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 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 According to the Virial theorem the energy must be liberated during the collapse of the cloud is
3 GM2 ∆E = J 10 RJ and then the luminosity due to the gravity is given by
5/2 ∆E 3/2 MJ Lff ' ' G tff RJ On the other hand thanks to Stefan-Boltzmann’s equation we may express the radiated luminosity as
2 4 Lrad = 4πR eσT
where we introduced the efficiency factor 0 < e < 1, to indicate the deviation from thermodynamic equilibrium
Adiabatic evolution process
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 and then the luminosity due to the gravity is given by
5/2 ∆E 3/2 MJ Lff ' ' G tff RJ On the other hand thanks to Stefan-Boltzmann’s equation we may express the radiated luminosity as
2 4 Lrad = 4πR eσT
where we introduced the efficiency factor 0 < e < 1, to indicate the deviation from thermodynamic equilibrium
Adiabatic evolution process
The Virial According to the Virial theorem the energy must be liberated Theorem and its during the collapse of the cloud is astronomical applications 2 Marco 3 GMJ Loreggia, ∆E = Anna 10 RJ Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter On the other hand thanks to Stefan-Boltzmann’s equation we may express the radiated luminosity as
2 4 Lrad = 4πR eσT
where we introduced the efficiency factor 0 < e < 1, to indicate the deviation from thermodynamic equilibrium
Adiabatic evolution process
The Virial According to the Virial theorem the energy must be liberated Theorem and its during the collapse of the cloud is astronomical applications 2 Marco 3 GMJ Loreggia, ∆E = Anna 10 RJ Sancassani, Giulia Zuin and then the luminosity due to the gravity is given by
The Formation of 5/2 Stars ∆E 3/2 MJ Lff ' ' G Fundamental tff RJ Plane and FP’s Tilt
Dark Matter Adiabatic evolution process
The Virial According to the Virial theorem the energy must be liberated Theorem and its during the collapse of the cloud is astronomical applications 2 Marco 3 GMJ Loreggia, ∆E = Anna 10 RJ Sancassani, Giulia Zuin and then the luminosity due to the gravity is given by
The Formation of 5/2 Stars ∆E 3/2 MJ Lff ' ' G Fundamental tff RJ Plane and FP’s Tilt On the other hand thanks to Stefan-Boltzmann’s equation we Dark Matter may express the radiated luminosity as
2 4 Lrad = 4πR eσT
where we introduced the efficiency factor 0 < e < 1, to indicate the deviation from thermodynamic equilibrium 4π 2/5 M = R9/2eσT 4 J G3/2 J
Then we arrive at the estimate required
T 1/4 M = 0, 03 M Jmin e1/2µ9/4
where T is expressed in kelvin and M is the mass of the sun.
Adiabatic evolution process
The Virial Theorem and its astronomical applications Equating the two expressions for the cloud’s luminosity Marco (Lff = Lrad ) and rearranging, we have Loreggia, Anna Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter Then we arrive at the estimate required
T 1/4 M = 0, 03 M Jmin e1/2µ9/4
where T is expressed in kelvin and M is the mass of the sun.
Adiabatic evolution process
The Virial Theorem and its astronomical applications Equating the two expressions for the cloud’s luminosity Marco (Lff = Lrad ) and rearranging, we have Loreggia, Anna Sancassani, 2/5 Giulia Zuin 4π M = R9/2eσT 4 J 3/2 J The G Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter Adiabatic evolution process
The Virial Theorem and its astronomical applications Equating the two expressions for the cloud’s luminosity Marco (Lff = Lrad ) and rearranging, we have Loreggia, Anna Sancassani, 2/5 Giulia Zuin 4π M = R9/2eσT 4 J 3/2 J The G Formation of Stars Then we arrive at the estimate required Fundamental Plane and FP’s Tilt T 1/4 M = 0, 03 M Dark Matter Jmin e1/2µ9/4
where T is expressed in kelvin and M is the mass of the sun. fragmentation ceases when the segments of the original cloud begin to reach the range of the solar mass object our estimate is relatively insensitive to other reasonable choise for T , e and µ
Conclusion
The Virial Theorem and its astronomical applications
Marco Loreggia, Anna Sancassani, In conclusion from this simple analysis we can observe that: Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter Conclusion
The Virial Theorem and its astronomical applications
Marco Loreggia, Anna Sancassani, In conclusion from this simple analysis we can observe that: Giulia Zuin fragmentation ceases when the segments of the original The cloud begin to reach the range of the solar mass object Formation of Stars our estimate is relatively insensitive to other reasonable Fundamental Plane and choise for T , e and µ FP’s Tilt
Dark Matter Conclusion
The Virial Theorem and its astronomical applications
Marco Loreggia, Anna Sancassani, In conclusion from this simple analysis we can observe that: Giulia Zuin fragmentation ceases when the segments of the original The cloud begin to reach the range of the solar mass object Formation of Stars our estimate is relatively insensitive to other reasonable Fundamental Plane and choise for T , e and µ FP’s Tilt
Dark Matter References:
The Virial Theorem and its astronomical applications
Marco 1 A. MAEDERr, Physics, Formation and Evolution of Rotating Stars, book Loreggia, editor by Springer, 2009 Anna Sancassani, 2 FRANCIS LEBLANC, An Introduction to Stellar Astrophysics, book editor Giulia Zuin by Wiley, 2010
The 3 BRADLEY W. CARROLL, DALE A. OSTLIE, An introduction to modern Formation of Astrophysics, secon edition, editor by Pearson Stars 4 RALF S. KLESSEN, ANDREAS BURKET, MATTHEW R.BATE, Fundamental Fragmentation of molecular clouds: the initial phase of a stellar cluster, Plane and FP’s Tilt article of The American Astronomical Society The Astrophysical Journal, 501:L205-L208, 1998 July 10 Dark Matter 5 Simulation and visualisation by MATTHEW BATE, University of Exeter UK The Virial Theorem and its astronomical applications Marco SECOND SECTION: Loreggia, Anna Sancassani, Giulia Zuin The Fundamental Plane The Formation of Stars and Fundamental Plane and FP’s Tilt
Dark Matter the problem of “Tilt” the instrumentations was not able to give appropriate informations about their kinematics
For example, with the introduction of CCD, in the second half of Seventies, the scientists were able to make more accurate photometric and kinematic measurements. . .
