galalactic dynamics- orbits in a logarithmic dark matter halo potential

Numerical Study In Galactic DynamicsOrbits in a Logarithmic Dark Matter Halo Potential

By: Pritam Kalbhor1

Guide: Dr. Kanak Saha2

1Department of Physics,University Pune,

Pune 411 007.

2Assistant Professor,Inter-University Centre for Astronomy and Astrophysics (IUCAA),

Pune 411 007.

P. Kalbhor (University of Pune) Galactic Dynamics Guided by Dr.K.Saha 1 / 29

Table of Contents


Introduction

Dark matter halos play a key role in current models of galaxyformation and evolution. It is important to understand orbits of starsin a galaxy in order to study evolution of galaxy.

We are going to use logarithmic model of potential forapproximating the potential of dark matter halo.We are using numerical method to solve differential equation ofmotion.Study orbits in static logarithmic potential and then rotatingpotential.


Numerical methods

To solve first order differential equation dydx = f(y, x) we use different

numerical methods

Euler’s method

y(x+ h) = f(x0) + hf ′(x0)

It is just truncation of Taylor series after first order term in h

Very bad method if we want stability in solution.Truncation error is of the order of h2. Not enough for stablesolution over time.After performing N steps error in the solution is increased to(Nh2)


Leapfrog method

Velocities are first calculated at time t+ h/2, these are used tocalculate the positions at time t+ h. In this way, the velocitiesleap over the positions, then the positions leap over the velocities.Initially x0(t = t0) and v0(t0 − h/2) are given, then Leap-Frogalgorithm is

v(t+ h/2) = v(t− h/2) + a(t)tr(t+ h) = r(t) + v(t+ h/2)t

It is second order accurate, better than the Euler’s method.Error after each time step is proportional to the O(h3).The main feature of Leapfrog method is time reversibility.


4th order Runge-Kutta’s methodUses slopes at multiple points in each time step to find solution atnext step.Fourth order accurate and error per step is of the order of h5,better than Leapfrog.It is Truncation of fifth order term of h in Taylor’s seriesexpansion.This method is fairly accurate in methods of Runge Kutta family.


Relative Error in above methods

I also worked Halley’s comet traejectory using Leapfrog and RK4 tocompare.


Logarithmic Potential

Φ =v2

0

2ln

(R2

c +R2 +z2

q2

)

Potential is Axisymmetric about z-axis.


Rc, the Core Radius defines how density is distributed near centerof galaxy.

q defines how flat the potential is.

v0, the circular velocity at R Rc matches with observations.


Stationary Points in Gravitational potentialFor multivariable functions we come across Hessiam Matrix to testthe stationary point. Hessian is nothing but matrix of secondorder differentiation of f(x1, x2, ..., xn)

Hij =∂2f

∂xi∂xj

Given a function f(x1, x2, ..., xn). condition for stationary point is~∇i = 0 for i = 0, 1, ..., n.The nature of the stationary point is determined by theeigenvalues of H

If all eigenvalues of H are ≥ 0 =⇒ local minimum.If all eigenvalues of H are ≤ 0 =⇒ local maximum.If H has both positive and negative eigenvalues, the stationarypoint is a saddle point.


Stationary Points in Gravitational potential (cont.)

Poisson Equation is ∇2Φ = 4πGρ ≥ 0 for for any mass distribution.This says that Gravitational potential has no maximum.

Density distributionStarting from Poisson’s Equation

ρL(R, z) =v2

0

4πGq2Φ

(2q2Φ + 1)R2

c +R2 + (2− q−2Φ )z2

(R2c +R2 + z2q−2

Φ )2

This density is physically valid only for qΦ > 0.7. Since densitybecomes negative near z−axis for |z| & 7Rc.

This implies there is also maximum in logarithmic potential.


