Newtonian and general relativistic orbits with small eccentricities on 2D

surfaces

Prof. Chad A. Middleton CMU Physics Seminar

February 6, 2014

Outline

•  The elliptical orbits of Newtonian gravitation

•  The 2D surfaces that generate Newtonian orbits with small eccentricities

•  Precessing elliptical orbits of GR with small eccentricities

•  The 2D surfaces that generate general relativistic orbits with small eccentricities

•  describes the curvature of

spacetime •  describes the matter &

energy in spacetime

Einstein’s theory of general relativity

Sean M. Carrol, Spacetime and Geometry: An Introduction to Einstein’s General Relativity (Addison Wesley, 2004)

Matter tells space

how to curve, space tells matter

how to move.

Gµ� =8�G

c4Tµ�

Gµ⌫

Tµ⌫

Consider a spherically-symmetric, non-rotating massive object… Embedding diagram (t = t0 , θ = π/2).. •  2D equatorial ‘slice’ of the 3D space at one

moment in time

Einstein’s theory of general relativity

where 2GM/c2 = 1 Is there a warped 2D surface that will yield the orbits of planetary motion?

z(r) = 2

s2GM

c2

✓r � 2GM

c2

◆

The Lagrangian in spherical-polar coordinates with a Newtonian potential •  is of the form.. •  where we set and a dot refers to a time derivative.

•  For the azimuthal-coordinate..

L =1

2m

⇣r2 + r2�2

⌘+

GmM

r

✓ = ⇡/2

@L

@�� d

dt

@L

@�= 0 yields r2� = ` -Kepler’s 2nd Law

•  For the radial-coordinate.. •  yields the equation of motion for an object of mass m.. *

•  Using the differential operator..

, * can be written in the form..

The Lagrange equations of motion

@L

@r� d

dt

@L

@r= 0

r � `2

r3+

GM

r2= 0

d

dt=

`

r2d

d�

•  For the radial-coordinate.. •  yields the conic sections..

•  where •  and is the eccentricity of the orbit.

The equation of motion

d2r

d�2� 2

r

✓dr

d�

◆2

� r +GM

`2r2 = 0

r(�) =r0

(1 + " cos�)

`2 = GMr0"

http://conic-sections-section.wikispaces.com/General+Infomation+Main

http://www.controlbooth.com/threads/cyc-color-wash-using-fresnels.30704/

-Kepler’s 1st Law

•  the exact solution..

•  for small eccentricities..

* Notice:

•  * yields an excellent approximation for the solar system planets!

The radial equation of motion

Planets r0 (m) ε % error

Mercury 5.79*1010 0.2056 4.227Venus 1.08*1010 0.0068 0.005Earth 1.50*1011 0.0167 0.028Mars 2.28*1011 0.0934 0.872Jupiter 7.78*1011 0.0483 0.233Saturn 1.43*1012 0.056 0.314Uranus 2.87*1012 0.0461 0.213Neptune 4.50*1012 0.01 0.01

r(�)ex

=

r0(1 + " cos�)

r(�)app ' r0(1� " cos�)% error =

|rex

� rapp

|rex

⇤ 100%

= "2 cos� ⇤ 100%

•  Setting and using for circular orbits...

Notice: •  Kepler’s 3rd Law is independent of m!

Kepler’s 3rd Law

r = r = 0 � = 2⇡/T

T 2 =

✓4⇡2

G

◆· r

3

M

M

m

~F

~v

~r

An object orbiting on a 2D cylindrically-symmetric surface •  is described by a Lagrangian of the form.. •  Now, for the orbiting object..

where for a rolling sphere, for a sliding object.

•  The orbiting object is constrained to reside on the surface.. z = z(r)

L =1

2m(r2 + r2�2 + z2) +

1

2I!2 �mgz

I = ↵mR2and !2

= v2/R2so

1

2

I!2=

1

2

↵mv2

↵ = 2/5↵ = 0

•  For the azimuthal-coordinate.. •  For the radial-coordinate.. •  yields the equation of motion for the orbiting object..

The Lagrange equations of motion

@L

@r� d

dt

@L

@r= 0

@L

@�� d

dt

@L

@�= 0

(1 + z02)r + z0z00r2 �˜2

r3+ gz0 = 0

yields

where

r2� = `/(1 + ↵)

˜⌘ `/(1 + ↵)g ⌘ g/(1 + ↵)

Compare the equation of motion for the orbiting object.. *

•  to the equation of motion for planetary orbits..

L.Q. English and A. Mareno, “Trajectories of rolling marbles on various funnels”, Am. J. Phys. 80 (11), 996-1000 (2012)

* will NOT yield Newtonian orbits on ANY cylindrically-symmetric surface, except in the special case of circular orbits.

The Lagrange equation of motion

(1 + z02)r + z0z00r2 �˜2

r3+ gz0 = 0

r � `2

r3+

GM

r2= 0

Compare the equation of motion for the orbiting object.. *

•  to the equation of motion for planetary orbits..

•  So, what about for nearly circular orbits?

* will NOT yield Newtonian orbits on ANY cylindrically-symmetric surface, except in the special case of circular orbits.

The Lagrange equation of motion

(1 + z02)r + z0z00r2 �˜2

r3+ gz0 = 0

r � `2

r3+

GM

r2= 0

•  Using the differential operator, the radial equation becomes.. •  For nearly circular orbits with small eccentricities..