Fundamental Plane of ellipticals - Introduction
The Virial Theorem and its astronomical Until 70’s, between scientists, elliptical galaxies wasn’t applications considered objects of intrinsic interest from the standpoint of Marco Loreggia, dynamic. Essentially for two reasons: Anna Sancassani, Giulia Zuin they was considered simple stellar systems, revolution The Formation of ellipsoid Stars
Fundamental Plane and FP’s Tilt
Dark Matter For example, with the introduction of CCD, in the second half of Seventies, the scientists were able to make more accurate photometric and kinematic measurements. . .
Fundamental Plane of ellipticals - Introduction
The Virial Theorem and its astronomical Until 70’s, between scientists, elliptical galaxies wasn’t applications considered objects of intrinsic interest from the standpoint of Marco Loreggia, dynamic. Essentially for two reasons: Anna Sancassani, Giulia Zuin they was considered simple stellar systems, revolution The Formation of ellipsoid Stars
Fundamental Plane and the instrumentations was not able to give appropriate FP’s Tilt informations about their kinematics Dark Matter Fundamental Plane of ellipticals - Introduction
The Virial Theorem and its astronomical Until 70’s, between scientists, elliptical galaxies wasn’t applications considered objects of intrinsic interest from the standpoint of Marco Loreggia, dynamic. Essentially for two reasons: Anna Sancassani, Giulia Zuin they was considered simple stellar systems, revolution The Formation of ellipsoid Stars
Fundamental Plane and the instrumentations was not able to give appropriate FP’s Tilt informations about their kinematics Dark Matter
For example, with the introduction of CCD, in the second half of Seventies, the scientists were able to make more accurate photometric and kinematic measurements. . . The second type parameters are, for example: the effective radius (Re), the effective surface brightness (Ie), the central velocity dispersion (σ0), the luminosity in various bands X (LX), the mean color (B − V) and the line-strenght indices of magnesium (Mg2)
Fundamental Plane of ellipticals - Some Parameters
The Virial Theorem and its astronomical applications There are two fundamental sets of elliptical galaxies basic Marco Loreggia, structural parameters: one is the set of shape parameters and Anna Sancassani, the second consists of the shape-independent parameters. We Giulia Zuin are interested in the second type The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter Fundamental Plane of ellipticals - Some Parameters
The Virial Theorem and its astronomical applications There are two fundamental sets of elliptical galaxies basic Marco Loreggia, structural parameters: one is the set of shape parameters and Anna Sancassani, the second consists of the shape-independent parameters. We Giulia Zuin are interested in the second type The Formation of Stars The second type parameters are, for example: the effective Fundamental radius (R ), the effective surface brightness (I ), the central Plane and e e FP’s Tilt velocity dispersion (σ0), the luminosity in various bands X Dark Matter (LX), the mean color (B − V) and the line-strenght indices of magnesium (Mg2) the surface brightness (Ie): the overall brightness of an extended astronomical object (a galaxy, a cluster, a nebula) can be measured by its apparent magnitude. This is a limited tool for our purposes because it is clear that, on equal apparent magnitude, an extended object will be harder to see than a star. We need an extra parameter: the surface brightness give an indication of how easily observable the object is the velocity dispersion (σ) is the statistical dispersion of velocity about the mean velocity of a group of objects
Fundamental Plane of ellipticals - Definition of some important parameters
The Virial Theorem and We recall the definition of some of these parameters, (we will its astronomical focus only on those that are involved in the following): applications
Marco Loreggia, the effective radius (Re) of a galaxy is the radius at which Anna Sancassani, one half of the total light of the system is emitted within this Giulia Zuin radius (spherical symmetry)
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter the velocity dispersion (σ) is the statistical dispersion of velocity about the mean velocity of a group of objects
Fundamental Plane of ellipticals - Definition of some important parameters
The Virial Theorem and We recall the definition of some of these parameters, (we will its astronomical focus only on those that are involved in the following): applications
Marco Loreggia, the effective radius (Re) of a galaxy is the radius at which Anna Sancassani, one half of the total light of the system is emitted within this Giulia Zuin radius (spherical symmetry)
The Formation of the surface brightness (Ie): the overall brightness of an Stars extended astronomical object (a galaxy, a cluster, a nebula) can Fundamental be measured by its apparent magnitude. This is a limited tool Plane and FP’s Tilt for our purposes because it is clear that, on equal apparent
Dark Matter magnitude, an extended object will be harder to see than a star. We need an extra parameter: the surface brightness give an indication of how easily observable the object is Fundamental Plane of ellipticals - Definition of some important parameters
The Virial Theorem and We recall the definition of some of these parameters, (we will its astronomical focus only on those that are involved in the following): applications
Marco Loreggia, the effective radius (Re) of a galaxy is the radius at which Anna Sancassani, one half of the total light of the system is emitted within this Giulia Zuin radius (spherical symmetry)
The Formation of the surface brightness (Ie): the overall brightness of an Stars extended astronomical object (a galaxy, a cluster, a nebula) can Fundamental be measured by its apparent magnitude. This is a limited tool Plane and FP’s Tilt for our purposes because it is clear that, on equal apparent
Dark Matter magnitude, an extended object will be harder to see than a star. We need an extra parameter: the surface brightness give an indication of how easily observable the object is the velocity dispersion (σ) is the statistical dispersion of velocity about the mean velocity of a group of objects Re - Ie: There is a correlation between the effective radius Re and the mean surface brightness within Re, Ie:
−a Re ∝ Ie (Kormendy, 1977)
In 1987, Djorgovsky and Davies completed the formula with the determination of the value a = 0, 83 ± 0, 08. So, fainter galaxies have an higher surface brightness and a smaller effective radius
