GalRotpy: an educational tool to understand and parametrize the rotation   curve and gravitational potential of disk-like galaxies

Andr\'es Granados; Daniel Torres; Leonardo Casta\~neda; Lady-J.; Henao-O.; Santiago Vanegas

arXiv:1705.01665·astro-ph.GA·August 3, 2020

GalRotpy: an educational tool to understand and parametrize the rotation curve and gravitational potential of disk-like galaxies

Andr\'es Granados, Daniel Torres, Leonardo Casta\~neda, Lady-J., Henao-O., Santiago Vanegas

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TL;DR

GalRotpy is an educational Python tool that helps users understand galaxy rotation curves by visualizing mass components and fitting parameters using MCMC, demonstrated on real galaxy data.

Contribution

It introduces a new Python-based visual and fitting tool for galaxy rotation curves, integrating multiple mass models and MCMC fitting in an educational context.

Findings

01

Successfully characterized two galaxies, NGC6361 and M33.

02

Results for M33 align with existing literature.

03

Provides an interactive way to learn galaxy dynamics.

Abstract

\textbf{GalRotpy} is an educational \verb+Python3+-based visual tool, which is useful to undestand how is the contribution of each mass component to the gravitational potential of disc-like galaxies by means of their rotation curve. Besides, \textbf{GalRotpy} allows the user to perform a parametric fit of a given rotation curve, which relies on a MCMC procedure implemented by using \verb+emcee+ package. Here the gravitational potential of disc-like galaxies is built from the contribution of a Miyamoto-Nagai potential model for the bulge/core and the thin/thick disc, an exponential disc, together with the NFW (Navarro-Frenk- White) potential or the Burkert (cored density profile) potential for the Dark Matter halo, where each contribution is implemented by using \verb+galpy+ package. We summarize the properties of each contribution to the rotation curve involved, and then describe how…

Tables4

Table 1. Table 1: Set of parameters involved in each of the contributions considered in GalRotpy . The given range for each parameter attempts to be a guide about which values may be used to run this program, particularly for dwarf galaxies and Milky way-like galaxies. When our results are far from the values presented in this table, most likely the results are unphysical.

Component

Parameter Range

Units

Bulge I

a = 0

0.0 < b < 0.5

0.1 < M < 1.0

k ​ p ​ c

k ​ p ​ c

10^{10} ​ M_{⊙}

Bulge II

0.01 < a < 0.05

0.5 < b < 1.5

1 < M < 5

k ​ p ​ c

k ​ p ​ c

10^{10} ​ M_{⊙}

Thin Disc

1 < a < 10

0.1 < b < 1.0

0.5 < M < 1.5

k ​ p ​ c

k ​ p ​ c

10^{11} ​ M_{⊙}

Thick Disc

1 < a < 10

0.1 < b < 15.0

0.5 < M < 1.5

k ​ p ​ c

k ​ p ​ c

10^{11} ​ M_{⊙}

Exponential Disc

2 < h_{r} < 6

1 < Σ_{0} < 15

k ​ p ​ c

10^{2} ​ M_{⊙} / p ​ c^{2}

NFW - Halo

0.1 < a < 30

0.1 < M_{0} < 10

k ​ p ​ c

10^{11} ​ M_{⊙}

Burkert - Halo

2 < a < 38

0.1 < ρ_{0} < 10

k ​ p ​ c

10^{6} ​ M_{⊙} / k ​ p ​ c^{3}

Table 2. Table 2: Set of parameters obtained using GalRotpy with their corresponding uncertainties for three different models for the rotation curve of NGC6361. For each dark halo distribution, the halo’s total mass M h subscript 𝑀 ℎ M_{h} and the concentration parameter X 𝑋 X are given for Δ c = 97.2 subscript Δ 𝑐 97.2 \Delta_{c}=97.2 .

Model I

Component

Parameter

Fit

68%

95%

NFW-Halo

a ​ (k ​ p ​ c)

M_{0} (\times 10^{11} M_{⊙})

ρ_{0} (\times 10^{8} M_{⊙} / k p c^{3})

X (\times 10)

M_{h} (\times 10^{11} M_{⊙})

4.93

3.04

2.02

5.27

9.14

{}_{- 0.19}^{+ 0.21}

{}_{- 0.15}^{+ 0.16}

{}_{- 0.14}^{+ 0.15}

{}_{- 0.14}^{+ 0.14}

{}_{- 0.38}^{+ 0.40}

{}_{- 0.37}^{+ 0.42}

{}_{- 0.28}^{+ 0.33}

{}_{- 0.27}^{+ 0.31}

{}_{- 0.27}^{+ 0.28}

{}_{- 0.72}^{+ 0.82}

Model II

Component

Parameter

Fit

68%

95%

Exponential Disc

h_{r} ​ (k ​ p ​ c)

