaleheight
(z0disc
).
of the
gaussians
boxy
residual
image
after
subtracting
one
from
the
other.
For the potential, errors of method are smaller
The resulting B/P height function is a non-separable
into
a conservative
estimate
of the
real
2 Kruit
the galactic
plane
der
1988).
These
are given
by maximum of t
given(van
by
, (x0of
, ythe
) ise↵ect
the
position
of peanut
the
0
function of position and is given by:
present in the image we adopt as our starting point. Given
2
2
or equal to the precision we ask for0.5
first gaussian and (x
1, y
1 ) the position of the maximum
✓
◆
✓
◆
the scope of this paper, which
is to zdemonstrate the generic
1
1
z
the
second
gaussian.
2 We fit these two two-dimensional gau
2
F
(z)
=
sech
, model, we find
(10) this
F
(x,
y,
z)
=
sech
.
(12)
e↵ect
of
a
peanut
bulge
on
a
galaxy
0
0
0.4
3 HEIGHT FUNCTIONS
USED
2z
z
0
2z0 (x, y)
z0 (x, y)
sians to values0 of the scaleheight
obtained from the simu
approximation more than satisfactory.
tion along y = 0 and x = 3 (which is where the maximu
In order to obtain
the three-dimensional
of a galaxy
For forces we get mostly smaller 0.3
errors than
and
The scaleheight
z 0 (x,y)
varies like two density
two-dimensional
-2
-2
✓ ◆ In the remainder of the paper,
disc from a two-dimensional image we need to assume a
of
the
scaleheight
occurs).
gaussians:
1
z
what we asked for but at some points we might
F
(z)
=
sech
, model.
(11)
height function, which defines how the density drops o↵ as
refer
to
this
as
our
fiducial
3.3
Boxy
Height
Function
⇡z
z
-4
-4
0
0
0.2
get slightly larger error.
a function of ✓
z from
plane 2z◆◆
=0. The height
✓ the equatorial
To obtain the scaleheights, we take cuts along the
2
(x x0 ) (z0 )(ywill yof
0 ) course a↵ect the
respectively.
function
and
the
scaleheight
The B/P bulge
might
at times
have
rather
boxy
isophotes.
z0 (x, y) =A exp
+
+
and
yaxes
and
fit
the
vertical
particle
distribution w
8 Fragkoudi et al. -6
2
2
-6
0.1
2
2
results,
and
we
therefore
need
to
use
the
height
function
This could be
due
to projection
e↵ects,We
whereby
the determine
peanut
-6
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
For
derivatives
of
forces
errors
are
slightly
larger
for
specific
a
sech-squared
function.
thereby
the var
✓
✓
◆◆
2 of the galaxy2 we are trying to
which best approximates
that
x (kpc)
x (kpc)
(x x1 )
(y y1 )
is3.2
projected
at Height
such
anFunction
angle bin
thatto
the
isophotes
appear
boxy
disc
Peanut
6
6
tion
of
z
from
bin
along
the
cut.
The results can
0
A
exp
+
+
z
,
no
B/P
0
no
B/P
model.
0
points,
also
due
to
having
to
introduce
softening.
2
2
(Athanassoula
&
Misiriotis
2002).
However,
boxy
isophotes
B/P
B/P 2
2
2
(a) Characteristic Diagram
(b) Comparing x1 orbits
(c) Comparing x1 orbits
0
4
6
8
10
seen in
Fig. 3.for
Inthe
thepeanut,
side-onweview
(panelthe
(b)) we see that t
The
height
function
can
be
either
constant or can
To obtain a height
function
examined
4
4
might
be
present
even
when
the
bar
is
seen
side-on,
i.e.
they
r (kpc)
(13)
scaleheight
y =cuts
0 behaves
like the co
change with position. In the case where it is constant with
particle distribution
alongalong
di↵erent
in x and approximately
y from the
might
indeed
be
due
to
a
vertical
thickening
of
the
bar
which
2
height A
functions:
(solid
Figure 5. Left: Characteristic diagram for the model created from the image of a simulated galaxy and two height functions,
one withFigure 6. Strength 2of non-axisymmetric forcings (QT ) as a function of radius, for models with di↵erent
where
is thesech
maximum
scaleheight of the peanut above
bination of two gaussians, except in the central region whe
2
creates
a
boxy
vertical density
distribution
(Patsis
et al.
Figure
5:
Plots
of
the
bar
strength
against
radius,
as
given
green line), the fiducial peanut setup (thick black solid line) and the fiducial boxy setup (dash-dotted blue line).disc
c
2014
RAS,
MNRAS
000,
1–17
(dashed-dotted black line) and one without (dashed red line) a B/P bulge. We plot the intersection of each orbit of the
x1 family withred line), sech (dashed
the
disc
scaleheight
(z
).
