Solution Of Perturbed 2-Body Delaunay Variable Problem

SUGGESTED PROBLEM:

Using Legendre polynomials. estimate the magnitude of the error in the modified Hamiltonian H  in terms of k and   m ₃. 

 (You can take  m ₃ =  mass of Jupiter = 1/ (1047.355 m ☉ ) 

 m ₂ = mass  of Earth  =  1/( 32930 m ☉ )   

 and a ₃ =  semi-major axis of Jupiter = 5.2 AU  

Make sure your reference Hamiltonian conforms with these.

 First obtain the perturbation term R in terms of Legendre functions:

R=  k ²  m ₃    [ 1/  r ₃ +  ½  r ²/ r ₃ ³ - 3/2  r ²/ r ₃ ³  cos ² S ] 

If we take  m ₃ =  mass of Jupiter,  m ₂ = mass  of Earth, and a ₃ =  semi-major axis of Jupiter we can calculate the first order perturbations in L, G,  ℓ and g using the reference Hamiltonian:

H   =  

 -  m ² / 2 L ²  -  k ²  m ₃    [1/  r ₃ +  ½  r ²/ r ₃ ³ - 3/2  r ²/ r ₃ ³  cos ² S ] 

We thereby obtain a functional Hamiltonian:

H  (L, G,  ℓ,  g, m , k ² , m ₃  ,  a ₃  ,  t)  

And can write out the differential equations to solve the problem.  One such equation would be:

dL/ dt =   - ¶ H   / ¶ ℓ         

  Integration yielding:

L  -  L _o =   ò ^t  _t o      F (ℓ)   dt    

Where F (ℓ)  =  F(L, G,  ℓ ,  g, constants, t) 

We  then substitute for each of the variables: L, G etc. Earth and Jupiter values, and also:    

 ℓ   -  ℓ  _o  ,   g   -   g  _o  ,   etc.  leaving everything else constant and taking the specific integral in each case.  Do this for L, G,  ℓ and g

Using the  mass values for Jupiter and Earth  expressed in terms  of solar (m ☉)

 

m ₃ =  1/ (1047.355 m ☉ )

m ₂ =  1/( 32930 m ☉ )   

   a ₃ =  5.2 AU  

On computation using the preceding, we get an error in the reference Hamiltonian:  

d H   =  k ²  m ₃    10   ^-2

For an error magnitude e  »   0.012

Further Analysis Discloses Gravitational Wave Event Has More Prosaic Explanation Than 'Parallel Universes'

 

                  Image from a recent Science & Astronomy Facebook post

Two parallel universes out of phase by an elemental tau t ₁  (10^-43 s)

 As with many advanced physics concepts the abstraction often precedes the discovery or what can be tied physically to it.  Such was the case with de Broglie waves before an actual experiment confirmed the existence of these “matter waves.” Thus, the experimental arrangement:

Leads to electrons scattered by atoms originating from crystal planes inside the nickel crystal, leaving patterns from which the de Broglie wavelength could be calculated according to:

In this case the dark fringes (as suggested  by Deutsch) arise from "shadow photons" originating from a parallel universe.  

By August, 2023 a much more complex detection strategy emerged. This entailed an Oak Ridge experiment, according to NBC News, with physicist Leah Broussard firing a beam of neutrons through a 15m (50-foot)  tunnel. At the end of the tunnel, the particles would strike an impenetrable wall with a neutron detector set up on the other end of the wall.

This entailed an Oak Ridge experiment, according to NBC News, with physicist Leah Broussard firing a beam of neutrons through a 15m (50-foot)  tunnel. At the end of the tunnel, the particles would strike an impenetrable wall with a neutron detector set up on the other end of the wall.

According to Broussard’s theory some of those particles would transform into mirror-image versions of themselves, passing through the wall. Then, according to the working hypothesis, if the neutron detector detected a neutron, it could prove the existence of a parallel dimension or universe.  How so?   Well, in order for the neutron to pass through the wall it has to oscillate into the mirror world-universe then back into our own   If just one neutron does make it to the other side and back it will change the game and mark the first experiment ever accessing a parallel universe via a portal.  (An artist's representation of this model is shown in the lower diagram.)

