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2
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•
Each proposal will be presented 3 times. (Each member of a
given team will lead 1 time.) Present the pros and then
potential cons of each proposal. Remember that you can
sway the rest of the class, and that they may not have read
a given proposal as well as you have.
•
After each proposal as been presented there will be a general
discussion.
•
Following general discussion, there will be a secret ballot
vote.
•
We will tally up the votes after class and send you the
winning result, along with a close runner-up.
NOAO Observing Proposal
Standard proposal
Date: September 26, 2013
Panel:
For office use.
Category: Star Clusters
An Abridged Tail: Mapping the Palomar 5 Tidal Stream
with DECam
PI: Marla Geha
Status: P Affil.: Yale University
Astronomy Department, New Haven, CT 06511 USA
Email: marla.geha@yale.edu
Phone: 203-432-5796
CoI:
CoI:
CoI:
CoI:
CoI:
Ana Bonaca
Kathryn Johnston
Nitya Kallivayalil
Andreas Küpper
David Nidever
Status:
Status:
Status:
Status:
Status:
T
P
P
P
P
Affil.:
Affil.:
Affil.:
Affil.:
Affil.:
FAX:
Yale University
Columbia University
University of Virginia
Columbia University
University of Michigan
Abstract of Scientific Justification (will be made publicly available for accepted proposals):
Palomar 5 (Pal 5) is a gravitationally disrupting Milky Way globular cluster exhibiting prominent
tidal tails. These tails show tantalizing evidence for stellar density variations. Such features can
form when a dark matter subhalo passes through the stream, heating stars and creating density
irregularities. However, variations are also a natural consequence of the cluster’s dissolution process,
with eddies and wakes predicted along the debris tail. At the depth of SDSS, the observed Pal
5 density variations are at the level of stochastic background variations, and cannot yet verify or
rule out either scenario. We propose to image the entire Pal 5 system with DECam to gzi=24,
two magnitudes fainter than the SDSS limit. Our goal is to create a high significance density map
along the entire stream to test the origin of density variations. We will map beyond the SDSS
footprint, providing improved constraints on the interaction history of Pal 5 with the Milky Way.
The FOV and sensitivity of DECam are well matched to this experiment. The proposed data will
yield unique insights into the clumpiness of the Milky Way’s dark matter halo, as well the physics
of cluster dissolution.
Summary of observing runs requested for this project
Run
1
2
3
4
5
6
Telescope
Instrument
CT-4m
DECam
No. Nights
Moon
Optimal months
Accept. months
3
dark
May - Jun
Apr - Jul
Scheduling constraints and non-usable dates (up to four lines).
NOAO Proposal
Page 2
This box blank.
Scientific Justification Be sure to include overall significance to astronomy. For standard proposals
limit text to one page with figures, captions and references on no more than two additional pages.
Stellar streams in the Milky Way halo provide irrefutable evidence that our Galaxy was formed, at
least in part, hierarchically via the tidal disruption of dwarf galaxies and globular clusters. Finding
and characterizing tidal streams is a crucial test for structure formation models. On global scales,
streams can constrain the radial profile, shape and orientation of the Milky Way’s dark-matter halo
(e.g., Koposov et al. 2010, Law & Majewski 2010). Streams are also useful probes of small-scale
dark matter structures. While debris from larger satellites such as the Sagittarius dwarf galaxy are
largely unaffected by small subhalos in the Milky Way (Johnston et al. 2002), long cold streams
from systems such as globular clusters are expected to suffer direct impacts from these ‘missing
satellites’. Impacts with dark matter subhalos can both dynamically heat a stream and create gaps
in surface density along the debris (Yoon et al. 2011, Carlberg 2012).