The
variance
of
the
gaussians
is
the
scaleheight
drops
below
that
of
the
outer
disc. For a c
0
0
0
Figure 4: The colour scale gives the difference between the two
The vertical solid black line indicates the radius at which the fiducial peanut is maximum and the vertical dashed black
line
indicates
2002).
This
tends
to
be
the
case
for
galaxies
with
weak
bars,
4
the y-axis as a function of its Jacobi constant EJ . The zero velocity curve (ZVC) of the case without a B/P bulge is plotted with thethe end of the bar.by
given by 2 , (x0 , y0 ) is the position of the maximum of the
along
= 3, x-shape
where the
peanut forming,
is maximum
where instead
of a xstrong
or peanut
boxy (end-on vie
-2
-2
models (with different height functions)
being
compared.
The
first
solid blue line (the ZVC for the case with and without a B/P bulge are very similar). Middle and Right: We plot a few periodic x1 orbits
Combes & Sanders 1981,
Buta & Block
2001), which
first gaussian and (x1 , y1 ) the position of the maximum of
Azimuthally
averaged
panel
(c)),
the behaviour
of z0 is still well approximated
isophotes
are
seen
(Athanassoula
2006).
row shows how the scaleheight
of the
different
height
in the
potential
without
a B/Pfunctions
bulge (red solid lines) and in the potential with a B/P bulge (black dashed lines). For-4 visual clarity we4290, which bears
-4
the
second
gaussian.
We fit these two two-dimensional gausa gaussian,
although
our
fit slightly
under
predicts
the val
measures
the non-axisymmetric
galaxy.
a striking morphological
resemblance to
theperturbation
two cases howeverinisthe
significantly
di↵erent.
By using
Tangential
force
To
model
a
boxy
bulge
we
use
a
height
function
which
Radial
force
two panels of
orbits,
only a few orbits. We see that in general the orbits in the B/P model are less extended than in theour simulated galaxy. Therefore the results presented in this
sians
to values
of will
thebescaleheight obtained from the simulathe Qb method of measuring bar
strength,
the bars
varies. The second row gives show
the potential,
and
theeach
thirdwith
and
the scaleheight.
-6
-6
We
show
the
bar
strength
for
two
flat
height
functions
(red
drops o↵ as of
sech-squared
with height from the z =0 plane,
-6
-4
-2
0
2
4 as6well as in
-6 previous
-4
-2and 0subsequent
2
4 sections
6
section,
corjudged
as
having
the
same
strength
(class
6),
and
hence
case without the B/P. This follows from the fact that the forces in the plane are reduced in the model with the B/P bulge.
tion
along
y
=
0
and
x
=
3
(which
is
where
the
maximum
For
the
potential
the
difference
between
height
functions
doesn’t
Along some cuts at x values intermediate between t
fourth row the x- and y- components of the force. In the first
x (kpc)
x (kpc)However, even weakly
respond to strongly
barred
galaxies.
the same e↵ect
on thefunction
disc, even though
it appears that
and
green
line), for our fiducial peanut
height
(solid
of the scaleheight occurs). In the remainder of the paper, we
✓
◆
(a) Characteristic Diagram
(b) Comparing
orbits
(c) Comparing
x1 orbits
1 barred
have
such xan
important
as
the
scaleheight
chose.
thewe
gas flow
towards the centre of the galaxy is hampered
galaxies will role
have
B/P
bulges,
albeit weaker ones, that
column we compare two “flat” height functions (both without a B/P
1
z
line)toand
for a boxy
height
(dashed
blue
2
refer
thisinasSection
our fiducial
model.
and therefore thethick
results black
will also apply
these galaxies
alwhere the function
peanut is maximum.
This to
is why
5.3
c y,
2014
RAS, MNRAS
000, 1–17
F
(x,
z)
=
sech
,
(14)
In
general,
thicker
disks
are
affected
more
by
the
choice
of
bulge), in the one we comparemodel,
a galaxy
with
a
peanut
bulge
to
2z
(x,
y)
z
(x,
y)
Tostrength,
obtain which
the scaleheights,
we take cuts along the x to a lesser
extent.
We
intend
to carry
out
a full sta- linewegives
introduce
measureofof the
bar
can
these families do not extend out to r=3
kpc , where
0
0
Figure
5. Left: Characteristic
diagram
for the model
created from the image ofthough
a simulated
galaxy
and two
height
functions,
one
with
line).
