<p><strong>tl;dr:</strong> It will be around the year 14,800 that Earth's eccentricity drops below that of Neptune's.</p>
<p>It's unusual to post two answers, but this one is sufficiently different that in this case I think it's better as the contents and comments they may attract may be quite different.</p>
<p>Lucky for us the one of the <a href="https://en.wikipedia.org/wiki/Jet_Propulsion_Laboratory_Development_Ephemeris#Latest_releases%5B8%5D" rel="nofollow noreferrer">JPL development ephemerides' latest releases</a> goes out beyond AD 17,000!  See Ryan S. Park et al 2021 AJ <strong>161</strong> 105 <a href="https://iopscience.iop.org/article/10.3847/1538-3881/abd414" rel="nofollow noreferrer">The JPL Planetary and Lunar Ephemerides DE440 and DE441</a>.</p>
<p>Also lucky for us the Python package <a href="https://rhodesmill.org/skyfield/" rel="nofollow noreferrer">Skyfield</a> can download and interpolate the ephemerides for us in an easy to use way.</p>
<p>However once we have state vectors (position and velocities) rather than orbital elements, we need to decide what &quot;eccentricity&quot; means. Orbital elements assume Kelperian trajectories and what we have is numerically integrated solutions to an n-body problem.</p>
<p>So I'll use an <em>ad hoc</em> definition <span class="math-container">$e = (r_{max} - r_{max}) / (r_{max} + r_{max})$</span> and look at two distances; heliocentric and barycentric.</p>
<p>We can't use osculating orbital elements since those are constantly changing within a single orbit.</p>
<p>Here are the results. Every 200 years I calculate Neptune's position twice a year (total of 332 times per orbit) and every 50 years I calculate Earth's position once a day (367 times per orbit) then apply the equation above to get the two eccentricities.</p>
<p>The result agrees qualitatively with my <a href="https://astronomy.stackexchange.com/a/47188/7982">other answer</a>, but since I show more accurate numerical data rather than ballparking the plots, I can now say that it will be around the year 14,800 that Earth's eccentricity drops below that of Neptune's.</p>
<p><a href="https://i.sstatic.net/KF3kK.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/KF3kK.png" alt="When will Earth's eccentricity drop below that of Neptune's?" /></a></p>
<pre><code>from skyfield import api
import numpy as np
import matplotlib.pyplot as plt

loaddata = api.Loader('~/Documents/fishing/SkyData')  # avoids multiple copies of large files

ts = loaddata.timescale() # include builtin=True if you want to use older files (you may miss some leap-seconds)
eph = loaddata('de441.bsp')

sun = eph['sun']
earth = eph['earth barycenter']
neptune = eph['neptune barycenter']

# Neptune
years_neptune = np.arange(2000, 17001, 200)
yearz = np.arange(0, 165.6, 0.5)
e_neptune_bary, e_neptune_helio = [], []
for year in years_neptune:
    y = year + yearz
    times = ts.utc(year + yearz, 1, 1)

    x_bary = neptune.at(times).position.km
    r_bary = np.sqrt((x_bary**2).sum(axis=0))
    e_bary = (r_bary.max() - r_bary.min()) / (r_bary.max() + r_bary.min())
    e_neptune_bary.append(e_bary)

    r_helio  = neptune.at(times).observe(sun).apparent().distance().km
    e_helio = (r_helio.max() - r_helio.min()) / (r_helio.max() + r_helio.min())
    e_neptune_helio.append(e_helio)

e_neptune_bary = np.array(e_neptune_bary)
e_neptune_helio = np.array(e_neptune_helio)

