The Virial Theorem and its astronomical The Virial Theorem and its astronomical applications Marco applications Loreggia, Anna Sancassani, Giulia Zuin Marco Loreggia, Anna Sancassani, Giulia Zuin The Formation of Stars Università degli Studi di Padova Fundamental Plane and FP’s Tilt Dark Matter Bi - national Heraeus Summer School Series I. Cosmology Heidelberg, August 17 - 25, 2013 Contents The Virial Theorem and its astronomical applications This work is divided in three parts: Marco Loreggia, Anna Sancassani, 1 The Formation of Stars Giulia Zuin The Formation of Stars 2 Fundamental Plane and FP’s Tilt Fundamental Plane and FP’s Tilt Dark Matter 3 Dark Matter The Virial Theorem and its astronomical applications Marco Loreggia, Anna Sancassani, Giulia Zuin FIRST SECTION: The Formation of Stars The formation of Stars Fundamental Plane and FP’s Tilt Dark Matter Introduction The Virial Theorem and its astronomical applications Marco Loreggia, Anna Sancassani, Giulia Zuin The Formation of Stars Fundamental Plane and FP’s Tilt Dark Matter Eagle Nebula Stellar sector ngc 2467 (source NASA) (source NASA) Purpose The Virial Theorem and its astronomical applications Marco Loreggia, Anna Sancassani, Giulia Zuin The This criterion is called the Jeans Formation of Stars mass from Sir James Jeans Fundamental (1877 − 1946) Plane and FP’s Tilt Dark Matter Front page of the original document Equation of Continuity if ρ is the density ∂ρ + div(ρv) = 0 of the medium with velocity v ∂t and the equation of Navier-Stokes. a is the acceleration due to external forces, P is the pressure dv 1 due to the mass of fluid and the ω = a − ∇P + ω∇2v dt ρ is the kinematic coefficient of viscosity. The Virial Theorem The Virial Theorem and its astronomical applications The basic equations which let to describe the hydrostatic Marco equilibrium are the Loreggia, Anna Sancassani, Giulia Zuin The Formation of Stars Fundamental Plane and FP’s Tilt Dark Matter and the equation of Navier-Stokes. a is the acceleration due to external forces, P is the pressure dv 1 due to the mass of fluid and the ω = a − ∇P + ω∇2v dt ρ is the kinematic coefficient of viscosity. The Virial Theorem The Virial Theorem and its astronomical applications The basic equations which let to describe the hydrostatic Marco equilibrium are the Loreggia, Anna Equation of Continuity Sancassani, Giulia Zuin if ρ is the density ∂ρ The + div(ρv) = 0 of the medium with velocity v Formation of ∂t Stars Fundamental Plane and FP’s Tilt Dark Matter The Virial Theorem The Virial Theorem and its astronomical applications The basic equations which let to describe the hydrostatic Marco equilibrium are the Loreggia, Anna Equation of Continuity Sancassani, Giulia Zuin if ρ is the density ∂ρ The + div(ρv) = 0 of the medium with velocity v Formation of ∂t Stars and the equation of Navier-Stokes. Fundamental Plane and a is the acceleration due to FP’s Tilt external forces, P is the pressure Dark Matter dv 1 due to the mass of fluid and the ω = a − ∇P + ω∇2v dt ρ is the kinematic coefficient of viscosity. 1 ∂P GM ∇P = ρg and = − r (1) ρ ∂r r 2 The Virial Theorem The Virial Theorem and its astronomical applications Marco Loreggia, Anna Now we suppose that the cloud has spherical simmetry, in Sancassani, Giulia Zuin which the viscous effects are negligible and that the body is in Hydrostatic Equilibrium. The Formation of Stars Fundamental Plane and FP’s Tilt Dark Matter The Virial Theorem The Virial Theorem and its astronomical applications Marco Loreggia, Anna Now we suppose that the cloud has spherical simmetry, in Sancassani, Giulia Zuin which the viscous effects are negligible and that the body is in Hydrostatic Equilibrium. The Formation of Stars 1 ∂P GM ∇P = ρg and = − r (1) Fundamental 2 Plane and ρ ∂r r FP’s Tilt Dark Matter The Virial Theorem The Virial Theorem and its astronomical applications Marco Loreggia, Anna Than it is possible to proof that for a cloud in Hydrostatic Sancassani, Giulia Zuin Equilibrium holds the next formula called the Virial Theorem. The Formation of 2K + U = 0 (2) Stars Fundamental Plane and where K is the total thermal energy (internal kinetic energy) of FP’s Tilt the star and U expresses the gravitational potential energy. Dark Matter the gravitational potential energy is approximately 3 G(M )2 U = − c (3) 5 Rc where Mc and Rc are the mass and the radius of the cloud. If the interstellar cloud is approximated as being isothermal and constant density ρ then the thermal energy may be written 3 K = Nk T (4) 2 B where N is the total number of particles contained in the cloud and kB is the Boltzmann constant. Jeans criterion The Virial Theorem and its astronomical If now we consider a spherical cloud of constant density, applications Marco Loreggia, Anna Sancassani, Giulia