Σ_{0} (\times 10^{3} M_{⊙} / p c^{2})

M_{⋆} (\times 10^{10} M_{⊙})

1.05

2.84

1.97

{}_{- 0.08}^{+ 0.08}

{}_{- 0.11}^{+ 0.11}

{}_{- 0.27}^{+ 0.26}

{}_{- 0.16}^{+ 0.15}

{}_{- 0.23}^{+ 0.22}

{}_{- 0.55}^{+ 0.49}

Burkert-Halo

a ​ (k ​ p ​ c)

ρ_{0} (\times 10^{8} M_{⊙} / k p c^{3})

X (\times 10)

M_{h} (\times 10^{12} M_{⊙})

6.79

1.24

4.51

1.49

{}_{- 0.96}^{+ 1.25}

{}_{- 0.25}^{+ 0.32}

{}_{- 0.40}^{+ 0.37}

{}_{- 0.28}^{+ 0.43}

{}_{- 1.69}^{+ 2.91}

{}_{- 0.45}^{+ 0.69}

{}_{- 0.70}^{+ 0.81}

{}_{- 0.46}^{+ 1.14}

Model III

Component

Parameter

Fit

68%

95%

Thin Disc

a (\times 10^{- 1} k p c)

b (\times 10^{- 1} k p c)

M_{⋆} (\times 10^{10} M_{⊙})

6.56

6.64

2.42

{}_{- 4.46}^{+ 4.48}

{}_{- 4.49}^{+ 4.46}

{}_{- 0.28}^{+ 0.28}

{}_{- 6.22}^{+ 6.58}

{}_{- 6.30}^{+ 6.58}

{}_{- 0.55}^{+ 0.59}

Burkert-Halo

a ​ (k ​ p ​ c)

ρ_{0} (\times 10^{8} M_{⊙} / k p c^{3})

X (\times 10)

M_{h} (\times 10^{12} M_{⊙})

5.70

1.53

4.87

1.11

{}_{- 0.61}^{+ 0.73}

{}_{- 0.26}^{+ 0.31}

{}_{- 0.33}^{+ 0.35}

{}_{- 0.15}^{+ 0.19}

{}_{- 1.12}^{+ 1.62}

{}_{- 0.47}^{+ 0.68}

{}_{- 0.63}^{+ 0.71}

{}_{- 0.25}^{+ 0.46}

Table 3. Table 3: Set of parameters obtained using GalRotpy with their corresponding uncertainties for two models for the rotation curve of M33. For each dark halo distribution the halo’s total mass M h subscript 𝑀 ℎ M_{h} and the concentration parameter X 𝑋 X are given for Δ c = 97.2 subscript Δ 𝑐 97.2 \Delta_{c}=97.2 .

Model I

Component

Parameter

Fit

68%

95%

Exponential Disc

h_{r} ​ (k ​ p ​ c)

Σ_{0} (\times 10^{2} M_{⊙} / p c^{2})

M_{⋆} (\times 10^{9} M_{⊙})

1.52

2.50

3.61

{}_{- 0.11}^{+ 0.10}

{}_{- 0.43}^{+ 0.37}

{}_{- 0.91}^{+ 0.96}

{}_{- 0.23}^{+ 0.20}

{}_{- 0.90}^{+ 0.66}

{}_{- 1.74}^{+ 1.89}

NFW-Halo

a (\times 10 k p c)

M_{0} (\times 10^{11} M_{⊙})

ρ_{0} (\times 10^{6} M_{⊙} / k p c^{3})

X (\times 10)

M_{h} (\times 10^{11} M_{⊙})

1.46

2.37

6.05

1.37

4.16

{}_{- 0.29}^{+ 0.42}

{}_{- 0.55}^{+ 0.91}

{}_{- 2.13}^{+ 2.96}

{}_{- 0.22}^{+ 0.24}

{}_{- 0.72}^{+ 1.11}

{}_{- 0.49}^{+ 1.02}

{}_{- 0.91}^{+ 2.45}

{}_{- 3.55}^{+ 6.88}

{}_{- 0.42}^{+ 0.48}

{}_{- 1.21}^{+ 2.86}

Model II

Component

Parameter

Fit

68%

95%

Exponential Disc

h_{r} ​ (k ​ p ​ c)

Σ_{0} (\times 10^{2} M_{⊙} / p c^{2})

M_{⋆} (\times 10^{9} M_{⊙})

1.35

4.74

5.47

{}_{- 0.11}^{+ 0.11}

{}_{- 0.15}^{+ 0.15}

{}_{- 0.82}^{+ 0.88}

{}_{- 0.21}^{+ 0.21}

{}_{- 0.30}^{+ 0.31}

{}_{- 1.52}^{+ 1.71}

Burkert-Halo

a ​ (k ​ p ​ c)

