Elliptical-like orbits on a warped spandex fabric

Transcription

1 Elliptical-like orbits on a warped spandex fabric Prof. Chad A. Middleton CMU Physics Seminar September 3, 2015 Submitted to the American Journal of Physics

2 Kepler s 3 Laws of planetary motion Kepler s 1 st Law.. The planets move in elliptical orbits with the sun at one focus. apocenter Kepler s 2 nd Law.. A line extending from the Sun to any planet sweeps out equal areas in equal times. M ~r Kepler s 3 rd Law.. T 2 = 4 2 G r3 M pericenter r( )= r 0 1+" cos m

3 Einstein s theory of general relativity G µ describes the curvature of spacetime T µ G µ = 8 G c 4 T µ describes the matter & energy in spacetime Matter tells space how to curve, space tells matter how to move. Sean M. Carrol, Spacetime and Geometry: An Introduction to Einstein s General Relativity (Addison Wesley, 2004)

4 The physics of the analogy Is there a 2D surface that will generate orbits that obey Kepler s 3 laws? For a marble orbiting on a warped elastic fabric, how does the period of the orbit relate to the radial distance? Can one generate elliptical-like orbits on the warped elastic surface? *C.A. Middleton, The 2D surfaces that generate Newtonian and general relativistic orbits with small eccentricities, Am. J. Phys. 83 (7), (2015) **C. A. Middleton and M. Langston, Circular orbits on a warped spandex fabric, Am. J. Phys. 82 (4), (2014) Sean M. Carrol, Spacetime and Geometry: An Introduction to Einstein s General Relativity (Addison Wesley, 2004)

5 The physics of the analogy Is there a 2D surface that will generate orbits that obey Kepler s 3 laws? -> Not exactly! * For a marble orbiting on a warped elastic fabric, how does the period of the orbit relate to the radial distance? -> See reference ** Can one generate elliptical-like orbits on the warped elastic surface? -> Yes, stay tuned! *C.A. Middleton, The 2D surfaces that generate Newtonian and general relativistic orbits with small eccentricities, Am. J. Phys. 83 (7), (2015) **C. A. Middleton and M. Langston, Circular orbits on a warped spandex fabric, Am. J. Phys. 82 (4), (2014) Sean M. Carrol, Spacetime and Geometry: An Introduction to Einstein s General Relativity (Addison Wesley, 2004)

6 The physics of the analogy Is there a 2D surface that will generate orbits that obey Kepler s 3 laws? -> Not exactly! * For a marble orbiting on a warped elastic fabric, how does the period of the orbit relate to the radial distance? -> See reference ** Can one generate elliptical-like orbits on the warped elastic surface? -> Yes, stay tuned! *C. A. Middleton, The 2D surfaces that generate Newtonian and general relativistic orbits with small eccentricities, Am. J. Phys. 83 (7), (2015) **C. A. Middleton and M. Langston, Circular orbits on a warped spandex fabric, Am. J. Phys. 82 (4), (2014) Sean M. Carrol, Spacetime and Geometry: An Introduction to Einstein s General Relativity (Addison Wesley, 2004)

7 The physics of the analogy Is there a 2D surface that will generate orbits that obey Kepler s 3 laws? -> Not exactly! * For a marble orbiting on a warped elastic fabric, how does the period of the orbit relate to the radial distance? -> See reference ** Can one generate elliptical-like orbits on the warped elastic surface? -> Yes, stay tuned! *C.A. Middleton, The 2D surfaces that generate Newtonian and general relativistic orbits with small eccentricities, Am. J. Phys. 83 (7), (2015) **C. A. Middleton and M. Langston, Circular orbits on a warped spandex fabric, Am. J. Phys. 82 (4), (2014) Sean M. Carrol, Spacetime and Geometry: An Introduction to Einstein s General Relativity (Addison Wesley, 2004)

8 Outline A marble rolling on a cylindrically symmetric surface (Lagrangian dynamics) The shape of the spandex fabric (Calculus of Variations) Small slope regime o Angular separation between successive apsides o Experiment Large slope regime o Angular separation between successive apsides o Experiment Elliptical-like orbits in GR

9 A marble rolling on a cylindrically symmetric surface is described by a Lagrangian of the form.. L = 1 2 m(ṙ2 + r 2 2 +ż 2 )+ 1 2 I!2 mgz Now, for the rolling marble.. I = 2 5 mr2 and! 2 = v 2 /R 2 so 1 2 I!2 = 1 5 mv2 Notice: The marble is constrained to reside on the fabric.. z = z(r)

10 The Lagrange equations of motion take the form * (1 + z 02 ) r + z 0 z 00 ṙ 2 r gz0 =0 define the differential operator d dt = 5` d 7r 2 d * becomes (1 + z 02 ) d2 r d 2 +(z0 z 00 2 r (1 + z02 )) dr d 2 = 5` 7r 2 r + 7g 5`2 z0 r 4 =0