Mg2 - σ0: The intensity of Magnesium increases with both the luminosity of galaxy (Faber, 1973) and the central velocity dispersion (Burstein et al. 1988; Bernardi et al. 2003)
Fundamental Plane of ellipticals - Some correlations among parameters of ellipticals
The Virial With the development of techniques in the Seventies, it was realized Theorem and its that many property of ellipticals was correlated by empirical relations: astronomical applications
Marco Loreggia, Anna Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter In 1987, Djorgovsky and Davies completed the formula with the determination of the value a = 0, 83 ± 0, 08. So, fainter galaxies have an higher surface brightness and a smaller effective radius
Mg2 - σ0: The intensity of Magnesium increases with both the luminosity of galaxy (Faber, 1973) and the central velocity dispersion (Burstein et al. 1988; Bernardi et al. 2003)
Fundamental Plane of ellipticals - Some correlations among parameters of ellipticals
The Virial With the development of techniques in the Seventies, it was realized Theorem and its that many property of ellipticals was correlated by empirical relations: astronomical applications
Marco Re - Ie: There is a correlation between the effective radius Re Loreggia, Anna and the mean surface brightness within Re, Ie: Sancassani, Giulia Zuin −a Re ∝ Ie (Kormendy, 1977) The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter So, fainter galaxies have an higher surface brightness and a smaller effective radius
Mg2 - σ0: The intensity of Magnesium increases with both the luminosity of galaxy (Faber, 1973) and the central velocity dispersion (Burstein et al. 1988; Bernardi et al. 2003)
Fundamental Plane of ellipticals - Some correlations among parameters of ellipticals
The Virial With the development of techniques in the Seventies, it was realized Theorem and its that many property of ellipticals was correlated by empirical relations: astronomical applications
Marco Re - Ie: There is a correlation between the effective radius Re Loreggia, Anna and the mean surface brightness within Re, Ie: Sancassani, Giulia Zuin −a Re ∝ Ie (Kormendy, 1977) The Formation of Stars In 1987, Djorgovsky and Davies completed the
Fundamental formula with the determination of the value Plane and a = 0, 83 ± 0, 08. FP’s Tilt
Dark Matter Mg2 - σ0: The intensity of Magnesium increases with both the luminosity of galaxy (Faber, 1973) and the central velocity dispersion (Burstein et al. 1988; Bernardi et al. 2003)
Fundamental Plane of ellipticals - Some correlations among parameters of ellipticals
The Virial With the development of techniques in the Seventies, it was realized Theorem and its that many property of ellipticals was correlated by empirical relations: astronomical applications
Marco Re - Ie: There is a correlation between the effective radius Re Loreggia, Anna and the mean surface brightness within Re, Ie: Sancassani, Giulia Zuin −a Re ∝ Ie (Kormendy, 1977) The Formation of Stars In 1987, Djorgovsky and Davies completed the
Fundamental formula with the determination of the value Plane and a = 0, 83 ± 0, 08. So, fainter galaxies have an higher FP’s Tilt surface brightness and a smaller effective radius Dark Matter Fundamental Plane of ellipticals - Some correlations among parameters of ellipticals
The Virial With the development of techniques in the Seventies, it was realized Theorem and its that many property of ellipticals was correlated by empirical relations: astronomical applications
Marco Re - Ie: There is a correlation between the effective radius Re Loreggia, Anna and the mean surface brightness within Re, Ie: Sancassani, Giulia Zuin −a Re ∝ Ie (Kormendy, 1977) The Formation of Stars In 1987, Djorgovsky and Davies completed the
Fundamental formula with the determination of the value Plane and a = 0, 83 ± 0, 08. So, fainter galaxies have an higher FP’s Tilt surface brightness and a smaller effective radius Dark Matter
Mg2 - σ0: The intensity of Magnesium increases with both the luminosity of galaxy (Faber, 1973) and the central velocity dispersion (Burstein et al. 1988; Bernardi et al. 2003) L- σ0: In 1976 Faber and Jackson discovered the following relation:
n L ∝ σ0
where 3 ≤ n ≤ 5 and, the σ0 is referred, generally, to the central region of radius Re/8
Fundamental Plane of ellipticals - Some correlations among parameters of ellipticals
The Virial Theorem and its astronomical applications Color - Magnitude: There is a correlation between the
Marco absolute magnitude an the color of galaxies; so Loreggia, Anna the brighter galaxies are more red than fainter Sancassani, Giulia Zuin galaxies
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter Fundamental Plane of ellipticals - Some correlations among parameters of ellipticals
The Virial Theorem and its astronomical applications Color - Magnitude: There is a correlation between the
Marco absolute magnitude an the color of galaxies; so Loreggia, Anna the brighter galaxies are more red than fainter Sancassani, Giulia Zuin galaxies
The Formation of L- σ : In 1976 Faber and Jackson discovered the Stars 0
Fundamental following relation: Plane and FP’s Tilt L ∝ σn Dark Matter 0
where 3 ≤ n ≤ 5 and, the σ0 is referred, generally, to the central region of radius Re/8 The innovation of this discover was the definition of the three dimensional space of three observable σ0, Re and Ie. In this space, the elliptical galaxies are not uniformly disposed, but are concentrated on a logarithmic plane, named fundamental plane, that is a relation like:
logRe = α logσ0 + β logIe + γ
The fundamental plane
The Virial Theorem and its astronomical In 1987 two groups of scientists introduced the effective radius applications in the last equation, and observed that, in this way, the value Marco Loreggia, of n become more exact Anna Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter The fundamental plane
The Virial Theorem and its astronomical In 1987 two groups of scientists introduced the effective radius applications in the last equation, and observed that, in this way, the value Marco Loreggia, of n become more exact Anna Sancassani, Giulia Zuin The innovation of this discover was the definition of the three The dimensional space of three observable σ , R and I . In this Formation of 0 e e Stars space, the elliptical galaxies are not uniformly disposed, but Fundamental Plane and are concentrated on a logarithmic plane, named fundamental FP’s Tilt plane, that is a relation like: Dark Matter
logRe = α logσ0 + β logIe + γ The coefficients depends slightly on the used photometric band.