Pal 5 – A Unique Probe of the Milky Way Potential: The Pal 5 tidal stream is thin and long,
spaning an impressive ∼ 30◦ in the SDSS (Odenkirchen et al. 2003, Carlberg et al. 2012). No other
globular cluster shows such prominent tidal tails at a comparable distance (∼23 kpc). The tails hint
at a pattern of stellar over- and under-densities which cannot be explained by reddening variations
alone. While some studies attribute density variations to subhalo encounters (e.g., Siegal-Gaskins
& Valluri 2008, Carlberg 2013), the physics of tidal disruption also impart such inhomogeneities
in the form of epicyclic overdensities (e.g., Küpper et al. 2008). Thus any interpretation requires
disentangling the effects of nature (internal dynamics) versus nurture (influence of the parent halo)
on a tidal stream.
Internal Dynamics versus Dark Matter Clumps? Internal and external processes are predicted to have different effects on a stream. Gaps induced by perturbations from passing dark
subhalos will be irregularly spaced and have larger amplitude as compared to internal cluster dynamics (Yoon et al. 2011). Internal effects are episodic over the phase and eccentricity of an orbit,
thus variations should appear regularly spaced along the debris (Küpper et al. 2012). There is tantalizing evidence that the more ‘regular’ overdensities close to the Pal 5 cluster (227◦ < α < 234◦ ;
Fig. 1) can be attributed to intrinsic stream dynamics (Küpper et al. 2012, Carlberg et al. 2012),
while a large gap at α > 234◦ may be dark matter induced (Carlberg 2009). However, at the SDSS
depth, the analysis requires significant smoothing which influences the number and position of the
recovered overdensities. Further, the signal is dominated by foreground Milky Way stars such that
the size and distribution of the gaps cannot be unambiguously identified. To robustly differentiate
between these processes, the density of the Pal 5 stream must be mapped to deeper magnitudes
(and therefore higher significance) than the current SDSS data allow.
At a magnitude limit of r ∼ 22, within the color-magnitude region occupied by Pal 5, we observe
70% Milky Way foreground stars with a stochastic variation of 10-20%, and 30% Pal 5 stars (based
on the SDSS data itself). The variation in the Milky Way foreground are comparable to that of
the predicted epicyclic variations. Deeper imaging increases the contrast between Milky Way and
Pal 5 stars, although it also increases the signal from unresolved background galaxies (Figure 3).
At r = 24, we predict 70% Pal 5 stars and 30% foreground, therefore ensuring that variations of
∼ 30% are significantly above the background fluctuations (Figure 2).
DECam as a Major Advance: We propose DECam imaging of the entire Pal 5 stream to
gzi = 24. Unlike any previous imager, the DECam FOV includes both the Pal 5 stream and
background regions in a single pointing. We will map the stream beyond the SDSS footprint in
both directions. These data will place definitive constraints on the physics of tidal streams and on
dark matter substructure in the Milky Way halo.
NOAO Proposal
Page 4
This box blank.
Figure 2: (Top) Stellar density profile based on a N-body model of Pal 5 in the stream coordinate system,
where x = 0 is the cluster center, and x increases along the trailing tail. Shown are two magnitude cuts:
g < 22 in gray, comparable to the SDSS coverage, and g < 24 in black, comparable to the proposed DECam
data. (Bottom) Corresponding stellar density maps at these two photometric depths. Deeper photometric
coverage will double the confidence in recovery of Pal 5 overdensities, while the expected increase in the
Milky Way foreground variations is marginal.
1.0
10.
1.0
FieldB (g−z,g−i) all
10.
1.0
FieldB all
10.
FieldB (g−z,g−i)−cut
16
4
Pal 5 isochrone
2
20
g
1
22
0
SDSS
24
−1
−1
g
18
g−z
3
Pal 5 isochrone
0
1
g−i
2
3
26
−1
g=24.0
0
1
g−i
2
3
−1
0
1
g−i
2
3
Figure 3: We will observe Pal 5 in the gzi filters to minimize unresolved background galaxy contamination.
