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Potential of Milky Way Potential of Milky Way given analytically given analytically XI B Bulgarian- ulgarian-S Serbian erbian A Astronomical stronomical C Conference onference XI 14-18.05.2018. - Belogradchik, 14-18.05.2018. -

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Potential of Milky Way Potential of Milky Way given analytically given analytically

XI XI B Bulgarian- ulgarian-S Serbian erbian A Astronomical stronomical C Conference

  • nference

14-18.05.2018. - Belogradchik, 14-18.05.2018. - Belogradchik, Milan Stojanović, Slobodan Ninković, Milan Stojanović, Slobodan Ninković, Nemanja Martinović, Miljana D. Jovanović, Nemanja Martinović, Miljana D. Jovanović, Gabrijela Marković Gabrijela Marković mstojanovic@aob.rs mstojanovic@aob.rs

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Potential of Milky Way Potential of Milky Way

given analytically given analytically

XI XI B Bulgarian- ulgarian-S Serbian erbian A Astronomical stronomical C Conference

  • nference

14-18.05.2018. - Belogradchik, 14-18.05.2018. - Belogradchik, Milan Stojanović, Slobodan Ninković, Milan Stojanović, Slobodan Ninković, Nemanja Martinović, Miljana D. Jovanović, Nemanja Martinović, Miljana D. Jovanović, Gabrijela Marković Gabrijela Marković mstojanovic@aob.rs mstojanovic@aob.rs

A NEW A NEW

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 3/23

Introduction Introduction

  • Milky Way is spiral galaxy, type: Sbc

Milky Way is spiral galaxy, type: Sbc

  • ~2 x 10

~2 x 1011

11 stars + dm,

stars + dm,

  • Diameter 30-55 kpc

Diameter 30-55 kpc

  • Sun’s distance 8.1

Sun’s distance 8.1 ± ± 0.4 kpc 0.4 kpc

  • Escape velocity 550 km/s

Escape velocity 550 km/s

❑⊙❑

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 4/23

Introduction Introduction

  • Milky Way is spiral galaxy, type: Sbc

Milky Way is spiral galaxy, type: Sbc

  • ~2 x 10

~2 x 1011

11 stars + dm

stars + dm

  • Diameter 30-55 kpc

Diameter 30-55 kpc

  • Sun’s distance 8.1

Sun’s distance 8.1 ± ± 0.4 kpc 0.4 kpc

  • Escape velocity 550 km/s

Escape velocity 550 km/s

❑⊙❑

  • Components:

Components:

– Disc (thin & thick)

Disc (thin & thick)

– Bulge

Bulge

– Dark matter halo

Dark matter halo

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 5/23

Introduction Introduction

  • Milky Way is spiral galaxy, type: Sbc

Milky Way is spiral galaxy, type: Sbc

  • ~2 x 10

~2 x 1011

11 stars + dm

stars + dm

  • Diameter 30-55 kpc

Diameter 30-55 kpc

  • Sun’s distance 8.1

Sun’s distance 8.1 ± ± 0.4 kpc 0.4 kpc

  • Escape velocity 550 km/s

Escape velocity 550 km/s

  • Mass ??

Mass ?? 0.5x10 0.5x1012

12

2x10 2x1012

12

❑⊙❑

  • Components:

Components:

– Disc (thin & thick)

Disc (thin & thick)

– Bulge

Bulge

– Dark matter halo

Dark matter halo

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 6/23

Inspiration – Inspiration –

Iocco et al. 2015 Iocco et al. 2015 Li 2016 Li 2016

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 7/23

Inspiration Inspiration

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 8/23

Iocco et al. 2015 Iocco et al. 2015

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 9/23

Milky Way potential Milky Way potential Four subsystems Four subsystems

  • Potential is assumed to be stationary and

Potential is assumed to be stationary and axially simmetric axially simmetric

– Bulge (b) - Hernquist model

Bulge (b) - Hernquist model

– Innermost dark matter (idm) – spherical symmetry

Innermost dark matter (idm) – spherical symmetry

– Disc (d) –

Disc (d) – PASA, vol. 32, 2015 PASA, vol. 32, 2015

– Outer dark corona (odm) –

Outer dark corona (odm) – Open Ast. Vol. 26, 2017 Open Ast. Vol. 26, 2017

  • For each of them, potential is given

For each of them, potential is given analytically by using only elementary functions analytically by using only elementary functions

  • Model parameters are specifjed by fjtting an

Model parameters are specifjed by fjtting an assumed rotation curve. assumed rotation curve.

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 10/23

Innermost dm and bulge Innermost dm and bulge

  • We use (modifjcation of) Hernquist model

We use (modifjcation of) Hernquist model

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 11/23

Innermost dm and bulge Innermost dm and bulge

  • We use (modifjcation of) Hernquist model

We use (modifjcation of) Hernquist model if if r

r1

1 ,

, r r2

2 → 0

→ 0

M=11, r=2.05 M=11, r=2.05

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 12/23

Innermost dm and bulge Innermost dm and bulge

  • We use (modifjcation of) Hernquist model

We use (modifjcation of) Hernquist model if if r

r1

1 ,

, r r2

2 → 0

→ 0

M=11, r=2.05 M=11, r=2.05

bulge bulge 10.5<M<13

10.5<M<13 0.3<r<0.5 0.3<r<0.5 M in 10 M in 109

9 solar masses

solar masses r in kpc r in kpc

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 13/23

Disc Disc

  • Observational constraints

Observational constraints

(Deg & Widrow 2013) (Deg & Widrow 2013)

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 14/23

Disc Disc

  • Observational constraints

Observational constraints

(Deg & Widrow 2013) (Deg & Widrow 2013)

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 15/23

Disc Disc

  • Observational constraints

Observational constraints

(Deg & Widrow 2013) (Deg & Widrow 2013)

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 16/23

Disc Disc

  • Observational constraints

Observational constraints

(Deg & Widrow 2013) (Deg & Widrow 2013)

  • We start with famous Miyamoto-Nagai formula

We start with famous Miyamoto-Nagai formula

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 17/23

Disc Disc

  • Observational constraints

Observational constraints

(Deg & Widrow 2013) (Deg & Widrow 2013)

  • We start with famous Miyamoto-Nagai formula

We start with famous Miyamoto-Nagai formula (Ninkovic 2015) (Ninkovic 2015)

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 18/23

Disc Disc

  • Three parameters are important:

Three parameters are important:

– Total mass

Total mass M M, exp. scale length R , exp. scale length Rd

d and scale len. b

and scale len. b

  • Since disc is fmattened it must be

Since disc is fmattened it must be a>>b a>>b and and γ γ1

1,

,γ γ2

2<0.5.

<0.5.

  • We have chosen

We have chosen a+b=2.1 a+b=2.1R Rd

d

and and 7.8x10 7.8x1010

10<M<9x10

<M<9x1010

10

  • And

And γ γ1

1=-0.3, γ

=-0.3, γ2

2=0.3, c

=0.3, c1

1=c

=c2

2=R

=Rd

d and

and a≈2R a≈2Rd

d

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Outer dark corona Outer dark corona

  • The assumed model density & cumulative mass

The assumed model density & cumulative mass

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 20/23

Outer dark corona Outer dark corona

  • The assumed model density & cumulative mass

The assumed model density & cumulative mass

  • We can now obtain potential

We can now obtain potential (Ninkovic 2017) (Ninkovic 2017)

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14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 21/23

What? (do we get when we What? (do we get when we combine everything) combine everything)

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Why? Why?

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Why? Why? Thank you! Thank you!