Measuring the Milky Way potential without dynamical models

Full text

Jorge Peñarrubia (IAA-CSIC)
in coll. w/ Sergey Koposov & Matt Walker
Measuring the Milky Way potential
without dynamical models
Ringberg 12th April 2012
Friday, 13 April 2012

Tidal Streams as tracers of the potential
Odenkirchen+09
Stars moving on very similar orbits
Strong constraints on MW potential
PAL 5
GD1
Koposov, Rix & Hogg 2010
Friday, 13 April 2012

ISSUES:
Stars moving on very similar orbits
Strong constraints on MW potential
JP, Martinez-Delgado, Rix+05
Mon
★ Complexity in the stream
★ Non-uniform maps
★ Membership probabilities
★ Phase-space mixing with age
Tidal Streams as tracers of the potential
Friday, 13 April 2012

ISSUES:
Stars moving on very similar orbits
Strong constraints on MW potential
JP, Martinez-Delgado, Rix+05
Mon
★ Complexity in the stream
★ Non-uniform maps
★ Membership probabilities
★ Phase-space mixing with age
Tidal Streams as tracers of the potential
Friday, 13 April 2012

Stars moving on very similar orbits
Strong constraints on MW potential
Analysis of integrals of motion!
ISSUES:
JP, Benson, Martinex-Delgado & Rix et al. 2006
Tidal Streams as tracers of the potential
★ Complexity in the stream
★ Non-uniform maps
★ Membership probabilities
★ Phase-space mixing with age
Friday, 13 April 2012

Stars moving on very similar orbits
Strong constraints on MW potential
Analysis of integrals of motion!
ISSUES:
JP, Benson, Martinex-Delgado & Rix et al. 2006
Tidal Streams as tracers of the potential
★ Complexity in the stream
★ Non-uniform maps
★ Membership probabilities
★ Phase-space mixing with age
requires 6D phase-space information
Friday, 13 April 2012

THE IDEA
Peñarrubia, Koposov & Walker (2012)
Φ
r
r
E=1
2v2+Φ(r)
f(E)=δ(E−E0)
H≡−
!f(E) ln f(E)dE =0
Friday, 13 April 2012

THE IDEA
Peñarrubia, Koposov & Walker (2012)
Φ
r
r
E=1
2v2+Φ(r)
∆H>0
˜
f(E)
Friday, 13 April 2012

THE IDEA
Peñarrubia, Koposov & Walker (2012)
Φ
r
r
E=1
2v2+Φ(r)
∆H>0
˜
f(E)
Friday, 13 April 2012

THE IDEA
Peñarrubia, Koposov & Walker (2012)
Φ
r
r
E=1
2v2+Φ(r)
∆H>0
˜
f(E)
“Biases in the calculus of orbital energy yields and
increase in the entropy of the energy distribution”
Friday, 13 April 2012

Entropy
Theorem:
“The entropy measured for stellar systems with separable energy distributions
increases under the presence of biases in the theoretical modelling of the
host’s gravity”
ε=−E+Φ∞
˜
f(ε,r)=f[ε−δΦ(r),r]=f[ε−δΦ(r)]g(r)
Separability condition
Relative energy
˜ε(r)=ε(r)+δΦ(r)
Energy Bias
Measured energy distribution:
˜
f(ε)=!f(ε−δΦ(r))g(r)d3r≈
f(ε)!"1−δΦ(r)f!(ε)
f(ε)+δΦ2(r)
2
f!! (ε)
f(ε)#g(r)d3r=
f(ε)!1−"δΦ#f!(ε)
f(ε)+!δΦ2"
2
f!! (ε)
f(ε)".
Friday, 13 April 2012