Data from the DECam Magellanic Clouds survey (SMASH) suggests that the gzi filters are optimal for
star-galaxy separation. (Left) Color-color diagram of all stellar-like objects in LMC fields, the stellar locus is
marked with a red-white dashed line. (Center ) CMD for all the photometric sources, including the “cloud”
of unresolved galaxies at g > 23. (Right) CMD after applying the stellar locus cut, which removes most of
the unresolved galaxies. The Pal 5 isochrone is overplotted in white for comparison.
Class Exercise: Evaluate my NOAO
Proposal
•
Some guiding questions:
•
Is the “big picture” question clear and well-framed? Or is it lost in
the details?
•
Is the sample (or target) justified? Why this particular target as
opposed to others?
•
Is this a timely investigation? Is the significance to astronomy of
the proposed program made clearly? Is there a clear discussion of
how it will further our understanding of an outstanding issue/
question?
•
Is the request for time (or desired depth of observations) justified?
•
Is the request for this particular Telescope justified? (e.g., FOV,
pixel scale or resolution, efficiency). Could the goals be better
achieved with another facility?
Structure of a proposal
• Title: steers reader in a particular direction
• Abstract: crucial for getting to top ~half of proposals (at this
point your proposal has been provisionally graded)
• Body of text: will often be glanced at rather than read, so must
be very easy to read
• Figures: need to convey the key points independent of the text
• Technical sections: will be checked for red flags
Final Project: Structure of your proposal
•
We have 2 hours of Directors Discretionary Time at APO
(night of May 8). Remote observing will be done in the
conference room, and presence is mandatory to receive
credit.
•
Proposal:
•
1 page of text: Science Justification and Technical
Description/Justification:
•
e.g., Title; first paragraph = abstract (What you propose to
do, what you will achieve); 2nd para = give some more
background; 3rd para = technical details/justification.
•
1 page of Figures, including an airmass chart, references.
http://35m-schedule.apo.nmsu.edu/2014-04-09.1/html/schedule-2014-05.html
Final Project:
•
Lab sessions this week can be used to provide support for
the final project.
•
I will be available Wednesday 1-3 PM and Thursday 12.15 1.30 PM as well.
How to Write an Observing Proposal
Step I: Generating Ideas
• Ideally: “I want to figure this out. What data do I need?”
• Often, particularly for students: “I have (or was given) these data.
What can I do with them?”
• Developing a sense of what the important questions are is one of
the most crucial, and most difficult, skills to develop
Example
•
How do galaxies convert gas into stars?
•
Merger sequence of massive galaxies has been extensively
studied. Seems to lead to “quenched” systems. Does the
merger sequence for dwarf galaxies proceed in the same way
as massive galaxies? Specific question for this proposal
•
Hypothesis: Dwarf galaxies have shallower potential wells
and may hold on to their gas differently than more massive
galaxies. Read papers, see if this is supported by models
•
Test: Measure star formation rates for pairs of dwarf
galaxies, compare to those of more massive galaxy pairs.
•
•
Broad science question
Has this been done before?
Proposal: We have identified a sample of dwarf galaxy pairs
in the field. Want to study their star formation rates as a
function of separation and mass ratio.
How to measure star formation rates? Read papers,
talk
to
people
Solution: Measure star formation rates by measuring H-alpha
Can survey data answer part of this
emission (which traces star formation).
question?
Develop the project - I
• What type of data are needed ? (spectra, optical images, radio data, ...)
• How many photons are needed ? How many objects ? What is the required
resolution (spatial and spectral) ? Etc etc
• What telescope / instrument is needed ?
• With all questions: aim for quantitative goal, e.g. a 5 sigma detection
• Tools: software to simulate your experiment: exposure time calculators,
mock observations, etc.
• Telescope: aim for smallest / least capable telescope that can do the job
•
Ideas:
•
Spectroscopy of M82 supernova.
•
Your own observing idea.
•
Build upon a UVa project from the APO schedule.
http://35m-schedule.apo.nmsu.edu/2014-04-09.1/html/schedule-2014-05.html
All the White Papers from the Decadal Review can be accessed at:
http://sites.nationalacademies.org/bpa/BPA_050603
This is a good place to get acquainted with the big picture questions
of the day/era.