ρ_{0} (\times 10^{7} M_{⊙} / k p c^{3})

X (\times 10)

M_{h} (\times 10^{11} M_{⊙})

6.61

2.66

2.52

2.39

{}_{- 0.71}^{+ 0.84}

{}_{- 0.52}^{+ 0.63}

{}_{- 0.20}^{+ 0.22}

{}_{- 0.23}^{+ 0.29}

{}_{- 1.27}^{+ 1.71}

{}_{- 0.89}^{+ 1.32}

{}_{- 0.37}^{+ 0.43}

{}_{- 0.40}^{+ 0.61}

Table 4. Table 4: Set of parameters reported by López Fune, Salucci, and Corbelli ( 2017 ) . Here Δ c = 97.2 subscript Δ 𝑐 97.2 \Delta_{c}=97.2 .

Model I

Component

Parameter

Fit

Uncertainty

Stellar-Gas-Halo

M_{⋆} (\times 10^{9} M_{⊙})

4.9

\pm 1.5

NFW-Halo

M_{h} (\times 10^{11} M_{⊙})

X

5.4

9.5

\pm 0.6

\pm .0.7

Model II

Component

Parameter

Fit

Uncertainty

Stellar-Gas-Halo

M_{⋆} (\times 10^{9} M_{⊙})

4.9

\pm 1.5

Burkert-Halo

a ​ (k ​ p ​ c)

ρ_{0} (\times 10^{6} M_{⊙} / k p c^{3})

M_{h}

9.5

12.3

3.0

\pm 0.6

\pm 1.0

\pm 0.8

Equations110

\nabla^{2} Φ (x) = 4 π Gρ (x),

\nabla^{2} Φ (x) = 4 π Gρ (x),

Φ (x) = - G \int d^{3} x^{'} \frac{ρ ( x ^{'} )}{∣ x - x ^{'} ∣},

Φ (x) = - G \int d^{3} x^{'} \frac{ρ ( x ^{'} )}{∣ x - x ^{'} ∣},

V_{c}^{2}(R)=R\frac{\partial\Phi}{\partial r}\bigg{|}_{r=R},

V_{c}^{2}(R)=R\frac{\partial\Phi}{\partial r}\bigg{|}_{r=R},

V_{c}^{2} = \frac{G}{R} M (\leq R),

V_{c}^{2} = \frac{G}{R} M (\leq R),

Φ = Φ_{T} = i \sum n Φ_{i},

Φ = Φ_{T} = i \sum n Φ_{i},

V_{c}^{2} = i \sum n V_{c (i)}^{2} .

V_{c}^{2} = i \sum n V_{c (i)}^{2} .

Φ_{P} (R, z) = - \frac{GM}{R ^{2} + z ^{2} + b ^{2}} = - \frac{GM}{r ^{2} + b ^{2}}

Φ_{P} (R, z) = - \frac{GM}{R ^{2} + z ^{2} + b ^{2}} = - \frac{GM}{r ^{2} + b ^{2}}

\nabla^{2} Φ (x) = 4 π G μ (R) δ (z),

\nabla^{2} Φ (x) = 4 π G μ (R) δ (z),

Φ_{K} (R, z) = - \frac{GM}{R ^{2} + ( a + ∣ z ∣ ) ^{2}},

Φ_{K} (R, z) = - \frac{GM}{R ^{2} + ( a + ∣ z ∣ ) ^{2}},

Φ_{M N} (R, z) = - \frac{GM}{R ^{2} + ( a + z ^{2} + b ^{2} ) ^{2}},

Φ_{M N} (R, z) = - \frac{GM}{R ^{2} + ( a + z ^{2} + b ^{2} ) ^{2}},

V_{c} (R) = R \frac{GM}{( R ^{2} + ( a + b ) ^{2} ) ^{3/2}}

V_{c} (R) = R \frac{GM}{( R ^{2} + ( a + b ) ^{2} ) ^{3/2}}

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{a ^{2}} + \frac{z ^{2}}{c ^{2}} = \frac{R ^{2}}{a ^{2}} + \frac{z ^{2}}{c ^{2}} = 1,

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{a ^{2}} + \frac{z ^{2}}{c ^{2}} = \frac{R ^{2}}{a ^{2}} + \frac{z ^{2}}{c ^{2}} = 1,

δ M (a) = 2 π Σ_{0} a δ a; δ Σ (a, R) = \frac{Σ _{0} δ a}{a ^{2} - R ^{2}},

δ M (a) = 2 π Σ_{0} a δ a; δ Σ (a, R) = \frac{Σ _{0} δ a}{a ^{2} - R ^{2}},

Σ (R) = \int_{R}^{\infty} d a \frac{Σ _{0} ( a )}{a ^{2} - R ^{2}} .

