The Lost Meaning of Jupiter's High-degree Love Numbers
Author
Benjamin Idini
Title
The Lost Meaning of Jupiter's High-degree Love Numbers
Description
Analytical solution to the gyrotidal effect applied to Jupiter's tidal response.
Category
Academic Articles & Supplements
Keywords
Solar system gas giant planets, Galilean satellites, Planetary interior, Tides
URL
http://www.notebookarchive.org/2022-03-anvqac8/
DOI
https://notebookarchive.org/2022-03-anvqac8
Date Added
2022-03-23
Date Last Modified
2022-03-23
File Size
31.78 kilobytes
Supplements
Rights
CC BY 4.0
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This file contains supplementary data for Benjamin Idini and David J. Stevenson, “The Lost Meaning of Jupiter's High-degree Love Numbers,” The Planetary Science Journal, 3(1) 2022 11. https://doi.org/10.3847/PSJ/ac4248.
The Lost Meaning of Jupiter’s High-degree Love Numbers
The Lost Meaning of Jupiter’s High-degree Love Numbers
Benjamin Idini & David Stevenson
California Institute of Technology
bidiniza@caltech.edu
bidiniza@caltech.edu
NASA’s Juno mission recently reported Jupiter’s high-degree (degree l, azimuthal order m = 4,2 ) Love number = 1.289 ± 0.063 (1σ), an order of magnitude above the hydrostatic obtained in a nonrotating Jupiter model. After numerically modeling rotation, the hydrostatic =1.743 ± 0.002 is still 7σ away from the observation, raising doubts about our understanding of Jupiter’s tidal response. Here, we use first-order perturbation theory to explain the hydrostatic result analytically. We use a simple Jupiter equation of state (n = 1 polytrope) to obtain the fractional change in when comparing a rotating model with a nonrotating model. Our analytical result shows that the hydrostatic is dominated by the tidal response at l = m = 2 coupled into the spherical harmonic l,m = 4,2 by the planet’s oblate figure. The l = 4 normalization in introduces an orbital factor into , where a is the satellite semimajor axis and s is Jupiter’s average radius. As a result, different Galilean satellites produce a different . We conclude that high-degree tesseral Love numbers (l > m, m ≥ 2) are dominated by lower-degree Love numbers and thus provide little additional information about interior structure, at least when they are primarily hydrostatic. Our results entail important implications for a future interpretation of the currently observed Juno . After including the coupling from the well-understood l = 2 dynamical tides (Δ ≈ -4%), Jupiter’s hydrostatic requires an unknown dynamical effect to produce a fractional correction Δ≈ -11% in order to fit Juno’s observation within 3σ. Future work is required to explain the required Δ.
k
42
k
42
k
42
k
42
k
42
k
42
k
42
2
(a/s)
k
42
k
42
k
42
k
2
k
42
k
42
k
42
Spherical Bessel functions
Spherical Bessel functions
In[]:=
Clear["Global`*"]j[l_]:=Nest&,,l/.xπ//Simplifydj[l_]:=DNest&,,l,x/.xπ//Simplifyddj[l_]:=DNest&,,l,{x,2}/.xπ//Simplify
l
(-x)
D[#,x]
x
Sin[x]
x
l
(-x)
D[#,x]
x
Sin[x]
x
l
(-x)
D[#,x]
x
Sin[x]
x
Tidal forcing coefficient, Equation (8):
Tidal forcing coefficient, Equation (8):
Replace the corresponding semimajor axis definition into U[l,m].
In[]:=
Rj=6.9911;aio=4.217;aeu=6.709;aga=10.704;U[l_,m_]:=LegendreP[l,m,0]
9
10
10
10
10
10
10
10
l
Rj
aio
1/2
4π(l-m)!
(2l+1)(l+m)!
Evaluation of coefficients in Equation (22):
Evaluation of coefficients in Equation (22):
In[]:=
Q[l_,m_]:=-f1[l_]:=dj[l]+j[l]f2[l_,q_]:=-(πddj[l]+(l+2)dj[l])f3a[l_]:=f3b[l_,q_]:=-f4[l_]:=(3(+)-1)f5[l_]:=Q[l+1,2]Q[l+2,2]f6[l_]:=Q[l-1,2]Q[l,2]c1[q_]:=f1[2]U[2,2]+f2[2,q]f4[2]U[2,2]//FullSimplify;c2[q_]:=f2[4,q]U[4,2]f6[4]//FullSimplify;c3[q_]:=f3a[2]U[2,2]+f3b[4,q]U[4,2]f6[4]+f3b[2,q]U[2,2]f4[2]//FullSimplify;c4[q_]:=f2[2,q]U[2,2]f5[2]//FullSimplify;c5[q_]:=f1[4]U[4,2]+f2[4,q]f4[4]U[4,2]//FullSimplify;c6[q_]:=f3a[4]U[4,2]+f3b[4,q]U[4,2]f4[4]+f3b[2,q]U[2,2]f5[2]//FullSimplify;
1/2
2
l
2
m
4-1
2
l
(l+1)
π
5q
2
π
(2l+1)
π
5q
2
π
l(2l+1)
π
1
2
2
Q[l,2]
2
Q[l+1,2]
3
2
3
2
Solution for A2 and A4, Equations (26) and (27):
Solution for A2 and A4, Equations (26) and (27):
It requires to manually replace the result of A2 and A4 into A22[q_] and A42[q_], respectively.
In[]:=
L1={c1[q],c2[q],c3[q]};L2={c4[q],c5[q],c6[q]};L1=L1/L1[[1]];L2=L2/L2[[1]];L2=L2-L1;A4=L2[[3]]/L2[[2]]//FullSimplify;A2=L1[[3]]-A4L1[[2]]//FullSimplify;A2A4
Out[]=
-10.+
231.989-33.5551q
15.4659+q(-4.49452+1.q)
Out[]=
24.9905+
-118.728+4247.83q
15.4659+q(-4.49452+1.q)
In[]:=
A22[q_]:=-9.999999999999995`+;A42[q_]:=24.990511656423635`+;dk2[q_]:=dk4[q_]:=-1qq=0.0892;
231.98888762845212`-33.55514203728285`q
15.465925841896812`+q(-4.4945180237141455`+1.`q)
-118.72806728735083`+4247.8339632653415`q
15.465925841896812`+q(-4.4945180237141455`+1.`q)
A22[q]j[2]-1
5j[2]-1
A42[q]j[4]-1
9j[4]
πj[3]
In[]:=
dk2[qq]dk4[qq]
Out[]=
1.11255
Out[]=
14.3953
Cite this as: Benjamin Idini, "The Lost Meaning of Jupiter's High-degree Love Numbers" from the Notebook Archive (2022), https://notebookarchive.org/2022-03-anvqac8
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