Loose Categories of Astronomers
•
Observers / Data Miners
-- Go to telescopes, take data to observe new objects/phenomena
-- Mine existing large databases to find new objects/phenomena
-- Test the predictions/ideas of the modelers/simulators/theorists
•
Modelers/Simulators/Theorists
-- Explain the observations of the Observers
-- Run computer simulations to explain new objects/phenomena
-- Use physics to explain new objects/phenomena
Intro to Numerical Simulations
We turn to numerical simulations when analytic techniques breakdown or are inaccurate.
However, numerical simulations themselves approximate because:
-- Numerical errors
-- Activity below resolution scale
-- Simplification of physics
Simulations are powerful if we understand the limitations and ask the appropriate questions:
-- Provide physical understanding of a system
-- Make testable predictions for a system
-- See how various of input assumptions affect final results
-- Test validity of analytic approximations and techniques
Simulating the Universe
show millenium simulation movie
1kpc = 3 x 10^19 m ~ 3300 ly
Can we find traces of such events
in our Local Group?
Milky Way
halo
GC’s
bulge
disk
8 kpc
open clusters
halo
~200 kpc
05.03.2007
Sun
25 kpc
Sagittarius
Magellanic Clouds
Mürren - Saas-Fee-Course - E.K. Grebel
2MASS
infrared
31 image
NFW Profile
NFW Profile
NFW Profile
•
Analytic calculations and numerical simulations suggest that
the density profiles of dark matter halos may contain useful
information about the cosmological parameters of the
universe.
•
These authors simulate the formation of 19 different systems
with scales ranging from dwarf galaxies to rich clusters.
•
Large cosmological simulations of a Lambda = 1 + CDM
universe.
NFW Halo
• Density profile well-described by (Navarro, Frenk & White 1997)
⇢s
⇢(r) =
(r/rs )(1 + r/rs )2
102
101
ρ/ρs
M/Ms
1
10-1
10-2
10-3
10-4
10-2
10-1
1
101
r/rs
http://background.uchicago.edu/~whu/presentations/trieste_print3.pdf
102
Lack of Concentration?
• NFW parameters may be recast into Mv , the mass of a halo out to
the virial radius rv where the overdensity wrt mean reaches
v = 180.
• Concentration parameter
rv
c⌘
rs
• CDM predicts c ⇠ 10 for M⇤ halos. Too centrally concentrated for
galactic rotation curves?
• Possible discrepancy has lead to the exploration of dark matter
alternatives: warm (m ⇠keV) dark matter, self-interacting
dark-matter, annihillating dark matter, ultra-light “fuzzy” dark
matter, . . .
http://background.uchicago.edu/~whu/presentations/trieste_print3.pdf
1996ApJ...462..563N
Cusp-core
problem:
Intro to Numerical Simulations
Computer simulations come in all shapes and sizes, but have a few common ideas:
1. Set-up a system you are interested in studying:
-- an asteroid
-- planetary system
-- interior of a star
-- star cluster
-- galaxy or system of galaxies
-- the universe
2. Add physics
-- Newtonian gravity
-- General relativity
-- Fluid Dynamics
-- Magnetic Fields
3. Allow system to evolve with time
-- Chose time step
-- Apply physics to system
-- Run for finite amount of time
4. Visualize results
Gravity
What does it mean to ‘include’ gravity in a simulation?
Newton’s Law of Gravity states that:
GM m
F =
r2
(Physics 101-style)
More specifically, for a collection of particles with mass m, the force on each particle is:
N
F (⇧x) =
j=1,i=j
Gmi mj
(x⇧i
3
|x⇧i x⇧j |
x⇧j )
For each particle, at each moment in time, we can determine the force from all other particles.
Calculate the acceleration (F=ma). For a small time step, advance each particle in space.