Notice: •  when , stationary elliptical orbits •  when , precessing elliptical orbits.

⌫ = 0.98where & are parameters.

The radial equation of motion

(1 + z02)d2r

d�2+ [z0z00 � 2

r(1 + z02)]

✓dr

d�

◆2

� r +g˜2z0r4 = 0

r(�) = r0(1� " cos(⌫�))

r0 ⌫" = 0.6

⌫ = 1

⌫ 6= 1

•  We find the solution, to 1st order in the eccentricity, when..

* For … •  Precessing elliptical orbits when n < 2 •  Stationary elliptical orbits, for certain radii, when n < 1

•  No stationary elliptical orbits when n = 1!

Michael Nauenberg, “Perturbation approximation for orbits in axially symmetric funnels”, to appear in Am. J. Phys.

The radial equation of motion

˜2 = gr30z00

z00(1 + z020 )⌫2 = 3z00 + rz000

z(r) / � 1

rn

•  To find the 2D surface that yields stationary elliptical orbits with small eccentricities for all radii, solve…

•  The solution for the slope of the surface is..

* Notice: •  * is independent of spin of orbiting object. •  When , * becomes the equation of an inverted cone with slope .

Gary D. White, “On trajectories of rolling marbles in cones and other funnels”, Am. J. Phys. 81 (12), 890-898 (2013)

where is an arbitrary integration constant

The 2D surface that generates Newtonian orbits

z0�1 + z02

�= 3z0 + rz00

dz

dr=

p2 (1 + r4)�1/2

= 0p2

•  Integrating yields the shape function...

•  where F(a,b) is an elliptic integral of the 1st kind.

Notice: •  This 2D surface will generate stationary elliptical

orbits with small eccentricities for all radii!

The 2D surface that generates Newtonian orbits

z(r) = �p2

✓� 1

◆1/4

F (sin�1(�r4)1/4,�1)

•  Setting and using for circular orbits...

•  Notice that when ..

•  and when ..

Kepler’s 3rd Law

T 2 =

2p2⇡2

g

!rp

1 + r4

r = r = 0 � = 2⇡/T

r4 ⌧ 1

r4 � 1

T 2 / r

T 2 / r3

- Kepler-like relation for that of an inverted cone.

- Kepler’s 3rd Law of planetary motion.

•  The eqn of motion for an object orbiting about a non-rotating, spherically-symmetric object of mass M in GR is..

*

•  where a dot refers to a derivative w.r.t. proper time. •  Using the differential operator, * becomes..

•  we choose a solution of the form.. where & are parameters.

Precessing elliptical orbits in GR with small eccentricities

r � `2

r3+

GM

r2+

3GM`2

c2r4= 0

d2r

d�2� 2

r

✓dr

d�

◆2

� r +GM

`2r2 +

3GM

c2= 0

r(�) = r0(1� " cos(⌫�)) r0 ⌫

•  We find the solution, to 1st order in the eccentricity, when..

Notice:

•  Deviation from closed elliptical orbits increases with decreasing . •  When , becomes complex: elliptical orbits not allowed •  When , & become complex: no circular orbits.

Precessing elliptical orbits in GR with small eccentricities

`2 = GMr0

✓1� 3GM

c2r0

◆�1

⌫2 = 1� 6GM

c2r0

r0r0 < 6GM/c2 ⌫

`r0 < 3GM/c2 ⌫

Planets r0 (m) ε 6GM/c2r0Mercury 5.79*1010 0.2056 1.53*10-7

Venus 1.08*1010 0.0068 8.19*10-7

Earth 1.50*1011 0.0167 5.90*10-8

Mars 2.28*1011 0.0934 3.88*10-8

Jupiter 7.78*1011 0.0483 1.14*10-8

Saturn 1.43*1012 0.056 6.19*10-9

Uranus 2.87*1012 0.0461 3.08*10-9

Neptune 4.50*1012 0.01 1.97*10-9

•  To find the 2D surface that yields precessing elliptical orbits with small eccentricities for all radii, solve…

•  The solution for the slope of the surface is..

Notice: •  dependent on central mass, M, and average radius of orbit, r0. •  depends on in both overall factor and in the power. •  Slope diverges as

where

The 2D surfaces that generates general relativistic orbits

z0(1 + z02)⌫ = 3z0 + rz00 with

dz

dr=

s2 + �

1� �· (1 + r2(2+�))�1/2

�� ! 1

� ⌘ 6GM/c2r0

⌫ =

r1� 6GM

c2r0

•  Slope that generates Newtonian stationary elliptical orbits..

•  Slope that generates the GR precessing elliptical orbits…

Notice: •  They agree when . •  GR offers a tiny correction for the

orbits of the solar system planets.

where

Compare the 2D surfaces…

dz

dr=

s2 + �

1� �· (1 + r2(2+�))�1/2

dz

dr=

p2 (1 + r4)�1/2

� ! 0

� ⌘ 6GM/c2r0

Planets r0 (m) ε βMercury 5.79*1010 0.2056 1.53*10-7

Venus 1.08*1010 0.0068 8.19*10-7

Earth 1.50*1011 0.0167 5.90*10-8

Mars 2.28*1011 0.0934 3.88*10-8

Jupiter 7.78*1011 0.0483 1.14*10-8

Saturn 1.43*1012 0.056 6.19*10-9

Uranus 2.87*1012 0.0461 3.08*10-9

Neptune 4.50*1012 0.01 1.97*10-9