An important property of the FP is the constant dispersion of the various involved observable: for example, the distribution of Re around the best-fit (with fixed Ie and σ0) has a dispersion that can change from 15% to 20%.
The fundamental plane
The Virial Theorem and its astronomical applications
Marco logRe = α logσ0 + β logIe + γ Loreggia, Anna Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter An important property of the FP is the constant dispersion of the various involved observable: for example, the distribution of Re around the best-fit (with fixed Ie and σ0) has a dispersion that can change from 15% to 20%.
The fundamental plane
The Virial Theorem and its astronomical applications
Marco logRe = α logσ0 + β logIe + γ Loreggia, Anna Sancassani, Giulia Zuin
The The coefficients depends slightly on the used photometric Formation of Stars band. Fundamental Plane and FP’s Tilt
Dark Matter The fundamental plane
The Virial Theorem and its astronomical applications
Marco logRe = α logσ0 + β logIe + γ Loreggia, Anna Sancassani, Giulia Zuin
The The coefficients depends slightly on the used photometric Formation of Stars band. Fundamental Plane and FP’s Tilt An important property of the FP is the constant dispersion of Dark Matter the various involved observable: for example, the distribution of Re around the best-fit (with fixed Ie and σ0) has a dispersion that can change from 15% to 20%. The Virial Theorem
The Virial Theorem and its astronomical applications It is possible to prove that the characteristic dynamical time of
Marco ellipticals and their collisionless relaxation time is of the same Loreggia, Anna order (Lynden-Bell 1967). In particular are both short with Sancassani, Giulia Zuin respect to the age of ellipticals.
The Formation of Stars ⇓ Fundamental Plane and FP’s Tilt Highly perturbed galaxies are presumably caught in a Dark Matter non-stationary phase. Stationarity is a sufficient condition for the validity of the Virial Theorem, and so for ellipticals the Virial Theorem holds. - G is the gravitational constant - M the mass - V 2 the mean quadratic velocity weighted by the mass - hRi the characteristic gravitational radius, weighted by the mass
The theoretical interpretation of Fundamental Plane
The Virial Theorem and So, assuming the validity of Virial Theorem, we can begin the its astronomical calculation. applications
Marco Loreggia, The scalar Virial Theorem, says that the kinetic energy T and Anna Sancassani, the potential energy Ω in a system are correlated in this way: Giulia Zuin
The 2T = −Ω Formation of Stars
Fundamental So it is true that: Plane and M D 2E FP’s Tilt G = V hRi Dark Matter - M the mass - V 2 the mean quadratic velocity weighted by the mass - hRi the characteristic gravitational radius, weighted by the mass
The theoretical interpretation of Fundamental Plane
The Virial Theorem and So, assuming the validity of Virial Theorem, we can begin the its astronomical calculation. applications
Marco Loreggia, The scalar Virial Theorem, says that the kinetic energy T and Anna Sancassani, the potential energy Ω in a system are correlated in this way: Giulia Zuin
The 2T = −Ω Formation of Stars
Fundamental So it is true that: Plane and M D 2E FP’s Tilt G = V hRi Dark Matter
- G is the gravitational constant - V 2 the mean quadratic velocity weighted by the mass - hRi the characteristic gravitational radius, weighted by the mass
The theoretical interpretation of Fundamental Plane
The Virial Theorem and So, assuming the validity of Virial Theorem, we can begin the its astronomical calculation. applications
Marco Loreggia, The scalar Virial Theorem, says that the kinetic energy T and Anna Sancassani, the potential energy Ω in a system are correlated in this way: Giulia Zuin
The 2T = −Ω Formation of Stars
Fundamental So it is true that: Plane and M D 2E FP’s Tilt G = V hRi Dark Matter
- G is the gravitational constant - M the mass - hRi the characteristic gravitational radius, weighted by the mass
The theoretical interpretation of Fundamental Plane
The Virial Theorem and So, assuming the validity of Virial Theorem, we can begin the its astronomical calculation. applications
Marco Loreggia, The scalar Virial Theorem, says that the kinetic energy T and Anna Sancassani, the potential energy Ω in a system are correlated in this way: Giulia Zuin
The 2T = −Ω Formation of Stars
Fundamental So it is true that: Plane and M D 2E FP’s Tilt G = V hRi Dark Matter
- G is the gravitational constant - M the mass - V 2 the mean quadratic velocity weighted by the mass The theoretical interpretation of Fundamental Plane
The Virial Theorem and So, assuming the validity of Virial Theorem, we can begin the its astronomical calculation. applications
Marco Loreggia, The scalar Virial Theorem, says that the kinetic energy T and Anna Sancassani, the potential energy Ω in a system are correlated in this way: Giulia Zuin
The 2T = −Ω Formation of Stars
Fundamental So it is true that: Plane and M D 2E FP’s Tilt G = V hRi Dark Matter
- G is the gravitational constant - M the mass - V 2 the mean quadratic velocity weighted by the mass - hRi the characteristic gravitational radius, weighted by the mass This parameter must take into account the structure of internal density of mass in the galaxy
We can define a parameter kV that satisfy:
2 D 2E σ0 = kv V
and that must reflects the kinematic structure of the galaxy
The theoretical interpretation of Fundamental Plane