Intro to Numerical Simulations: N-body Simulations
Of the four fundamental forces, gravity is by far the weakest. Yet on large
distances it dominates all other interactions owing to the fact that it is always
attractive. Most gravitational systems are well approximated by an ensemble of
point masses moving under their mutual gravitational attraction and range
from planetary systems (such as our own) to star clusters, galaxies, galaxy
clusters and the universe as a whole.
Gravitational encounters are inefficient for re-distributing kinetic energy,
such that many such encounters are required for relaxation, i.e. equipartition
of kinetic energy. Gravitational systems, where this process is potentially
important over their lifetime are called ‘collisional’ as opposed to
‘collisionless’ stellar systems
Collisional systems usually have a high dynamic age (tdyn short compared
to their lifetime) and high density, and include globular star clusters and
galactic centers. The majority of stellar systems, however, are collisionless.
Intro to Numerical Simulations - N-Body Simulations
In N-body approach, one follows orbits of representative mass elements, aka particles.
- Start with initial positions and velocities of particles.
- Compute gravitational potential.
- Compute accelerations for each particle.
- For each time step, advance each particle
- Repeat
Intro to Numerical Simulations - N-Body Simulations
simulated vs. observed galaxies mergers
Classical N-body problem: http://adsabs.harvard.edu/abs/2003gmbp.book.....H
N-Body simulations review article: http://adsabs.harvard.edu/abs/2011EPJP..126...55D
N-Body Simulations
Largest numerical simulations have N = 109 particles, but employ other ways to increase run time and accuracy.
We will discuss several approaches to the N-Body problem:
1. The 3-body restricted problem
2. Direct Summation or ‘Particle-Particle‘ codes
3. Tree Codes (aka Barnes-Hut Algorithm or Mesh Codes)
4. Particle-Mesh (PM) algorithm
5. Particle-Particle-Particle-Mesh (P3M)
5. Adaptive P3M
N-Body Simulations - History
The first N-body simulation in astrophysics was analog.
who needs computers???
1/r2 force modeled with
N = 74 lightbulbs!
N-Body Simulations - Toomre & Toomre
The first galaxy simulation on the computer was done by Toomre & Toomre (1972)
The solved the 3-body restricted problem for interacting galaxies
2 massive particles plus 120 ‘massless’ test particles
Retrograde encounter
Prograde encounter
N-Body Simulations - Toomre & Toomre
These early simulations highlighted generic features of
galaxy interactions confirmed by more modern studies
Numerical simulations of the Antennae galaxies (NGC
4038/39) within four decades. From top to bottom:
restricted simulation of Toomre & Toomre (1972); first
self-consistent simulation of the Antennae by Barnes
(1988); hydrodynamic run of Mihos et al. (1993);
recent models with SPH by Karl et al. (2010) and with
AMR by Teyssier et al. (2010). Improvements in both
the techniques and the set of parameters allowed the
models to get closer and closer to the observational data
http://ned.ipac.caltech.edu/level5/Sept11/
Duc/Duc2.html
This visualization of a galaxy collision supercomputer simulation shows the entire collision sequence, and
compares the different stages of the collision to different interacting galaxy pairs observed by NASA's
Hubble Space Telescope.
Credit: NASA, ESA, and F. Summers (STScI)
Simulation Data: Chris Mihos (Case Western Reserve University) and Lars Hernquist (Harvard University)
http://hubblesite.org/newscenter/archive/releases/2008/16/video/d/
MW-M31 collision!
This scientific visualization of a computer simulation depicts the inevitable collision between our Milky Way galaxy and the Andromeda galaxy (also
known as Messier 31). NASA Hubble Space Telescope observations indicate that the two galaxies, pulled together by their mutual gravity, will crash
together in a near-head-on collision about 4 billion years from now. The thin disk shapes of these spiral galaxies are strongly distorted and irrevocably
transformed by the encounter. Around 6 billion years from now, the two galaxies will merge to form a single elliptical galaxy.
http://hubblesite.org/newscenter/archive/releases/2012/20/video/a/
MW-M31 collision!
