Upload
others
View
5
Download
0
Embed Size (px)
Citation preview
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