# Earth
years_earth = np.arange(2000, 17001, 50)
days = np.arange(367)
e_earth_bary, e_earth_helio = [], []
for year in years_earth:
    times = ts.utc(year, 1, days)

    x_bary = earth.at(times).position.km
    r_bary = np.sqrt((x_bary**2).sum(axis=0))
    e_bary = (r_bary.max() - r_bary.min()) / (r_bary.max() + r_bary.min())
    e_earth_bary.append(e_bary)

    r_helio  = earth.at(times).observe(sun).apparent().distance().km
    e_helio = (r_helio.max() - r_helio.min()) / (r_helio.max() + r_helio.min())
    e_earth_helio.append(e_helio)

e_earth_bary = np.array(e_earth_bary)
e_earth_helio = np.array(e_earth_helio)

if True:
    fig, (ax1, ax2, ax3) = plt.subplots(3, 1)
    ax1.plot(years_neptune, e_neptune_bary, label='barycentric')
    ax1.plot(years_neptune, e_neptune_helio, label='heliocentric')
    ax1.set_xlim(1950, 17050)
    ax1.set_ylim(0, None)
    ax1.legend()
    ax1.set_ylabel('eccentricity')
    ax2.plot(years_earth, e_earth_bary, label='barycentric')
    ax2.plot(years_earth, e_earth_helio, label='heliocentric')
    ax2.set_xlim(1950, 17050)
    ax2.set_ylim(0, None)
    ax2.legend()
    ax2.set_ylabel('eccentricity')
    ax3.plot(years_neptune, e_neptune_bary, label='Neptune barycentric')
    ax3.plot(years_earth, e_earth_helio, label='Earth heliocentric')
    ax3.set_xlim(1950, 17050)
    ax3.set_ylabel('eccentricity')
    ax3.set_xlabel('Year')
    ax3.set_ylim(0, None)
    ax3.legend()
    fig.suptitle('DE441, e = rmax-rmin / rmax+rmin', fontsize=14)
    plt.show()
</code></pre>

CC BY-SA 4.0
2021-10-24T03:35:13.170Z
2021-10-24T03:35:13.170Z
uhoh
<p>Earth's Eccentricity varies between about 0.000055 and 0.0679.  This is the first <a href="https://en.wikipedia.org/wiki/Milankovitch_cycles#Orbital_eccentricity" rel="nofollow noreferrer">Milankovich cycle.</a>  When Earth' eccentricity is at its lowest, it is lower than that of Neptune.</p>
<blockquote>
<p>The Earth's orbit approximates an ellipse. Eccentricity measures the departure of this ellipse from circularity. The shape of the Earth's orbit varies between nearly circular (with the lowest eccentricity of 0.000055) and mildly elliptical (highest eccentricity of 0.0679).<sup>4</sup> Its geometric or logarithmic mean is 0.0019. The major component of these variations occurs with a period of 413,000 years (eccentricity variation of ±0.012). Other components have 95,000-year and 125,000-year cycles (with a beat period of 400,000 years). They loosely combine into a 100,000-year cycle (variation of −0.03 to +0.02). The present eccentricity is 0.017 and decreasing.</p>
</blockquote>
<hr />
<p><sup>4</sup>Laskar J, Fienga A, Gastineau M, Manche H (2011). &quot;La2010: A New Orbital Solution for the Long-term Motion of the Earth&quot; <a href="http://www.aanda.org/articles/aa/pdf/2011/08/aa16836-11.pdf" rel="nofollow noreferrer">PDF</a>. Astronomy &amp; Astrophysics. 532 (A889): A89. arXiv:<a href="https://arxiv.org/abs/1103.1084" rel="nofollow noreferrer">1103.1084</a></p>

