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. - 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 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. - 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
Introduction Introduction ● Milky Way is spiral galaxy, type: Sbc Milky Way is spiral galaxy, type: Sbc ● ~2 x 10 ~2 x 10 11 stars + dm, 11 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 ❑⊙❑ 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 ● Components: ● ~2 x 10 Components: ~2 x 10 11 stars + dm 11 stars + dm – Disc (thin & thick) Disc (thin & thick) ● Diameter 30-55 kpc Diameter 30-55 kpc – Bulge Bulge ● Sun’s distance 8.1 Sun’s distance 8.1 ± ± 0.4 kpc 0.4 kpc – Dark matter halo Dark matter halo ● Escape velocity 550 km/s Escape velocity 550 km/s ❑⊙❑ 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 ● Components: ● ~2 x 10 Components: ~2 x 10 11 stars + dm 11 stars + dm – Disc (thin & thick) ● Diameter 30-55 kpc Disc (thin & thick) Diameter 30-55 kpc – Bulge Bulge ● Sun’s distance 8.1 Sun’s distance 8.1 ± ± 0.4 kpc 0.4 kpc – Dark matter halo Dark matter halo ● Escape velocity 550 km/s Escape velocity 550 km/s ❑⊙❑ ● Mass ?? Mass ?? 0.5x10 12 0.5x10 12 2x10 12 2x10 12 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 5/23
Inspiration – Inspiration – Iocco et al. 2015 Iocco et al. 2015 Li 2016 Li 2016 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 6/23
Inspiration Inspiration 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 7/23
Iocco et al. 2015 Iocco et al. 2015 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 8/23
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. 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 9/23
Innermost dm and bulge Innermost dm and bulge ● We use (modifjcation of) Hernquist model We use (modifjcation of) Hernquist model 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 if if r r 1 , r r 2 → 0 1 , 2 → 0 M=11, r=2.05 M=11, r=2.05 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 r 1 , r r 2 → 0 1 , 2 → 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 10 9 solar masses 9 solar masses r in kpc r in kpc 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 12/23
Disc Disc ● Observational constraints Observational constraints (Deg & Widrow 2013) (Deg & Widrow 2013) 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) 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) 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) ● We start with famous Miyamoto-Nagai formula We start with famous Miyamoto-Nagai formula 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 (Ninkovic 2015) (Ninkovic 2015) 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 17/23
Disc Disc ● Three parameters are important: Three parameters are important: – Total mass Total mass M M , exp. scale length R , exp. scale length R d and scale len. b d 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 1 ,γ γ 2 <0.5. γ 1 , 2 <0.5. ● We have chosen We have chosen a+b=2.1 a+b=2.1R R d and and 7.8x10 7.8x10 10 <M<9x10 10 d 10 <M<9x10 10 ● And And γ γ 1 =-0.3, γ 2 =0.3, c 1 =c 2 =R d and a≈2R a≈2R d 1 =-0.3, γ 2 =0.3, c 1 =c 2 =R d and d 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 18/23
Outer dark corona Outer dark corona ● The assumed model density & cumulative mass The assumed model density & cumulative mass 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 19/23
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) 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 20/23
What? (do we get when we What? (do we get when we combine everything) combine everything) 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 21/23
Why? Why? 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 22/23
Why? Why? Thank you! Thank you! 14-18.05.2018.-Belogradchik XI BSAC - Potential of Milky Way given analytically - Stojanović et al. 23/23
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