The Virial Theorem and its astronomical If Re is the effective radius of the elliptical galaxy, we can applications define a parameter k that satisfy: Marco R Loreggia, Anna Sancassani, hRei = kR hRi Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter We can define a parameter kV that satisfy:
2 D 2E σ0 = kv V
and that must reflects the kinematic structure of the galaxy
The theoretical interpretation of Fundamental Plane
The Virial Theorem and its astronomical If Re is the effective radius of the elliptical galaxy, we can applications define a parameter k that satisfy: Marco R Loreggia, Anna Sancassani, hRei = kR hRi Giulia Zuin
The This parameter must take into account the structure of Formation of Stars internal density of mass in the galaxy
Fundamental Plane and FP’s Tilt
Dark Matter and that must reflects the kinematic structure of the galaxy
The theoretical interpretation of Fundamental Plane
The Virial Theorem and its astronomical If Re is the effective radius of the elliptical galaxy, we can applications define a parameter k that satisfy: Marco R Loreggia, Anna Sancassani, hRei = kR hRi Giulia Zuin
The This parameter must take into account the structure of Formation of Stars internal density of mass in the galaxy Fundamental We can define a parameter k that satisfy: Plane and V FP’s Tilt 2 D 2E Dark Matter σ0 = kv V The theoretical interpretation of Fundamental Plane
The Virial Theorem and its astronomical If Re is the effective radius of the elliptical galaxy, we can applications define a parameter k that satisfy: Marco R Loreggia, Anna Sancassani, hRei = kR hRi Giulia Zuin
The This parameter must take into account the structure of Formation of Stars internal density of mass in the galaxy Fundamental We can define a parameter k that satisfy: Plane and V FP’s Tilt 2 D 2E Dark Matter σ0 = kv V
and that must reflects the kinematic structure of the galaxy M D E G = V 2 M = c σ2R hRi 2 0 e
−1 where c2 = (GkV kR ) .
2 Finally, we know that L = c1IeRe where Ie is the mean 2 superficial effective brightness (Ie = L/2πRe ) so we insert this information in the above equation for M. The final result is:
M −1 R = (c c−1) σ2I−1 e 1 2 L 0 e
The theoretical interpretation of Fundamental Plane
The Virial Theorem and Now, substituting the hR i = k hRi and σ2 = k V 2 in the its e R 0 v astronomical virial equation: applications
Marco Loreggia, Anna Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter 2 M = c2σ0Re
−1 where c2 = (GkV kR ) .
2 Finally, we know that L = c1IeRe where Ie is the mean 2 superficial effective brightness (Ie = L/2πRe ) so we insert this information in the above equation for M. The final result is:
M −1 R = (c c−1) σ2I−1 e 1 2 L 0 e
The theoretical interpretation of Fundamental Plane
The Virial Theorem and Now, substituting the hR i = k hRi and σ2 = k V 2 in the its e R 0 v astronomical virial equation: applications
Marco Loreggia, M D 2E Anna G = V Sancassani, hRi Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter 2 Finally, we know that L = c1IeRe where Ie is the mean 2 superficial effective brightness (Ie = L/2πRe ) so we insert this information in the above equation for M. The final result is:
M −1 R = (c c−1) σ2I−1 e 1 2 L 0 e
The theoretical interpretation of Fundamental Plane
The Virial Theorem and Now, substituting the hR i = k hRi and σ2 = k V 2 in the its e R 0 v astronomical virial equation: applications
Marco Loreggia, M D 2E 2 Anna G = V M = c2σ0Re Sancassani, hRi Giulia Zuin
The −1 Formation of where c2 = (GkV kR ) . Stars
Fundamental Plane and FP’s Tilt
Dark Matter so we insert this information in the above equation for M. The final result is:
M −1 R = (c c−1) σ2I−1 e 1 2 L 0 e
The theoretical interpretation of Fundamental Plane
The Virial Theorem and Now, substituting the hR i = k hRi and σ2 = k V 2 in the its e R 0 v astronomical virial equation: applications
Marco Loreggia, M D 2E 2 Anna G = V M = c2σ0Re Sancassani, hRi Giulia Zuin
The −1 Formation of where c2 = (GkV kR ) . Stars Fundamental Finally, we know that L = c I R2 where I is the mean Plane and 1 e e e FP’s Tilt 2 superficial effective brightness (Ie = L/2πRe ) Dark Matter The final result is:
M −1 R = (c c−1) σ2I−1 e 1 2 L 0 e
The theoretical interpretation of Fundamental Plane
The Virial Theorem and Now, substituting the hR i = k hRi and σ2 = k V 2 in the its e R 0 v astronomical virial equation: applications
Marco Loreggia, M D 2E 2 Anna G = V M = c2σ0Re Sancassani, hRi Giulia Zuin
The −1 Formation of where c2 = (GkV kR ) . Stars Fundamental Finally, we know that L = c I R2 where I is the mean Plane and 1 e e e FP’s Tilt 2 superficial effective brightness (Ie = L/2πRe ) so we insert this Dark Matter information in the above equation for M. The theoretical interpretation of Fundamental Plane
The Virial Theorem and Now, substituting the hR i = k hRi and σ2 = k V 2 in the its e R 0 v astronomical virial equation: applications
Marco Loreggia, M D 2E 2 Anna G = V M = c2σ0Re Sancassani, hRi Giulia Zuin
The −1 Formation of where c2 = (GkV kR ) . Stars Fundamental Finally, we know that L = c I R2 where I is the mean Plane and 1 e e e FP’s Tilt 2 superficial effective brightness (Ie = L/2πRe ) so we insert this Dark Matter information in the above equation for M. The final result is:
M −1 R = (c c−1) σ2I−1 e 1 2 L 0 e The parameter c2 indeed depends on the mass and the −1 velocity dispersion in the galaxy (because c2 = (GkV kR ) ). So, c2 is constant if we assume that all galaxies are homologous in this sense too.