Few will be surprised here to learn that up to now this attempt at parallel universe detection has not been confirmed. Indeed, it appears to have been debunked, i.e.

ORNL's Portal to a Parallel Universe Myth Debunked

Flash forward and yet another means of parallel universe detection seems to have emerged, this one based on complicated gravitational wave signals from massive black holes.  As per a Facebook post from the Science and Astronomy FB group we read:

Scientists believe they may have detected a mysterious signal that could have come from a parallel universe — possibly traveling through a cosmic wormhole before reaching Earth. The unusual gravitational wave, known as GW190521, doesn’t behave like any black hole collision we’ve seen before, showing no normal “chirp” and lasting just a fraction of a second. This strange signal is now fueling bold new theories about hidden realities and dimensions beyond our own.

Well a "strange signal fueling bold new theories about hidden realities beyond our own" is one thing, but proving them is another thing. That being the case let's back up and clarify a few points of reference. Wormholes are hypothetical objects which connect one distant point in space from another, or two universes, and were first proposed by Albert Einstein and Nathan Rosen in 1935.  But let us be clear we have not detected any such entities up to now.

Black holes, on the other hand have been detected, in particular black hole collisions using LIGO ( Laser Interferometer Gravitational Wave Observatory)  e.g.

Navograv Astrophysics Team Makes History With Detection Of Black Hole Collision Using Gravitational Waves

The latest effort focuses on a gravitational wave event denoted GW190521 and the paper can be accessed at the link below:

GW190521: A Binary Black Hole Merger with a Total Mass of | Phys. Rev. Lett.

According to one (minority) take the gravitational wave event detected by the LIGO-Virgo collaboration on 21 May 2019, is a good candidate for an actual wormhole detection. The signal is unusual in its extremely short duration, lasting around 0.1 seconds, according to the team.

That team writes:

"A particularly compelling aspect of GW190521 is the lack of precursor signal prior to the main burst, which hints the possibility that it might not originate from a standard coalescence process of BBHs [binary black holes]," the team writes. "We hypothesize that GW190521 might represent a single, isolated GW echo pulse from the wormhole, which is the postmerger remnant of BBHs in another universe and connected to our universe through a throat."

Adding for good effect:

"The postmerger ringdown signal passes through the throat of wormhole and penetrate the photon sphere barrier in the side of our universe, and could then be detected in our universe as a short duration burst lacking a pre-merger phase."

Modeling the wormhole scenario, the team found that it could produce a signal-to-noise ratio comparable to the observed event. However, according to their Bayesian analysis it did not currently fit the event better than the binary black hole merger scenario proposed by the LIGO-Virgo collaboration. 

To cut a somewhat involved explanation and hypothesis short, the team believes that the detection of the gravitational wave event  GW231123 on November 23, 2023, shares a similar burst-like short duration nature with GW19052. And this could be evidence for a parallel universe, carefully phrased as:

 "an event motivating intensive investigation on the nature of its source"

 Further exercising caution, probably recalling the Oak Ridge fiasco, the team concludes.  

"A systematic model comparison test including various possible sources for such short duration GW signals might be necessary for better understanding the physical origins of corresponding GW events."

Lastly, we need to recognize there is a more plausible, non-exotic explanation for the signal oddity,

GW190521 as a dynamical capture of two nonspinning black holes | Nature Astronomy

See Also:

• Revisiting The Concept Of Parallel Universes
• And:
• Parallel Worlds Are Real And You Exist In Millions Of Them

And:

GW190521 The Impossible Black Hole

And:

Video | Numerical simulation of a heavy black-hole merger (GW190521) | LIGO Lab | Caltech

And:

Unusual Gravitational Wave May Be Sign of Wormhole Linking Universes : ScienceAlert

The Perturbed 2-Body Problem And The Role Of The Delaunay Variables In Solutions

 