Equation of motion in Static Logarithmic Potential

Hamiltonian for axisymmetric potential is

H =12(p2

R + p2z

)+ Φeff (R, z)

Effective potential is Φeff ≡ Φ(R, z) + L2z

2R2

Lz is conserved quantity.Equations of motion are,

R = −∂2Φeff

∂R2

z = −∂2Φeff

∂z2


Effective Potential

Φeff =v2

0

2ln

(R2

c +R2 +z2

q2

)+

L2z

2R2

Equipotential contours for Φeff at Rc = 1, Lz = 0.2, v0 = 1 is shown nfigure


Minimun in effective potential

Condition for mininum of the potential is ∇Φeff = 0

∂Φeff

∂R=∂Φ∂R− L2

z

R3= 0 ;

∂Φ∂z

= 0

The second condition satisfies anywhere in the equatorial plane and

first condition satisfies at Rmin =√

Lz2v0

(Lzv0

+√

L2z

v20

+ 4R2c

)


Some Concepts

Zero Velocity Curve (ZVC)

Curve at which velocity of the particle becomes zero for given fixedenergy. Outside region is forbidden, since particle cannot cross ZVCwith energy less than energy of ZVC.


Contants of motion and Integrals of motion

Contants of motion satisfies C[x(t1), v(t1); t1] = C[x(t2), v(t2); t2] forany t1 and t2While Integral of motion satisfies I[x(t1), v(t2)] = I[x(t2), v(t2)]

Surface of SectionAlso called as Poincare Section. It is 2 dimentional section ofmultidimensional phase space.

When intersection of the orbit with the (R, pR) plane occurs onreasonably smooth curves rather than in the entire allowed regionin the plane, this suggests that there could be some other integralsof motion.


Orbits in a Static Logarithmic Potential

Three regions for different orbits.

Orbits in equatorial plane.

Orbits in a meridional plane with R Rc and Lz = 0

Orbits in a meridional plane with R Rc and finite Lz

Orbits in equatorial plane.z = 0 is called as equatorial plane. In this plane we get effectivepotential to be

Φeff(z=0) =v2

0

2ln(R2

c +R2)

+L2

z

2R2

Potential in equatorial plane is dependent on R only.


Orbits in a meridional plane with Lz = 0Box orbits are possible for R Rc and zero angular momentum in zdirection. At high enough energy orbit transforms into planer looporbit.


Figure: loop orbit, E=1.22 Figure: Surface of section


Orbits in a meridional plane with finite Lz

For R Rc and finite Lz we get 3 dimentional loop orbit. It is plottedin R-z plane generally since angular velocity is constant so it reducesthe one degree of freedom.


Rotating Potential

Rotating potentials are (barred, spiral or triaxial object) are commenin astrophysics.

We use slowly rotating planer logarithmic potential.

Planer logarithmic potential is v202 ln

(R2

c + x2 + y2

q2

)Angular velocity of rotation is Ωb called as pattern speed.

Hamiltonian is HJ = H − ~Ωb · ~L

HJ is an integral of motion called as Jacobi integral or Jacobienergy.

EJ =12

∣∣∣~r∣∣∣2 + Φeff

Where, Φeff = Φ(~r)− 12

∣∣∣ ~Ωb × ~r∣∣∣2


We have been shown EJ is constant over time numerically also effectivepotential in rotating frame.


Equations of motion in rotating logarithmic potential

When we find Hamiltonian equations for give rotating potential we gotthese equations.

~x = −∇xΦ + Ω2bx+ 2Ωby

~y = −∇yΦ + Ω2by − 2Ωbx

~z = −∇zΦ

Note that we choosen axis of rotation along z-axis. We solved theseequations numerically using RK-4 method.


Choosing pattern speed

Since we are using slowly rotating potential, we can choose patternspeed to be 5-10% of the maximum value of angular velocity Ω.

Ω(R) =vc

R=

v0

(√R2

c +R2)


Orbits in rotating potential


Conclusion

We studied Numerical methods to deal with orbital dynamics.There are two main types of orbits (Box and Loop) possible.We also looked into Surface of sections for each orbit whichsuggests that there must be some other isolating integral whichmay not be expresible in terms of phase space coordinates.We studied how oribits changes under constant rotation oflogarithmic potential.All these study is the basic for modelling the galaxy, sinceunderstanding the orbits is necessary element to study dynamicsof galaxies. This study will further going to help us to studyevolution of galaxies.


Thank You !


galalactic dynamics- orbits in a logarithmic dark matter halo potential

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