Entropy
Measured Entropy
˜
H=−!dε˜
f(ε) ln[ ˜
f(ε)] =
H+!δΦ"!dεf!(ε)[1 + ln f(ε)]
−
!δΦ"2
2!dεf(ε)"f!(ε)
f(ε)#2
−
!δΦ2"
2!dεf##(ε)[1 + ln f(ε)].
1) !dεf!(1 + ln f)="fln f#Φ∞
0=0,
2) !dεf!! (1 + ln f)=−!dεf"f!
f#2.
˜
H=H+!δΦ2"−!δΦ"2
2!dεf(ε)"f!(ε)
f(ε)#2≡H+σ2
Φ
2σ2
ε
≥0
Theorem:
“The entropy measured for stellar systems with separable energy distributions
increases under the presence of biases in the theoretical modelling of the
host’s gravity”
Friday, 13 April 2012

Entropy
Measured Entropy
˜
H=H+!δΦ2"−!δΦ"2
2!dεf(ε)"f!(ε)
f(ε)#2≡H+σ2
Φ
2σ2
ε
• Entropy increases for
• Adding a constant value to the potential does not yield an increase in entropy
• Changes in entropy will be stronger for “cold” distributions
δΦ=δΦ(r)!=0
Theorem:
“The entropy measured for stellar systems with separable energy distributions
increases under the presence of biases in the theoretical modelling of the
host’s gravity”
Friday, 13 April 2012

Tests
f(ε)=1/!2πσ2
εexp[−(ε−εorb)2/(2σ2
ε)
Unbiased (true) energy distribution
Φ(r)=Φ0ln(d2
0+r2)
Unbiased (true) Potential
Friday, 13 April 2012

Tests
δϕ0=+0.01
δϕ0=-0.01
Friday, 13 April 2012

Tests
f(ε)=1/!2πσ2
εexp[−(ε−εorb)2/(2σ2
ε)
Unbiased (true) energy distribution
Φ(r)=Φ0ln(d2
0+r2)
Unbiased (true) Potential
rapo =5d0
σε= 10−3Φ0
HGauss =1/2[ln(2πσ2
ε) + 1]
Friday, 13 April 2012

Tests
Unbiased (true) energy distribution
Unbiased (true) Potential
rapo =5d0
σε= 10−3Φ0
∆H=1/2(σΦ/σε)2
f(ε)=1/!2πσ2
εexp[−(ε−εorb)2/(2σ2
ε)
Φ(r)=Φ0ln(d2
0+r2)
Friday, 13 April 2012

Energy biases
1. Potential parameters
2. Functional form of the potential
3. Gravity model
˜
Φ(r)=2Φ0!y+y3
3+y5
5+... +"(N−1)/2
k=0 y2k+1/(2k+ 1)#+Φ0ln d2
0
limN→∞ ˜
Φ=Φ0ln(r2+d2
0)=Φ
Friday, 13 April 2012

Energy biases
1. Potential parameters
2. Functional form of the potential
3. Gravity model
˜
Φ(r)=2Φ0!y+y3
3+y5
5+... +"(N−1)/2
k=0 y2k+1/(2k+ 1)#+Φ0ln d2
0
limN→∞ ˜
Φ=Φ0ln(r2+d2
0)=Φ
Entropy can be used to distinguish between different
potential parametrizations
Friday, 13 April 2012

Energy biases
1. Potential parameters
2. Functional form of the potential
3. Gravity model
Example 1: Dirac’s cosmology
ED=H2
0t2!1
2"dr
dt #2
+G
G0
Φ(r)−"dr
dt
·
r
t#$+1
2H2
0r2;
Gmpme
e2!10−39 !e2
mec3t;
at t=H0-1
δΦD=±[−H0(dr/dt ·r)+1/2H2
0r2].
Lynden-Bell (1982)
Friday, 13 April 2012

Energy biases
1. Potential parameters
2. Functional form of the potential
3. Gravity model
Example 1: Dirac’s cosmology
ED=H2
0t2!1
2"dr
dt #2
+G
G0
Φ(r)−"dr
dt
·
r
t#$+1
2H2
0r2;
Gmpme
e2!10−39 !e2
mec3t;
at t=H0-1
δΦD=±[−H0(dr/dt ·r)+1/2H2
0r2].
Lynden-Bell (1982)
Friday, 13 April 2012