The theoretical interpretation of Fundamental Plane
The Virial Theorem and its astronomical applications If we assume that every galaxy has the same luminosity Marco Loreggia, profile (i.e. we assume that all galaxies are “homologous” in Anna Sancassani, this sense) the parameter c1 is constant Giulia Zuin 2 2 (because Ie = L/2πR and L = c1IeR and if we substute...) The e e Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter The theoretical interpretation of Fundamental Plane
The Virial Theorem and its astronomical applications If we assume that every galaxy has the same luminosity Marco Loreggia, profile (i.e. we assume that all galaxies are “homologous” in Anna Sancassani, this sense) the parameter c1 is constant Giulia Zuin 2 2 (because Ie = L/2πR and L = c1IeR and if we substute...) The e e Formation of Stars
Fundamental Plane and The parameter c2 indeed depends on the mass and the FP’s Tilt −1 velocity dispersion in the galaxy (because c2 = (GkV kR ) ). Dark Matter So, c2 is constant if we assume that all galaxies are homologous in this sense too. The theoretical interpretation of Fundamental Plane
The Virial Theorem and its astronomical applications
Marco Loreggia, A priori, the resulting equation Anna Sancassani, Giulia Zuin −1 −1 M 2 −1 Re = (c1c2 ) σ0Ie The L Formation of Stars
Fundamental is not a defined locus in the space (σ0, Re, Ie). For every point Plane and FP’s Tilt of this three-dimensional space, the values of c2 and M/L can
Dark Matter vary a lot. The FP from the Virial Theorem
A B Re ∝ σ0 Ie with A = 2 B = −1
The theoretical interpretation of Fundamental Plane
The Virial Theorem and its astronomical applications −1 −1 M 2 −1 Marco Re = (c1c2 ) σ0Ie Loreggia, L Anna Sancassani, Giulia Zuin But, if the galaxies all belong to the same homologous family The and the ratio M/L is constant from galaxy to another with the Formation of Stars change in mass, we have that the equation below should define Fundamental univocally the physical characteristic of the galaxy: Plane and FP’s Tilt
Dark Matter The theoretical interpretation of Fundamental Plane
The Virial Theorem and its astronomical applications −1 −1 M 2 −1 Marco Re = (c1c2 ) σ0Ie Loreggia, L Anna Sancassani, Giulia Zuin But, if the galaxies all belong to the same homologous family The and the ratio M/L is constant from galaxy to another with the Formation of Stars change in mass, we have that the equation below should define Fundamental univocally the physical characteristic of the galaxy: Plane and FP’s Tilt
Dark Matter The FP from the Virial Theorem
A B Re ∝ σ0 Ie with A = 2 B = −1 The problem of “Tilt”
The Virial Theorem and its astronomical applications The PROBLEM:
Marco Loreggia, Anna the experimental evidences say that the galaxies, Sancassani, Giulia Zuin in the (σ0, Re, Ie) space, are concentrated in a The Formation of plane that is considerably distant from the plane Stars
Fundamental that the Virial Theorem return! Plane and FP’s Tilt
Dark Matter This gap between the theoretical and experimental coefficients is also named “the tilt of fundamental plane” The problem of “Tilt”
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 The problem of “Tilt”
The Virial Theorem and its astronomical applications
Marco Loreggia, In the 1987 it was determined, in photometric band B, the Anna ¯ Sancassani, right coefficients of the plane: A = 1, 39 ± 0, 15 and Giulia Zuin B¯ = −0, 9 ± 0, 1. The Formation of Stars A B Fundamental To be more precise, when the two planes Re ∝ σ0 Ie and Plane and A¯ B¯ FP’s Tilt Re ∝ σ0 Ie are in logarithmic form, they appear to be ◦ Dark Matter tilted by an angle of ∼ 15 in all three variables. the ratio M/L is not constant with the change in mass of galaxy and the ellipticals are not an homologous family of galaxies in this sense
(an explanation can be the existence of Dark Matter . . . )
The problem of “Tilt”
The Virial Theorem and its astronomical applications Marco This problem is a consequence of a wrong assumed Loreggia, Anna hypothesis: Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter The problem of “Tilt”
The Virial Theorem and its astronomical applications Marco This problem is a consequence of a wrong assumed Loreggia, Anna hypothesis: Sancassani, Giulia Zuin
The the ratio M/L is not constant with the change in mass of Formation of Stars galaxy and the ellipticals are not an homologous family of
Fundamental galaxies in this sense Plane and FP’s Tilt
Dark Matter (an explanation can be the existence of Dark Matter . . . ) The problem of “Tilt”
The Virial Theorem and its astronomical applications The conclusion is that there are three important still open
Marco problems: Loreggia, Anna Sancassani, Giulia Zuin i) why is the FP so thin (i.e. why is that relation so tight); The Formation of Stars ii) why is the FP “tilted”
Fundamental (i.e. why elliptical galaxies don’t seem to obey the Virial Plane and FP’s Tilt Theorem); Dark Matter iii) why does the FP itself exists (i.e. what causes the properties of elliptical galaxies to be so “systematic”) References:
The Virial Theorem and 1 SCHNEIDER P. T., Extragalactic astronomy and cosmology: an its introduction, Berlin, Springer, 2006. astronomical applications 2 PIZZELLA A., Corso di astrofisica delle galassie 1, Dispensa, 2006, http://www.astro.unipd.it/pizzella/corso/2006/dispense/ Marco Loreggia, dispense_FP_1_0.pdf (last access 26/07/13) Anna 3 FERRARI A., Stelle, galassie e universo: fondamenti di astrofisica, Milano, Sancassani, Giulia Zuin Springer, 2011. 4 BINNEY J. - MERRIFIELD M., Galactic astronomy, Princeton, Princeton The University Press, 1998. Formation of Stars 5 Galaxy scaling relations: origins, evolution and applications :