CC BY-SA 4.0
2021-10-21T04:18:06.090Z
Without saying something about what Neptune's orbit will do in the long term, this is not yet an answer. The title of the question is "**Neptune's eccentricity stability**" and in the body the OP asks "**Will Neptune's orbital eccentricity** remain lower than earth's forever?" **But this only talk about Earth's orbit** likely because a Wikipedia page for Earth's orbit was easy to find. This is a helpful comment only. `-1`
uhoh
CC BY-SA 4.0
2021-10-21T11:21:46.060Z
@uhoh indeed. This doesn't answer my question, only gives a lovely background of the question if you ask it about Earth
איתי מרלוב
CC BY-SA 4.0
2021-10-21T13:33:53.037Z
The very first sentence here does answer the question, though.
Rory Alsop
CC BY-SA 4.0
2021-10-22T02:27:08.973Z
@RoryAlsop the question is about the *time variation* of Neptune's eccentricity; "Will Neptune's orbital eccentricity remain lower than earth's forever?" The first sentence in this answer takes Neptune's eccentricity as a constant, exactly the opposite of what's asked.
uhoh
CC BY-SA 4.0
2021-10-22T20:22:01.080Z
No, it doesn't say anything of the sort @uhoh. You have interpreted it to say it's a constant, but that is just your assumption.
Rory Alsop
CC BY-SA 4.0
2021-10-22T21:49:30.633Z
@RoryAlsop there are four lights
uhoh
CC BY-SA 4.0
2021-10-20T19:43:53.230Z
2021-10-21T04:15:56.740Z
2021-10-21T04:15:56.740Z
James K
<p>Answer supported with sources to</p>
<blockquote>
<p>Will Neptune's orbital eccentricity remain lower than earth's forever?</p>
</blockquote>
<p><strong>tl;dr:</strong> No, in about 10,000 years Earths' will go below Neptune's, which stays above it's current value for the next ~400,000 years.</p>
<hr />
<p>It is hard to consider &quot;forever&quot; for two reasons:</p>
<ol>
<li>&quot;forever&quot; is a long time</li>
<li>While numerical integration can be arbitrarily accurate, the inputs needed to model the solar system (e.g. exact positions and velocities at <span class="math-container">$t_0$</span> and masses) can't be ascertained, and the calculation integrating from now until &quot;forever&quot; (or at least until the Sun expands and eats the Earth) is extremely long.</li>
</ol>
<p><a href="https://astronomy.stackexchange.com/a/47138/7982">@JamesK's answer</a> cites a source that suggests Earth's eccentricity oscillates at several frequencies and will range between about 5.5×10<sup>-5</sup> and 6.8×10<sup>-2</sup>.</p>
<p><a href="https://link.springer.com/article/10.1007/s10509-020-03855-w" rel="nofollow noreferrer">The orbital evolution of the Sun–Jupiter–Saturn–Uranus–Neptune system on long time scales</a> (also <a href="https://www.researchgate.net/publication/343911430_The_orbital_evolution_of_the_Sun-Jupiter-Saturn-Uranus-Neptune_system_on_long_time_scales" rel="nofollow noreferrer">here</a>) show Neptune staying <em>fairly low</em> in eccentricity for 2 million years, <strong>but never going anywhere near as low as Earth is expected to.</strong></p>
<h3>No. Not &quot;forever&quot; and not even for very long!</h3>
<p>So for the next two million years or so, Neptune's eccentricity will remain in the same ballpark, whereas Earth's will repeatedly dip way below it.</p>
<hr />
<p>From <a href="https://link.springer.com/article/10.1007/s10509-020-03855-w" rel="nofollow noreferrer">here</a> showing that Neptune will stay above it's current 0.008 eccentricity for at least 400,000 years.</p>
<p><a href="https://i.sstatic.net/b5wgT.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/b5wgT.png" alt="enter image description here" /></a></p>
<p><strong>below:</strong> Cropped from <a href="https://en.wikipedia.org/wiki/File:MilankovitchCyclesOrbitandCores.png" rel="nofollow noreferrer">Milankovitch Cycles Orbit and Cores</a> and annotated to show Neptune's minimum for the next 400 kiloyears.</p>
<p>It looks like in very roughly only 10,000 years Earth's eccentricity will dip below that of Neptune's.</p>
<p><a href="https://i.sstatic.net/34iRn.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/34iRn.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2021-10-25T04:02:23.827Z
Thank you for the informative answer! If Neptune's eccentricity won't stay as it is forever, what's its lowest and highest?
איתי מרלוב
CC BY-SA 4.0
2021-10-25T06:27:25.750Z
@איתימרלוב my answer explains why "forever" can't be addressed. What you're asking can't be answered.
uhoh
CC BY-SA 4.0
2021-10-25T15:02:21.013Z
If this cannot be answered, can you at least tell me which planet in our solar system excluding Mercury through Mars has the eccentricity that varies the least overtime?
איתי מרלוב
CC BY-SA 4.0
2021-10-25T20:42:02.460Z
@איתימרלוב That's a **different question than what you've asked here**, so it's better to post that a new question. But "over time" is too vague just as "forever" is impossible. The DE411 ephemeris that I used in [this answer](https://astronomy.stackexchange.com/a/47189/7982) spans the years -13,200 to 17,191 so we have reliable information there. Outside of that one has to do a lot of poking around looking at various papers that might not agree with each other. That's probably more work than it's worth for me, but if you post as a new question then others may want to give it a try.
uhoh
CC BY-SA 4.0
2021-10-24T02:22:36.897Z
2021-10-24T02:22:36.897Z
uhoh
<p>Will Neptune's orbital eccentricity remain lower than earth's forever?
It currently is at around 0.008, compared to Earth's 0.0167.</p>