http://oponet.stsci.edu/summers/files/viz/mw-m31-m33/mw_m31_dh_hammer-1440x720.mov
also show larger MW-M31 movie
ess calculations can now reach more than 109 particles [7–10]. This
Since these early works, N has nearly doubled every two years in accordance
hese rather
dissimilar N -body problems. The significant increase in
with Moore’s law. Latest collisional calculations have reached 10^6 particles,
arallel computers.
and latest collisionless calculations = 10^9 particles.
tware algoen this drat challenges
ystems, and
employed,
nt, and dis. Our focus
portant role
r goal is to
he many inof N -body
e apologise
iew. We do
nd its many
g up initial
ooks in the
ve excellent
n collisional
cover
Themany
significant increase in N in the last decade was driven by the
Fig. computers.
1. The increase in particle number over the past
collisionless
usage of parallel
Newtonian Gravity
Newton’s Law of Gravity:
N
F (⇧x) =
j=1,i=j
Gmi mj
(x⇧i
3
|x⇧i x⇧j |
x⇧j )
ASTR 120 style:
GM m
F =
r2
Newton’s First Law: A body acted on by no forces moves
with a uniform velocity in a straight line.
Newton’s Second Law:
d⌃
v
F⌃ij = m
dt
ASTR 120 style:
F=ma
To understand the dynamical state of a stellar system, we
need to solve the equations of motion:
For particle i, the equations of motion are:
dvk,i
=G
dt
N
j=1,i=j
dxk,i
= vk,i
dt
mj
(x⌥i x⌥j )2
(k = 1,2,3)
This corresponds to a closed set of 6N
equations, and a total of 6N unknowns
(x, y, z, vx ,vy ,vz)
Intro to Numerical Simulations - N-Body Simulations
In N-body approach, one follows orbits of representative mass elements, aka particles.
- Start with initial positions and velocities of particles.
- Compute accelerations for each particle.
- For each time step, advance each particle
- Repeat
The accurate time integration of close encounters is the most difficult part of collisional
N-body methods, while for collisionless N-body methods force softening alleviates this
problem substantially.
Intro to Numerical Simulations - N-Body Simulations
We will assume that particles are ‘collisionless’, don’t need to worry about physics of stars colliding.
mean free path:
time between collisions:
1
=
n⇥
tcoll
Galaxy
{
n=
stellar density pc-3
= cross section pc2
v=
typical velocity km s-1
v
Globular Cluster
tcoll~ 1021 years
tcoll~ 1019 years
This is a long time...thus, direct collisions can be ignored.
Time Integration:
ation of close encounters is the most difficult part of collision
ethods
force
(see
§3.4)
problem
substan
Simple
Eulersoftening
method which
updates
thealleviates
position and this
velocity
for a
givenemployed
particle by time
step ∆ttypes
via:
ethods
in both
of N -body methods. Let us begi
ich updates the position and velocity for a given particle by tim
x(t + ∆t) = x(t) + ẋ ∆t
ẋ(t + ∆t) = ẋ(t) + a(t) ∆t.
htforward, this scheme performs very poorly in practice. The Eu
∆t and the errors are proportional to ∆t2 . We can significant
cost either by increasing the expansion order and thus the acc
ctly using a low-oder scheme. We now compare and contrast a po
rder leapfrog integrator, which is heavily used in collisionless N me, which has become the integrator of choice for collisional app
Time Integration:
ation of close encounters is the most difficult part of collision
ethods
force
(see
§3.4)
problem
substan
Simple
Eulersoftening
method which
updates
thealleviates
position and this
velocity
for a
givenemployed
particle by time
step ∆ttypes
via:
ethods
in both
of N -body methods. Let us begi
ich updates the position and velocity for a given particle by tim
x(t + ∆t) = x(t) + ẋ ∆t
ẋ(t + ∆t) = ẋ(t) + a(t) ∆t.
htforward, this scheme performs very poorly in practice. The Eu
∆t and the errors are proportional to ∆t2 . We can significant
cost either
by increasing the expansion order and thus the acc
Just a taylor expansion to order ∆t.