Fundamental proceedings. . . , Germany, 18-20 November 1996, 1997. Plane and 6 BUSARELLO G. et al., The relations between the Virial Theorem and the FP’s Tilt fundamental plane of elliptical galaxies, in Astron.Astrophys., 320, p. Dark Matter 415-420. 7 CIOTTI L. et al.,The tilt of fundamental plane of ellipitical galaxies: I. Exploring dynamical and structural effects, in Mon.Not.Roy.Astron.Soc., 282 (1996), p. 1-12. 8 BORRIELLO A. et al., The fundamental plane of ellipticals: I. The dark Matter connection, in Mon.Not.Roy.Astron.Soc., 341 (2003), p. 1109-1120. The Virial Theorem and its astronomical applications
Marco Loreggia, Anna Sancassani, Giulia Zuin THIRD SECTION:
The Formation of Stars The Dark Matter Fundamental Plane and FP’s Tilt
Dark Matter Evidence for an unseen component in spiral galaxies
The Virial Theorem and its astronomical applications
Marco Loreggia, In 1970s Vera Rubin was studying HII regions in spiral Anna Sancassani, galaxies, and she was able to plot their velocities around Giulia Zuin the galactic centre as a function of their distance from it.
The Formation of Stars Problem: she found that the rotational speeds of the Fundamental Plane and clouds did not decrease with increasing distance from the FP’s Tilt galactic centre and, in some cases, even increased Dark Matter somewhat. Evidence for an unseen component in spiral galaxies
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 Possible answers: even the stars in the galaxy are embedded in a large halo of unseen matter or Newton’s law of gravity does not hold true for large distances. Dark matter halo in spiral Galaxies
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 Why dark matter is dark
The Virial Theorem and its astronomical applications
Marco Loreggia, With ordinary matter, electromagnetism affects everything Anna Sancassani, from chemistry to luminosity to electric and magnetic Giulia Zuin fields and even the pressure of stellar winds; thus The electromagnetism plays an important role in determining Formation of Stars the arrangement of ordinary matter, which is often Fundamental irregular. Plane and FP’s Tilt
Dark Matter Why dark matter is dark
The Virial Theorem and its astronomical applications With ordinary matter, electromagnetism affects everything Marco from chemistry to luminosity to electric and magnetic Loreggia, Anna fields and even the pressure of stellar winds; thus Sancassani, Giulia Zuin electromagnetism plays an important role in determining
The the arrangement of ordinary matter, which is often Formation of Stars irregular. Fundamental Except through gravitation, dark matter does not Plane and FP’s Tilt interact (or interacts only very weakly) with itself or with Dark Matter ordinary matter. Indeed, that’s why it’s dark: to emit light it would have to interact via the electromagnetic force. Why dark matter is dark
The Virial Theorem and With ordinary matter, electromagnetism affects everything its astronomical from chemistry to luminosity to electric and magnetic applications fields and even the pressure of stellar winds; thus Marco Loreggia, electromagnetism plays an important role in determining Anna Sancassani, the arrangement of ordinary matter, which is often Giulia Zuin irregular. The Formation of Except through gravitation, dark matter does not Stars interact (or interacts only very weakly) with itself or with Fundamental Plane and ordinary matter. Indeed, that’s why it’s dark: to emit light FP’s Tilt it would have to interact via the electromagnetic force. Dark Matter Because electromagnetism plays no role in the distribution of dark matter, however, dark matter forms large, smooth, spherical clumps, usually filled by ordinary galaxies plus hot gas or plasma, which it has trapped and retained solely through gravitation. Gravitational Lensing
The Virial Theorem and its astronomical applications Mass curves the space around it, bending the paths along Marco Loreggia, which rays of light travel. Anna Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter Gravitational Lensing
The Virial Theorem and its astronomical applications →The more mass and the closer to the center of mass, Marco the more space bends, and the more the image of a Loreggia, Anna distant object is displaced and distorted. Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter Gravitational Lensing
The Virial Theorem and its astronomical →The more mass and the closer to the center of mass, applications the more space bends, and the more the image of a Marco Loreggia, distant object is displaced and distorted. Anna Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter
Thus: measuring distortion (or shear) is key to measuring the mass of the lensing object itself. Homemade gravitational lens effect
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 Homemade gravitational lens effect
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 Homemade gravitational lens effect
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 Gravitational Lensing
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 1 1An international team of astronomers using the NASA/ESA Hubble Space Telescope has discovered a ghostly ring of dark matter that was formed long ago during a titanic collision between two massive galaxy clusters. It is the first time that a dark matter distribution has been found that differs substantially from the distribution of ordinary matter. Gravitational Lensing - Strong Lensing
