Discovery of Neptune and Planetary Perturbation

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Discovery of Neptune and Planetary Perturbation

• Tse Hon Ning
• Wong Mau Fung

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Content

• Introduction
• Forward problem - Uranus Perturbation
• Inverse problem Orbital element of Neptune
• Very brief story of Pluto
• Summary

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Introduction

• The discovery of Neptune

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Uranus

• Discovered in 1781
• 1821, irregularities in Uranuss orbit were
  observed.
• Radial distance from the Sun
• Heliocentric longitude

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The discrepancy in HL of Uranus,
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The main features of

• Typical magnitude 50 100s of arc
• Approximate period 110 yrs
• Secular variation appeared
• ?fraise through 0 at 1777

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Possible reasons for the irregularities

• Observational error?
• Correction of Newtons law of gravitation for
  distance object?
• Perturbed by an undiscovered planet?

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Perturbed by an undiscovered planet!!
John Couch Adams
Urbain Le Verrier
?f?orbital elements of the unknown planet
(inverse problem!)
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• Neptune was found in 1864!!!

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The forward problem in celestial mechanics

• The perturbation of Uranus

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Forward Problem
model parameters ? data
Orbital elements of Neptune
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• Formulation
• Equations of motion
• u radial displacement
• v tangential displacement
• The homogeneous solutions of u, v
• Brief calculation
• Interpretation
• The inhomogeneous solutions of u, v
• Brief calculation
• Interpretation
• The complete solution of
• Finding constants by fitting
• Physical Interpretation
• The effect of the 2 solutions

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Formulation

• Approximations
• angle of inclination of orbits ignored
• (U0.77 N1.78)
• Eccentricities are ignored
• (U0.0471 N0.0085)

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??f remaining discrepancy in HL after subtracted
other planetary perturbations ? all other planets
can be ignored ? 3 bodies problem (Sun, Uranus,
Neptune)
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Inertial frame U,N no interaction

• Unperturbed
• Coplanar circular orbits
• Radii RU, RN known semimajor axes

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Inertial frame U,N no interaction
Units Length AU Time year Mass solar mass
TU, TN orbital period of U and N O2p/T
orbital frequency t t-t0 t0 1822 time of
conjunction
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Rotating frame U,N no interaction

• Rotating with U
• angular velocity of N - O
• ? OU gt ON
• O OU ON 3.666 x 10-2

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Rotating frame U,N no interaction
Polar co-ordinates of U ?U(t), fU(t)
Unperturbed solution ?U(0)(t) RU fU(0)(t) OUt
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• From 3 bodies problem

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The homogeneous solutions of u, v

• Linear, second-order DEs describing a
  non-dissipative system
• u, v 2 coupled oscillators
• ? Finding the normal modes!

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The homogeneous solutions of u, v

• Frequencies of the normal mode 0, OU
• I) freq. 0

Where ai are constants.
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The homogeneous solutions of u, v

• II) freq. OU

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Interpretation on the homo. solutions

• The 2 homo. Solutions represent
• the difference between 2 nearby Kepler orbits
• Denote by semimajor axis a and eccentricity e
• Unperturbed orbit a RU e 0

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Interpretation on the homo. solutions

• 2 independent classes of nearby Kepler orbits
• a RU ?a e 0
• a RU e ?e ? 0

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Interpretation on the homo. solutions

• a RU ?a e 0
• Circular orbit of slightly larger radius
• ? slightly lower frequency
• Circular ? u constant
• Slightly different frequency ? v has a term
  linear in t

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Interpretation on the homo. solutions

• a RU ?a e 0

• remark
• Freq. 0 Freq. OU

u constant
linear in t
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Interpretation on the homo. solutions
Keplers third law
Varying about O OU R RU
Putting
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Interpretation on the homo. solutions

• II) a RU e ?e ? 0
• Period independent of eccentricity
• ? frequency no changed OU
• ? u, v also have freq. OU
• Remark for Freq. OU

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Interpretation on the homo. solutions
Keplers second law
Varying about O OU R RU
Putting
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Interpretation on the homo. solutions

• ?The homo. solutions contain the info. given by
  Keplers laws.