11 The equation of motion for a rolling marble on a cylindrically symmetric surface (1 + z 02 ) d2 r d 2 +(z0 z 00 2 r (1 + z02 )) dr For elliptical-like orbits with small eccentricities d 2 r + 7g 5`2 z0 r 4 =0 2 r( )=r 0 (1 " cos( )) where is the precession parameter 360 v = 1 v > 1 v < Inserting the approximate solution s into the equation of motion yields = 3z z00 0 r 0 z z

12 The slope of the spandex fabric Technique: 1. Construct potential energy (PE) integral functional of spandex fabric. i. Elastic PE of the spandex. ii. Gravitational PE of the spandex. iii. Gravitational PE of the central mass. 2. Apply Calculus of Variations. The elastic fabric-mass system will assume the shape which minimizes the total PE of the system. i. and iii. first considered in Comment on The shape of the Spandex and orbits upon its surface, Am. J. Phys. 70 (10), (2002)

13 The slope of the spandex fabric The Euler-Lagrange equation can be integrated once and takes the form.. rz 0 apple1 1 p 1+z 02 = (M + 0 r 2 ) where we defined the parameter.. g 2 E

14 The slope of the spandex fabric The Euler-Lagrange equation can be integrated once and takes the form.. rz 0 apple1 1 p 1+z 02 = (M + 0 r 2 ) where we defined the parameter.. g 2 E mass of central object areal mass density Modulus of elasticity

15 rz 0 apple1 The angular separation & the slope of the spandex fabric = 360 s 1 p 1+z 02 Small slope regime.. z 0 (r) 1 Large slope regime.. z 0 (r) 1 so z0 0 (1 + z02 0 ) 3z0 0 + r 0z0 00 = (M + 0 r 2 ) so Angular separation between successive like-apsides The slope of the spandex fabric 1 p =1 1 1+z 02 2 z02 + o(z 04 ) 1 p = 1 1 p 1+z 02 z 0 = 1 1+1/z 02 z 0 (1 1 2z 02 + o(1/z04 ))

16 Elliptical-like orbits on the spandex fabric 4 ft. diameter trampoline frame o styrofoam insert for zero pre-stretch o truck tie down around perimeter Camera mounted directly above, ramp mounted on frame. Position determined every 1/60 s and average radius, r ave, calculated per. from r max to r max can be measured. can be calculated via s z0 0 (1 + z02 0 = 360 ) 3z0 0 + r 0z0 00 The three orbits imaged here were found to have angular separations between successive apocenters of = 213.5, 225.7, and eccentricities of " =0.31, 0.29, 0.31

17 Elliptical-like orbits in the small slope regime For z 0 (r) 1, the slope of the spandex surface takes the form 1/3 2 z 0 (r) ' (M +. 0 r 2 ) 1/3 r Plugging this into the equation determining the angular separation yields a theoretical value of Φ deg Central Mass kg kg kg kg r 0 m = 220 h 1+ 2 (M + 0 r 2 )/r 0 2/3 i 1/2 apple r 2 (M + 0 r 2 ) 1/2

18 Elliptical-like orbits in the large slope regime For z 0 (r) 1, the slope of the spandex surface takes the form z 0 (r) ' 1+ M r Plugging this into the equation determining the angular separation yields a theoretical value of = 360 Φ deg v " u t (1 + M/r 0) 1+ (3 + 2 M/r 0 ) Central Mass kg kg kg kg M r 0 2 # r 0 m

19 Elliptical-like orbits in the large slope regime Notice: For a large central mass and a very small average radial distance, the angular separation yields a limiting behavior lim M/r 0 1 ' 360 p 2 M r 0 when M/r 0 > p 2, > 360!

20 Elliptical-like orbits with small eccentricities in GR The equation of motion for an object of mass m orbiting about a spherically symmetric object of mass M, in the presence of a cosmological constant (or vacuum energy), Λ.. * define the differential operator.. * becomes.. r + GM r 2 `2 3GM`2 r c 2 r 4 3 c2 r =0 d d = ` r 2 d d = ` r 2 d 2 r 2 d 2 r dr d 2 + GM `2 r2 r + 3GM c 2 c 2 3`2 r5 =0

21 Elliptical-like orbits with small eccentricities in GR The orbital equation of motion d 2 r 2 d 2 r dr d 2 + GM `2 r2 r + 3GM c 2 c 2 3`2 r5 =0 For elliptical-like orbits with small eccentricities r( )=r 0 (1 " cos( ))

22 Precessing elliptical orbits in GR with small eccentricities Case I: Λ = 0 We find the solution, to 1 st order in the eccentricity, when.. `2 = GMr =1 6GM c 2 r 0 3GM c 2 r 0 1 Notice: Innermost stable circular orbit When r 0 < 6GM/c 2 r ISCO, becomes complex: elliptical-like orbits not allowed! When r 0 < 3GM/c 2, & ` become complex: no circular orbits.