The
thick
grey
vertical
theanother
position
Figure
4: In the top two
panels
we
show
a
number
of
(dashed-dotted black line) and one without (dashed red line) a B/P bulge. We tistical
plot thestudy
intersection
each orbit
of the xstrength
of theofe↵ects
of di↵erent
B/P bulges
capture the reduction in bar strength
is vertical particle distribution with
1 family with
and y-when
axesa B/P
andbulge
fit the
one without a peanut, in the third
one we e↵ect
compare
a boxy bulge to
the peanut’s
is maximum.
height
function.
where the scaleheight is a top-hat function,
maximum
(i.e.
of the peanut) and the
the y-axis
as a function
its Jacobi
constant
. The
velocity curve
(ZVC)
of the
case without
B/Pgalaxy
bulge iselsewhere.
plotted with
thethe length
a potential without a B/P
bulge
(solidof red
line)
andEJin
a zero
potential
with
a the
B/P
on
models
ofpeanut
theirahost
present.
a
sech-squared
function.
We
thereby
determine
the
variaFig. 5(a) gives the value
solid blue line (the ZVC for the case with and without a B/P bulge are very similar). Middle and Right: We plot a few periodic x1 orbits
a model with a flat height functionThe
andcharacteristic
in the fourthdiagram
columninwe
vertical
thinFor2dashed
length
ofwiththe
bar.
We
see
bulge (dashed black line).
In the
bottom
panel
show
in the potential
without
a B/P bulge
(red we
solid lines)
and the
in the characteristic
potential with a B/P bulge (black
dashed lines).
visual clarityline
we gives the
tion
of
z 0 from
bin to bin along the cut. The results can be
(
(iii)
A
model
the
fiducial
boxy
height
function
at which
orbit intersects
the y-axis
of its Ja(i)
A
model
with
a
sech
and
sech
law
height
function
compare a model with a peanut
heightthefunction
to a model
withas
a a function
bulge
show two panels of orbits, each with only a few orbits. We see that in general the orbits in the B/P
model
are
less
extended
than
in
the
that
when
using
two
different
flat
height
functions,
the
bar
z
|x| 6 xmax & |y| 6 ymax
We
also plot
QT for our fiducial
boxy
height
function.
diagram
of
the
seen
in
Fig.
3.
In
the
side-on
view
(panel
(b))
we
see
that
the
In
previous
work
by
Laurikainen
&
Salo
(2002)
the
0
cobian
energy
(i.e.
energy
in
the
rotating
frame
of
reference;
case without the B/P. This follows from the fact that the forces in the plane are reduced in the model with the B/P bulge.
z
(x,
y)
=
,
(15)
0
Wethe
see again
that where the boxy
bulge is maximum,
QT0 behaves approximately like the comboxy height function.!
disc
e↵ect
of
position-independent
height
functions
and
height
scaleheight
along
y
=
For
the
forces
the
greatest
differences
are
found
in
interior
strength is not significantly affected,
which is consistent with
z0
otherwise.
Sinceis B/P
bulges reduce the forces in the z=0 plane of the galaxy, the
Binney & Tremaine (2008)). The Jacobian energy
given
is flattened due to the decrease in the strength of the forces
functions
which
only
vary
as
a
function
of
radius
was
bination
of two gaussians, except in the central region where
The results show that B/P bulges
reduce the potential and forces
inFig.
the plane.
However
we
also see
that where the top-hat
the
results
in
the
first
column
of
4.
On
the
other
hand,
region
of
the
analytic
bar,
where
most
of
the
mass
is
extent units.
and shape of the
periodic
orbits
is
also
significantly
affected:
B/P
examined,
and
these
were
found
not
to
change
Q
in
a
in arbitrary units, due to the mass being in arbitrary
b
bulge
disc
model, these families do not extend out to r=3 kpc , where
and
where
z
gives
the
scaleheight
of
the
disc
and
z
the
scaleheight
drops
below
that
of
the
outer
disc.
For
a
cut
boxy
function
ends,
Q
exceeds
the
values
of
the
sech
and
0
T
0
in the plane of the galaxy by aThe
significant
amount
to orbit
40% for
for a givenbulges
significant way. For realistic height functions such as the
the peanut’s
e↵ect
is maximum.
higher the
energy(up
of an
intersection
the
bar
i.e.
the
effect
bars
have
on
their
host
2 the
reduce the extent
of the
periodic
concentrated.
Differences
can
be
quite
significant
for
forces.
2 strength,
sech curves. This is presumably
due x
to =
the3,steep
dropthe
of peanut is maximum (end-on view,
gives the scaleheight of the boxy bulge. This is quite a simalong
where
exponential, sech, or sech models, they found that Qb was
The characteristic diagram in Fig. 5(a) gives the value
the forces). If we do not modelalong
a B/P
bulge
when
it
is
present,
we
the y-axis the larger the velocity of that they
orbit.change
Hence, the shape
scaleheight.