Energy biases
1. Potential parameters
2. Functional form of the potential
3. Gravity model
Example 2: QMOND
gM=gNν(r)≡gN!1
2+"1
4+a0
gN#,
gN=−GM(<r)/r2,
ΦM(r)=!∞
rgM(r")r
.";
Friday, 13 April 2012

Energy biases
1. Potential parameters
2. Functional form of the potential
3. Gravity model
Example 2: QMOND
gM=gNν(r)≡gN!1
2+"1
4+a0
gN#,
gN=−GM(<r)/r2,
ΦM(r)=!∞
rgM(r")r
.";
Friday, 13 April 2012

Energy biases
1. Potential parameters
2. Functional form of the potential
3. Gravity model
Example 3: f(R) gravity theories
A=!d4x√−g[f(R)+Lm];
f(R)=f0Rn
Ricci curvature
f(R)=R+2Λ
ΛCDM:
ΦR=1/2(ΦN+Φc)
Φc(r)=−4πG!1
r"r
0dr!ρ(r!)r!2#r
rc$β
+"∞
r
dr!ρ(r!)r!#r
rc$β%.
β=0
Newton
β=0.82
Fit rotation curves with NO DM
Cappozziello et al (2007)
Friday, 13 April 2012

Energy biases
1. Potential parameters
2. Functional form of the potential
3. Gravity model
Example 3: f(R) gravity theories
A=!d4x√−g[f(R)+Lm];
f(R)=f0Rn
Ricci curvature
f(R)=R+2Λ
ΛCDM:
Cappozziello et al (2007)
ΦR=1/2(ΦN+Φc)
Φc(r)=−4πG!1
r"r
0dr!ρ(r!)r!2#r
rc$β
+"∞
r
dr!ρ(r!)r!#r
rc$β%.
β=0
Newton
β=0.82
Fit rotation curves with NO DM
Friday, 13 April 2012

The Minimum Entropy Method
1. Phase-space catalogue: {X,Y,Z,Vx,Vy,Vz}i ; i=1,2,...,N*
2. Calculate Ei=1/2(Vx2 + Vy2+Vz2)i + Φ(Xi,Yi,Zi)
3. Calculate f(E), H
4. Look for Φ that minimizes H
it is a simple statistical technique for constraining simultaneously the MW
gravitational potential and testing different gravity theories directly from
phase-space surveys and without adopting dynamical models.
Friday, 13 April 2012

Tidal debris
the energy distribution of tidal debris is not separable
JP+10
JP+06, Eyre & Binney 08
Friday, 13 April 2012

Di=!fi(ε) ln "fi(ε)
f(ε)) #dε≡−Hi+Hc,i;
Kullback-Leiblar (or KL) divergence
JP, Benson, Rix et al. 06
Hc,i =−!fi(ε) ln f(ε)dε
Crossed entropy
Distributions are
separable if Di=0
Tidal debris
the energy distribution of tidal debris is not separable JP+06, Eyre & Binney 08
Friday, 13 April 2012

Tidal debris
H=−!f(ε) ln f(ε)dε=−α!fl(ε) ln f(ε)dε−(1 −α)!ft(ε) ln f(ε)dε
≡αHl+ (1 −α)Ht+αDl+ (1 −α)Dt≡#H$l,t +#D$l,t;
H=!H"l,t +!D"l,t;
minimum if δΦ=0 maximum if δΦ=0
maximum in H δΦ~0
!H"!
l=!D"!
l=0
minimum in H δΦ~0
!H"!
l=−!D"!
l
Friday, 13 April 2012

Summary
• “The true Milky Way potential is that that minimizes the entropy
measured for stellar systems with separable energy distributions”
• Best targets: Tidal debris of satellites/clusters with low dynamical
masses
• Future work: Gaia errors? MW background?
Friday, 13 April 2012

Friday, 13 April 2012