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The inhomogeneous solutions of u, v
y
Expressing the force in ?
U
x
S
N
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The inhomogeneous solutions of u, v

• First term acceleration of U toward N
• Second term acceleration of S toward N
• ?1st 2nd acceleration of U relative to S
• Substitute the ?terms
• where

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The inhomogeneous solutions of u, v
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The inhomogeneous solutions of u, v

• Fr and Ffare periodic,
• They contain harmonics at freq. nO
• ? Fourier Series!

odd
even
Fourier coeff.
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The inhomogeneous solutions of u, v
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The inhomogeneous solutions of u, v

• 2 coupled oscillators, driven by forces with
• freq. nO
• ? the inhomo. soln also hv freq. nO
• Let , sub. Into
• The coeff.
• Then

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Interpretation of the inhomogeneous solution

• for
  n2
• Hence n2 term dominate
• Sub into the solution to get
• The period and phase nearly agree with observed
  data
• But amplitude too large!!

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The complete solution

• The complete solution sum of the homogeneous and
  inhomogeneous solutions
• determined by initial values and time
  derivatives of u and v
• But no reason to prefer the data at the initial
  time than any other time

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The complete solution

• Thus use linear least-squares-fitting, minimize
• With many data points
• The best fit is found to be

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Partial contribution of the homogeneous and
inhomogeneous solution
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Partial contribution of the homogeneous and
inhomogeneous solution

• Magnitudes of these sinusoidal terms
• As their frequencies are approximately equal
• the complete solution would show beats.
• The discussed time has small beat amplitude of
  , but will get larger and larger in later
  time, since adding, not cancelling of the
  solutions.

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Partial contribution of the homogeneous and
inhomogeneous solution
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Sum of the contribution
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Inverse problem in celestial mechanics

• Orbital element of Neptune

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Inverse problem

• To Determine orbital elements from the historical
  data on .
• Fit the data to the complete solution
• Situation
• Neptunes mass unknown
• Frequency of Neptune unknown
• Time of conjunction unknown

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Inverse problem

• is proportional to Neptune mass
• Use ratio
• assumed Neptune mass/ true
  Neptune mass
• Frequency of the Neptune
• Thus D is now a function of seven variables

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Inverse problem

• To minimize with respect to
  all seven variables ?determine the Neptunes
  orbital elements
• Given , we can find best and
  analytically and define
• But can be found numerically only!!!!

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Inverse problem

• By modern knowledge , expected value of and
  are known
• Try the results close to the expected values of
  them
• Thus , and can be determined, results
• Root mean square is smaller than in the
  forward problem

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The best fit is incorrect?

• The inhomogeneous solution
• True frequency leads to a driving force slightly
  under resonance
• But the fitted frequency corresponds to a force
  slightly above resonance
• and not individually constrained by the
  data.
• But we just need to show resonance!!

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Inverse problem (phase ambiguity)

• Second harmonic (n2) ? phase ambiguity
• i) after displacing N by 180 degrees
• ( )
• ii) the inhomogeneous solution would be
  completely unchanged
• iii) thus another good fit can be found by
  the local minimum

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Inverse problem (phase ambiguity)

• Thus there are two solutions for the heliocentric
  longitude of Neptune, diametrically opposite each
  other
• And they are not too different in terms of the
  quality of the fit to the data

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Brief review on the model

• forward problem orbital elements?

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Brief review on the model

• Inverse problem ?orbital elements
• Result agrees but have phase ambiguity

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Brief story of Pluto

• Immense distance from Earth
• Discovered until 1930(by Lowell)
• Lowell believe planet X exists, based on
  calculations done with study of the motions of
  Uranus and Neptune
• Then set up Lowell Observatory
• Tombaugh used photographs

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Brief story of Pluto

• Any shifting of object against backdrop of the
  stars ? present of planetary body
• Pluto finally found on 18, Feb, 1930.
• However, pluto too small to affect the orbit of
  Neptune
• Continue efforts ? turned out to be error in
  calculations when Voyager 2 was used.

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Whether a planet exists?

• How bright the planet is?
• How fast it was moving at the time of search?
• How accurate the predictions are from the
  celestial mechanics calculations?
• How good quality is the observations?

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Summary

• Ephemerides have been improving
• But a consistent one could not be prepared based
  on all available data
• Two planets have already been discovered based on
  the motion of Uranus
• Are there more?

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The End?