23 Precessing elliptical orbits in GR with small eccentricities Case I: Λ = 0 The angular separation between successive apocenters takes the form r ISCO r 0 * = where 1/2 r ISCO 6GM/c 2 expand about r 0 r ISCO + r where r r ISCO * becomes lim ' 360 p 6 r 6GM/c 2 r GM/c 2 r

24 Precessing elliptical orbits in GR with small eccentricities Case I: Λ = 0, The behavior of the angular separation in GR, near the innermost stable circular orbit lim ' 360 p r GM/c 2 6 r 6GM/c 2 r of the marble, in the large slope regime Notice: lim r 0 M ' 360 p 2 M r 0 Both expressions diverge in the limit of vanishingly small distances. α plays the role of G/c 2 ; both set the scale of their respective theories.

25 Precessing elliptical orbits in GR with small eccentricities Case II: Λ 0 We find the solution, to 1 st order in the eccentricity, when.. 1 3GM `2 = Gr 0 1 c 2 (M 2 4 r 0 3 r3 0 0 ) 4 2 6GM 3GM 3 =1 c r3 0 0 r 0 c 2 r 0 (M r3 0 0) The angular separation between successive apocenters takes the form 1/2 apple 6GM 1 3GM/c 2 4 r 0 3 = c r3 0 0 r 0 1 6GM/c 2 r 0 (M r3 0 0) 1/2

26 Precessing elliptical orbits in GR with small eccentricities Case II: Λ 0, The behavior of the angular separation in GR, about a static, spherically symmetric massive object in the presence of a constant vacuum energy.. 1/2 apple 6GM 1 3GM/c 2 4 r 0 3 = c r3 0 0 r 0 1 6GM/c 2 r 0 (M r3 0 0) 1/2 of the marble, in the small slope regime = 220 h 1+ 2 (M + 0 r 2 )/r 0 2/3 i 1/2 apple r 2 (M + 0 r 2 ) 1/2 Notice: The areal mass density, σ 0, of the spandex fabric plays the role of a negative vacuum energy density, -ρ 0, of spacetime.

27 Conclusion We find good agreement between theory and experiment for the angular separation between successive apsides,. In the large slope regime, > 360 for small radii and large central mass. diverges in the limit of vanishing small distances for o the marble in the large slope regime. o a particle near the innermost stable circular orbit. The areal mass density, σ 0, of the spandex fabric plays the role of a negative vacuum energy density, -ρ 0, of spacetime.

28 Einstein s theory of general relativity Consider a static, spherically symmetric, massive object Embedding diagram (t = t 0, θ = π/2).. 2D equatorial slice of the 3D space at one moment in time s 2GM 2GM z(r) =2 c 2 r c 2 Does a warped spandex fabric yield orbits that obey Kepler s 3 laws? where 2GM/c 2 = 1

29 Circular orbits on a warped spandex fabric, revisited Small slope regime.. T 3 = g Large slope regime 3/2 1 p 2 r 2 (M + 0 r 2 ) 1/2 T 3 s Central Mass 0 kg kg kg kg kg r ave 2 M ΠΣ 0 r ave m 2 kg 1 2 T = /2 5g r (M + r) 1/2 T s Central Mass kg kg kg kg r ave MΑ r ave 1 2 m 1 2 Chad A. Middleton and Michael Langston, Circular orbits on a warped spandex fabric, Am. J. Phys. 82 (4), (2014)

30 Elastic PE of the spandex fabric Elastic PE of a differential concentric ring of the fabric of unstretched width dr du e = 1 2 apple(p dr 2 + dz 2 dr) 2 dr Define the modulus of elasticity, E.. E = appledr 2 r Integrating the differential segment over the whole fabric, the total elastic PE of the fabric is U e = Z R 0 E r( p 1+z 02 1) 2 dr p dr 2 + dz 2 dr dz

31 Gravitational PE of the spandex fabric Gravitational PE of a differential concentric ring of the fabric.. du g,s = dm s g z The mass of the differential ring is a constant under stretching.. dm s = 0 2 rdr = (z 0 ) 2 r p dr 2 + dz 2 where, are the unstretched, variable areal mass densities. 0 (z 0 ) Integrating the differential segment over the whole fabric, the total gravitational PE of the fabric is.. U g,s = Z R g rz dr

32 Gravitational PE of the central mass U g,m = Mg z(0) = Z R 0 Mg z 0 (r) dr Notice: we approximate the central mass as being point-like.

33 The total PE of the spandex-central mass system U = U e + U g,s + U g,m = Z R 0 f(z,z 0 ; r) dr where we defined the functional.. f(z,z 0 ; r) E r( p 1+z 02 1) g rz Mg z 0 To minimize the total PE, subject to the 0 =0

34 The shape equation for the elastic fabric The Euler-Lagrange equation takes the form.. d dr applerz 0 apple 1 1 p 1+z 02 Mg 2 E = 0 g E r which can be integrated.. rz 0 apple1 1 p 1+z 02 = (M + 0 r 2 ) where we defined the parameter.. g 2 E