Σ (R) = \int_{R}^{\infty} d a \frac{Σ _{0} ( a )}{a ^{2} - R ^{2}} .

Σ_{0} (a) = - \frac{2}{π} \frac{d}{d a} \int_{a}^{\infty} d R \frac{R Σ ( R )}{R ^{2} - a ^{2}} .

Σ_{0} (a) = - \frac{2}{π} \frac{d}{d a} \int_{a}^{\infty} d R \frac{R Σ ( R )}{R ^{2} - a ^{2}} .

Σ_{k} (R) =

Σ_{k} (R) =

Φ_{k} (R, z) =

Φ (R, z) =

Φ (R, z) =

f_{1} (R^{'}, k, R, z) := Σ (R^{'}) J_{0} (k R^{'}) R^{'} J_{0} (k R) exp (- ∣ z ∣ k) .

f_{1} (R^{'}, k, R, z) := Σ (R^{'}) J_{0} (k R^{'}) R^{'} J_{0} (k R) exp (- ∣ z ∣ k) .

δ Φ = - 2 π G Σ_{0} (a) s i n^{- 1} (\frac{2 a}{+ + -}) δ a,

δ Φ = - 2 π G Σ_{0} (a) s i n^{- 1} (\frac{2 a}{+ + -}) δ a,

Φ (R, z) = 4 G \int_{0}^{\infty} f_{2} (R, z; a) d a

Φ (R, z) = 4 G \int_{0}^{\infty} f_{2} (R, z; a) d a

f_{2} (R, z; a) := s i n^{- 1} (\frac{2 a}{+ + -}) \frac{d}{d a} \int_{a}^{\infty} \frac{R ^{'} Σ ( R ^{'} ) d R ^{'}}{R ^{'2} - a ^{2}} .

f_{2} (R, z; a) := s i n^{- 1} (\frac{2 a}{+ + -}) \frac{d}{d a} \int_{a}^{\infty} \frac{R ^{'} Σ ( R ^{'} ) d R ^{'}}{R ^{'2} - a ^{2}} .

Φ (R, 0) = - 4 G \int_{0}^{\infty} \frac{d a}{R ^{2} - a ^{2}} \frac{d}{d a} \int_{a}^{\infty} \frac{R ^{'} Σ ( R ^{'} ) d R ^{'}}{R ^{'2} - a ^{2}} .

Φ (R, 0) = - 4 G \int_{0}^{\infty} \frac{d a}{R ^{2} - a ^{2}} \frac{d}{d a} \int_{a}^{\infty} \frac{R ^{'} Σ ( R ^{'} ) d R ^{'}}{R ^{'2} - a ^{2}} .

Σ_{d} (R) = Σ_{0} exp (- R / h_{r}),

Σ_{d} (R) = Σ_{0} exp (- R / h_{r}),

\int_{a}^{\infty} \frac{R ^{'} Σ _{0} exp ( - R ^{'} / h _{r} )}{R ^{'2} - a ^{2}} d R^{'} = Σ_{0} a K_{1} (a / h_{r})

\int_{a}^{\infty} \frac{R ^{'} Σ _{0} exp ( - R ^{'} / h _{r} )}{R ^{'2} - a ^{2}} d R^{'} = Σ_{0} a K_{1} (a / h_{r})

Φ_{E D} (R, 0) = - π G Σ_{0} R (I_{0} (y) K_{1} (y) - I_{1} (y) K_{0} (y)),

Φ_{E D} (R, 0) = - π G Σ_{0} R (I_{0} (y) K_{1} (y) - I_{1} (y) K_{0} (y)),

V_{c} (R) = 4 π G Σ_{0} h_{r} y^{2} (I_{0} (y) K_{0} (y) - I_{1} (y) K_{1} (y)),

V_{c} (R) = 4 π G Σ_{0} h_{r} y^{2} (I_{0} (y) K_{0} (y) - I_{1} (y) K_{1} (y)),

M_{d} = 2 π h_{r}^{2} Σ_{0},

M_{d} = 2 π h_{r}^{2} Σ_{0},

I = \int_{0}^{\infty} d r j (r)

I = \int_{0}^{\infty} d r j (r)

I (R) = 2 \int_{0}^{\infty} d z j (r)

I (R) = 2 \int_{0}^{\infty} d z j (r)

I (R) = 2 \int_{R}^{\infty} \frac{j ( r ) r d r}{r ^{2} - R ^{2}},

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Full text

GalRotpy: an educational tool to understand and parametrize the rotation curve and the gravitational potential of disc-like galaxies

Andrés Granados

[email protected]

Departamento de Física