Errors
are proportional
to ∆t^2.
ctly using
a low-oder
scheme.
We now compare and contrast a po
rder leapfrog integrator, which is heavily used in collisionless N me, which has become the integrator of choice for collisional app
Time Integration:
We can significantly improve on this by increasing the expansion
order (accuracy) or by integrating a ‘near-by’ Hamiltonian
exactly using a low-order scheme.
State-of-the-art is the Hermite 4th order (collisional).
Leapfrog (collisionless).
Leapfrog is a Symplectic integrator. Exactly solve an
approximate Hamiltonian. As a consequence, the
numerical time evolution preserves certain conserved
quantities exactly, such as the total angular
momentum.
Leapfrog
integrator in
IDL:
pro leapfrog, x, v, xsol, vsol, dt, F_DERIVATIVE = FdXdT
;+
; NAME:
;
leapfrog
;
; PURPOSE:
;
Applies the second-order leapfrog method to solve ODE system,
;
giving a single step in evolution of the ODE solution trajectory.
;
; CALLING:
;
leapfrog, x, v, xsol, vsol, dt, F_DERIV=
;
; INPUTS:
;
x & v = array of initial conditions for ODE at time Tx.
;
dt
= time step desired.
;
; KEYWORDS:
;
F_DERIVATIVE = string, name of the function giving derivative array,
;
the right hand side of ODE system (default is "FdXdT").
;
Form is:
;
dXdT = FdXdT( xhalf )
;
; OUTPUTS:
;
xsol and vsol= solution of ODE at new time dt.
;
; PROCEDURE:
;
The 2-order leapfrog method
; HISTORY:
;h = dt/2.0D
xhalf
ahalf
vsol
xsol
=
=
=
=
dblarr(6)
dblarr(6)
dblarr(6)
dblarr(6)
xhalf
ahalf
vsol
xsol
=
=
=
=
x + v*h
call_function( FdXdT, xhalf, v)
v + ahalf*dt
x + h * (v + vsol)
END
Intro to Numerical Simulations - N-Body Simulations
In N-body approach, one follows orbits of representative mass elements, aka particles.
- Start with initial positions and velocities of particles.
- Compute accelerations for each particle.
- For each time step, advance each particle
- Repeat
The accurate time integration of close encounters is the most difficult part of collisional Nbody methods, while for collisionless N-body methods force softening alleviates this
problem substantially.
N-Body
Simulations
-Gravitational
Softening
Intro to Numerical Simulations - N-Body Simulations
If time step is too big or if particles get too close together, acceleration errors can be large.
In N-body approach, one follows orbits of representative mass elements, aka particles.
N
xj
- Start with initial positions ḡ
and=
velocities
of
particles.
Gm
i
i
i
- Compute accelerations for each particle.
|xj
xi
xi |3
- For each time step, advance each particle
The softening parameter turns particles from infinite point sources into ‘softened’ objects with finite radius.
- Repeat
ḡi =
N
Gmi
i
(|xj
xj xi
xi |2 + 2 )3/2
The accurate time integration of close encounters is the most difficult part of collisional Nsoftening parameter
body methods, while for collisionless N-body methods force softening alleviates this
problem substantially.
(formally this form is know as a ‘Plummer potential’)
N-Body Simulations -- Resolution
An N-body simulation has several different resolution limits:
Force Resolution: Set by gravitational softening.
Determines smallest physical scales on which simulation is reliable.
Mass Resolution: Set by particle mass.
Determine minimum mass scale which can be studied
Time Resolution: Set by time step.
Needs to match force softening!
High force resolution requires high time resolution
Because time steps are finite, its possible to get large integration errors if particles get close.
-> gravitational softening reduces this problem.
NFW paper:
Hydrodynamics
Hydrodynamics describes the time-dependent or stationary flow of fluids or gases.
Euler’s Equations (Conservation of Mass/Momentum)
Hydrodynamic simulation of a supersonic jet-stream
injected into a homogeneous medium.