                             Perturbation diagram to for Delaunay variable analysis

The perturbed two-body problem begins by mathematically defining the perturbation in terms of the quantities in the triangle above. This actually shows  how the presence of a third body (say m ₃  in the accompanying diagram) perturbs or disturbs the motion of m ₂. which is in orbit about a central mass m ₁. (And of course: m ₁  >>  m ₂  )  Then we can write for the perturbation (N.B.  m  is the reduced mass, m =   1/ (1/m ₂   + 1/ m ₃):

r'' =  -  m r  / r³   -   G m ₃  ((r -  r ₃  /  D ³  ) +   r ₃   /   r ₃  ³

=    -  m r  / r³   -  Ñ R

Where:   Ñ R  =   G m ₃  (r -  r₃  /  D ³  ) +   r ₃   /   r ₃  ³

And:  R  =   - G m ₃  (1/  D    +   r ₂ ·  r ₃   /   r ₃  ³)

Then, adding the perturbations we get:

r'' = -  m r  / r³   -  Ñ R  =  Ö( x'' ²  +  y'' ²  +  z'' ²  )

For which a set of coordinates: q _1,2,3   =  x, y, z are defined, and a set of momenta p _1,2,3   =  x', y', z'    associated with them.  Hence:

dq _i  / dt  = ¶ H  / ¶ p _i        and:     dp _i  / dt  = -¶ H  / ¶ q _i 

Leading to the Hamiltonian:  

H  =   ½ å ³ _i=1  P _i ²    -  m / r  -  R

If this form exists, then a, e, i  etc. must be functions of the time and also further computations may be needed, i.e.

But instead of functions with a, e, i one can make use of the Delaunay variables: L', G', H', respectively. Where:

L'  =  - ¶ H  / ¶ ℓ 

G' =  -   ¶ H  / ¶ g 

H’  =  -   ¶ H  / ¶ h

These Delaunay variables are defined by the equations:

ℓ  =   n(t  - T)

L =  Ö(m a)

G  =   L    Ö (1 -   e ²)

H =   G cos i

g   =  w

h  =   Ω

Where a is the orbit's semi-major axis,  Ω  is the longitude of the ascending node,  is the arguments of the perihelion, e is the eccentricity of the orbit and i is the inclination of orbit to the ecliptic plane.

In this case, the previous Hamiltonian: 

H  =   ½ å ³ _i=1  P _i ²    -  m / r  -  R

Transforms to:

H  (L,  ℓ  )  =   -  m ² / 2 L ²  -  k ²  m ₃    {1/  D (L,  ℓ  )  +   r ·  r ₃   /   r ₃  ² }

 A huge simplification arrives  if R = 0., which then reduces to the 2-body problem, which we've examined in previous blog posts.   Note that R in the second Hamiltonian is expressed:

R   =   k ²  m ₃    {1/  D (L,  ℓ  )  +   r ·  r ₃   /   r ₃  ² }

Now, for the case at hand (consult diagram), we may write:

D ²     =  r  ²   +   r ₃  ²    -   2 r ·  r ₃  cos S

 And:  r ·  r ₃  =   r   r ₃  cos S

Assume now that r ₃  is greater than r (perturbing an inner planet or body by an outer one):

1 /D      =  1/r [ 1 + (r / r ₃)  ²  -  2 (r / r ₃)  cos S] ^1/2

Then the above can be rewritten as the sum:

½ å ^¥ _{r= 0}  (r / r ₃) ⁱ  P _i 

=   1/  r ₃  [ P _o   +   (r / r ₃)  P ₁   + (r / r ₃)  ²  P ₂    +  ...]

The P _i   are functions of the angle S and are called Legendre polynomials.  In this case, the first three may be written:

P _o   =   1,      P ₁     =   cos S,    P ₂     =   ½ (3  cos ² S   -   1) 

Suggested Problem:

Using Legendre polynomials. estimate the magnitude of the error in the modified Hamiltonian H  in terms of k and   m ₃. 

 (You can take  m ₃ =  mass of Jupiter = 1/ (1047.355 m ☉ ) 

 m ₂ = mass  of Earth  =  1/( 32930 m ☉ )   

 and a ₃ =  semi-major axis of Jupiter = 5.2 AU  

(N.B.   ☉  denotes solar units)

Make sure your reference Hamiltonian conforms with these.