Zuin The Formation of Stars Fundamental Plane and FP’s Tilt Dark Matter If the interstellar cloud is approximated as being isothermal and constant density ρ then the thermal energy may be written 3 K = Nk T (4) 2 B where N is the total number of particles contained in the cloud and kB is the Boltzmann constant. Jeans criterion The Virial Theorem and its astronomical If now we consider a spherical cloud of constant density,the applications gravitational potential energy is approximately Marco Loreggia, 2 Anna 3 G(Mc ) Sancassani, U = − (3) Giulia Zuin 5 Rc The Formation of where Mc and Rc are the mass and the radius of the cloud. Stars Fundamental Plane and FP’s Tilt Dark Matter Jeans criterion The Virial Theorem and its astronomical If now we consider a spherical cloud of constant density,the applications gravitational potential energy is approximately Marco Loreggia, 2 Anna 3 G(Mc ) Sancassani, U = − (3) Giulia Zuin 5 Rc The Formation of where Mc and Rc are the mass and the radius of the cloud. Stars If the interstellar cloud is approximated as being isothermal Fundamental Plane and and constant density ρ then the thermal energy may be written FP’s Tilt 3 Dark Matter K = Nk T (4) 2 B where N is the total number of particles contained in the cloud and kB is the Boltzmann constant. Then by the Virial Theorem the condition of collapse (−U > 2K) becomes 3 GM2 k TM c > 3 B c (6) 5 Rc µmH If we suppose constant the initial mass density of the cloud ρ0 we have 1/3 3Mc Rc = (7) 4πρ0 Jeans criterion The Virial Theorem and N is just its astronomical M applications N = (5) µmH Marco Loreggia, where µ is the molecular weight mean and mH is the mass of Anna Sancassani, hydrogen atom. Giulia Zuin The Formation of Stars Fundamental Plane and FP’s Tilt Dark Matter If we suppose constant the initial mass density of the cloud ρ0 we have 1/3 3Mc Rc = (7) 4πρ0 Jeans criterion The Virial Theorem and N is just its astronomical M applications N = (5) µmH Marco Loreggia, where µ is the molecular weight mean and mH is the mass of Anna Sancassani, hydrogen atom. Giulia Zuin Then by the Virial Theorem the condition of collapse The Formation of (−U > 2K) becomes Stars 2 Fundamental 3 GMc kBTMc Plane and > 3 (6) FP’s Tilt 5 Rc µmH Dark Matter Jeans criterion The Virial Theorem and N is just its astronomical M applications N = (5) µmH Marco Loreggia, where µ is the molecular weight mean and mH is the mass of Anna Sancassani, hydrogen atom. Giulia Zuin Then by the Virial Theorem the condition of collapse The Formation of (−U > 2K) becomes Stars 2 Fundamental 3 GMc kBTMc Plane and > 3 (6) FP’s Tilt 5 Rc µmH Dark Matter If we suppose constant the initial mass density of the cloud ρ0 we have 1/3 3Mc Rc = (7) 4πρ0 Jeans criterion The Virial Theorem and its astronomical applications Marco Loreggia, Then after substitution we obtain the condition to initiate the Anna Sancassani, collapse of the cloud and is know as the Jeans criterion Giulia Zuin 3 1 The ! 2 ! 2 Formation of 5kBT 3 Stars M > =: MJ (8) Fundamental µmH G 4πρ0 Plane and FP’s Tilt where Jeans mass M is defined. Dark Matter J Jeans criterion The Virial Theorem and its astronomical applications Marco Loreggia, Anna Sancassani, From the expression of the radious in the previous formula we Giulia Zuin can get the so called Jeans radius The Formation of 1/2 Stars 15kBT RJ = (9) Fundamental 4πGµmH ρ0 Plane and FP’s Tilt Dark Matter 3π 1 1/2 1 tff = ≈ √ 32 Gρ0 Gρ0 where ρ0 is the initial density of the body. The free-fall timescale The Virial Theorem and its astronomical applications Marco Loreggia, Anna Other important parameter which we have to introduced is the Sancassani, Giulia Zuin free-fall timescale. The Formation of Stars Fundamental Plane and FP’s Tilt Dark Matter The free-fall timescale The Virial Theorem and its astronomical applications Marco Loreggia, Anna Other important parameter which we have to introduced is the Sancassani, Giulia Zuin free-fall timescale. The 1/2 Formation of 3π 1 1 Stars tff = ≈ √ Fundamental 32 Gρ0 Gρ0 Plane and FP’s Tilt where ρ0 is the initial density of the body. Dark Matter Homologous Collapse The Virial Theorem and So far, the instruments present here let to an Homologous its astronomical Collapse applications Marco Loreggia, Anna Sancassani, Giulia Zuin The Formation of Stars Fundamental Plane and FP’s Tilt Dark Matter (ii) only a few numbers of cloud let to the formation of stars Problems and observations The Virial Theorem and its astronomical applications Marco (i) stars frequently tend to form Loreggia, Anna in group Sancassani, Giulia Zuin The Formation of Stars Fundamental Plane and FP’s Tilt Dark Matter Problems and observations The Virial Theorem and its astronomical applications Marco (i) stars frequently tend to form Loreggia, Anna in group