Theboxy/peanut
forces are increased
in(c)),
such the
a waybehaviour
that the
galaxy,
is significantly
reduced
where
the
a↵ected by less than
5%, which
is consistent with our
own
of some
example
plified model of the boxy bulge, with only two free paramepanelbulge
of z0 is still well approximated by
at which
the orbitperiodic
intersects theorbits,
y-axis as afor
function
of its Ja- by removing
torque
becomes
very
large
at
this
point.
As
a
consequence,
results. This can be seen in Fig. 6, where we see that for
will introduce large errors intothe
ourxmodel.
1 orbits will be more elongated along the x-axis. We
cobian
energy (i.e.
energy
in the rotating
frame
of reference;
ters, its strength (height) and length (set by L=2xmax ). The
a gaussian,
although
our fit slightly under predicts the value
isgeneral,
maximum.
loops which are present
when
a
B/P
bulge
is
not
present
(upper
right
In
the
largest
differences
due
to
Qb is overestimated
by 16% compared
to the fiducial
peanut
an
equivalent
scaleheight,
the
height
functions
of
sech
and
Binney
& IC
Tremaine
is given
can already see therefore from Fig. 5(a) that the x1 orbits in
(a)
4290 (2008)). The Jacobian energy
(b) gtr116
thickness of the box, i.e. ymax , is set by the width of the bar.
the scaleheight.
case. We investigated the boxyofheight
function as an altersech2 will produce
very
similar of
bar height
strengths, which
will
figure)
in arbitrary units, due to the mass being in arbitrary units.
the
choice
function
or
scaleheight,
native to using the peanut height function,
therecuts
is oneat x values intermediate between the
the model without a B/P bulge will be more elongated than
Alongsince
some
The fiducial boxy height function has a height equal to
tend to become even more similar the thinner the disc.
The higher the energy of an orbit for a given intersection
parameter
less to model. However we conclude that even if
Figure
7.the
Visual
between
IC orbit.
4290Hence,
and gtr116. The
are found in the central parts of
the
disk.
those in the model with a B/P bulge. Figs. 5(b) and 5(c)
along
y-axiscomparisson
the larger the velocity
of that
one is merely interested in QT , a simple boxy height function
(ii) A model with the fiducial peanut height function
the
x
orbits
will
be
more
elongated
along
the
x-axis.
We
two
galaxies
have
strong
morphological
similarities
and
the
same
1
2014 RAS,
MNRAS 000, 1–17
where we plot some of the orbits of the x1 family in the two
is not a good approximation to ca peanut
function.
Fragkoudi et al.
6
QT
y (kpc)
y (kpc)
Results of height function study
y (kpc)
Figure 4. Errors from not modelling a B/P when it is present:
The top row gives the scaleheight along the bar major axis for the setups we are comparing in the plots. The second row gives the
di↵erence for the potential, the third row gives the di↵erence for Fx , and the fourth row gives the di↵erence for Fy . The green line
represents the ellipse fitted to the outer isophote of the bar.
First Column: Di↵erence between the sech and sech-squared setup. Second Column: Di↵erence between the fiducial peanut height function
and a sech-squared height function. Third Column: Di↵erence between a boxy height function and a sech-squared height function. Fourth
Column: Di↵erence between a boxy height function and our fiducial peanut height function. Not including a peanut or a boxy bulge
where there is one will induce large errors in the potential and forces. A boxy height function is not a good approximation for a peanut
height function. For details see the text, Sect. 4.1
c 2014 RAS, MNRAS 000, 1–19
0.7
6
y (kpc)
8
Potential
can already see therefore from Fig. 5(a) that the x1 orbits in
In the same figure we also plot
QT for our fiducial B/P
(b) gtr116
(a) IC 4290
evolution in disc galaxies: They drive gas to the
centre of the galaxy (Athanassoula 1992b) which
can create disc-like bulges (Kormendy & Kennicutt
2004, Athanassoula 2005) and can provide a
reservoir of fuel for AGN activity (Shlosman et al
1990). However most barred galaxies contain
Boxy/Peanut bulges, which have a significant
effect on their host galaxy:!
Forces
the forces in the plane of the galaxy!
• They reduce
• They reduce the extent of periodic orbits, and
alter their shape!
• They reduce the bar strength in the regions
where the B/P is maximum.!
All this suggests that B/P bars will have a
significant effect on the ability of the bar to
drive gas to the centre of its galaxy, thus
altering the mass distribution in the central
regions. !