The Virial Theorem and its astronomical Lensing of this type are called strong lensing. applications (A very massive object or collection of objects, like a Marco Loreggia, nearby galaxy cluster and the dark matter that encloses it). Anna Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter
,→ The visible distortion is a direct measure of the mass of the lens and points to its center. Gravitational Lensing - Weak Lensing
The Virial Theorem and its astronomical applications There are also weak lensing, work in the same way of Marco strong lensing, exept that the shear is too subtle to be Loreggia, Anna seen sirectly. Most of the apparent shear isn’t distortion at Sancassani, Giulia Zuin all: a galaxy has its own distinct shape, and we often see
The it from an angle that makes it too elongated. Formation of Stars Faint additional distortions in a collection of distant Fundamental galaxies can be calculated statistically, and the average Plane and FP’s Tilt shear due to the lensing of some massive object in front of Dark Matter them can be computed. Gravitational Lensing - Weak Lensing
The Virial Theorem and its astronomical There are also weak lensing, work in the same way of applications strong lensing, exept that the shear is too subtle to be Marco Loreggia, seen sirectly. Most of the apparent shear isn’t distortion at Anna Sancassani, all: a galaxy has its own distinct shape, and we often see Giulia Zuin it from an angle that makes it too elongated. The Formation of Faint additional distortions in a collection of distant Stars galaxies can be calculated statistically, and the average Fundamental Plane and shear due to the lensing of some massive object in front of FP’s Tilt them can be computed. Dark Matter
Yet, to calculate the mass of the lens from average shear, one has to know its center. Mapping the Dark Matter
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 Possible DM Candidates - Barionic Dark Matter
The Virial Theorem and its astronomical applications
Marco Loreggia, MACHOs (MAssive Compact Halo Objects) : Anna Sancassani, - Brown Dwarfs Giulia Zuin - Low mass, Faint red Stars The - White Dwarfs Formation of Stars - Neutron Stars Fundamental - Black Holes Plane and FP’s Tilt
Dark Matter → Detection Method: Gravitational Lensing Possible DM Candidates - Nonbarionic Dark Matter
The Virial Theorem and its WIMPs (Weakly Interacting Massive Particles) : astronomical applications neutral particles formed during the Big Bang,
Marco passing through massive particles without Loreggia, Anna interacting and which exert and experience Sancassani, Giulia Zuin only gravitational (and possibly weak) forces
The Formation of Stars i.e., photinos, neutrinos, gravitinos, axions... only
Fundamental neutrinos have been detected. Plane and FP’s Tilt Dark Matter → Detection Methods and Projects: The Amanda Project (Antartica Muon and Neutrino Detector Array), The Cryogenic Dark Matter Search, The DAMA Experiment (Particle Dark Mattre Searches with Highly Radiopure Scintillators at Gran Sasso). Composition of the Universe
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 Dark Matter role in Galaxies Formation
The Virial Theorem and its astronomical applications
Marco We essentially have accurate data on the distribution of Loreggia, Anna galaxies over most of the evolution of the universe. Sancassani, Giulia Zuin Although the measurements at earlier epochs have larger errors, due to smaller data sets, their accuracy and power The Formation of to constrain theoretical models is quite remarkable. Stars
Fundamental Plane and FP’s Tilt In 2006, the Chicago scientists based their supercomputer Dark Matter simulations on the assumption that galaxies form in the center of dark-matter halos. Dark Matter role in Galaxies Formation
The Virial Theorem and its astronomical According to this scheme, gravity causes the dark matter applications in these regions to collapse into halos. These halos provide Marco Loreggia, a central location where normal matter consisting of Anna Sancassani, hydrogen, helium and a small amount of heavier elements Giulia Zuin would collect in gaseous form. Once this gas had cooled The and condensed, it achieved sufficient density for star Formation of Stars formation to begin on a galactic scale.
Fundamental Plane and FP’s Tilt
Dark Matter CDM Model - Cluster Formation
The Virial Theorem and CDM Model: asserts that cold dark matter is present in the its astronomical Universe in the form of long filaments, which build a “cosmic applications web” on a big scale. The model also postulates that in the Marco Loreggia, intersections of these filaments there are cluster of galaxies. Anna Sancassani, Giulia Zuin
The Formation of Stars
Fundamental Plane and FP’s Tilt
Dark Matter Galaxy Clusters - Filaments of DM
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
2 2A July 2012 study of the galaxy clusters Abell 222 and Abell 223 found they are connected by a dark matter filament, shown here. The blue shading and the yellow contours indicate the density of matter. References:
The Virial Theorem and its astronomical applications
Marco 1 IAN MORRISON, Introduction to Astronomy and Cosmology, Wiley, 2008 Loreggia, Anna Sancassani, 2 STEVE KOPPES, Simulations show dark matter’s role in galaxy formation, Giulia Zuin The University of Chicago Chronicle, http://chronicle.uchicago.edu/060713/darkmatter.shtml The Formation of Stars 3 Un filamento di materia oscura tra due ammassi di galassie, Le Scienze, Fundamental 06/07/2012, www.lescienze.it/news/2012/07/06/news/filamento_ Plane and FP’s Tilt materia_oscura_cluster_galassie-1133317/
Dark Matter 4 Nuove ipotesi sulla materia oscura, Le Scienze, 10/06/2013