<earth><neptune><eccentric-orbit><eccentricity><earths-orbit>
Neptune's eccentricity stability
איתי מרלוב
# Neptune's eccentricity stability

(Asked by: username_0 on 2021-10-20)

Will Neptune's orbital eccentricity remain lower than earth's forever?
It currently is at around 0.008, compared to Earth's 0.0167.

**Tags:** earth, neptune, eccentric-orbit, eccentricity, earths-orbit


---

## 3 answers

### Answer by username_1 on 2021-10-24. Score: 2

tl;dr: It will be around the year 14,800 that Earth's eccentricity drops below that of Neptune's.
It's unusual to post two answers, but this one is sufficiently different that in this case I think it's better as the contents and comments they may attract may be quite different.
Lucky for us the one of the JPL development ephemerides' latest releases goes out beyond AD 17,000!  See Ryan S. Park et al 2021 AJ 161 105 The JPL Planetary and Lunar Ephemerides DE440 and DE441.
Also lucky for us the Python package Skyfield can download and interpolate the ephemerides for us in an easy to use way.
However once we have state vectors (position and velocities) rather than orbital elements, we need to decide what "eccentricity" means. Orbital elements assume Kelperian trajectories and what we have is numerically integrated solutions to an n-body problem.
So I'll use an ad hoc definition $e = (r_{max} - r_{max}) / (r_{max} + r_{max})$ and look at two distances; heliocentric and barycentric.
We can't use osculating orbital elements since those are constantly changing within a single orbit.
Here are the results. Every 200 years I calculate Neptune's position twice a year (total of 332 times per orbit) and every 50 years I calculate Earth's position once a day (367 times per orbit) then apply the equation above to get the two eccentricities.
The result agrees qualitatively with my other answer, but since I show more accurate numerical data rather than ballparking the plots, I can now say that it will be around the year 14,800 that Earth's eccentricity drops below that of Neptune's.

from skyfield import api
import numpy as np
import matplotlib.pyplot as plt

loaddata = api.Loader('~/Documents/fishing/SkyData')  # avoids multiple copies of large files

ts = loaddata.timescale() # include builtin=True if you want to use older files (you may miss some leap-seconds)
eph = loaddata('de441.bsp')

sun = eph['sun']
earth = eph['earth barycenter']
neptune = eph['neptune barycenter']

# Neptune
years_neptune = np.arange(2000, 17001, 200)
yearz = np.arange(0, 165.6, 0.5)
e_neptune_bary, e_neptune_helio = [], []
for year in years_neptune:
    y = year + yearz
    times = ts.utc(year + yearz, 1, 1)

    x_bary = neptune.at(times).position.km
    r_bary = np.sqrt((x_bary**2).sum(axis=0))
    e_bary = (r_bary.max() - r_bary.min()) / (r_bary.max() + r_bary.min())
    e_neptune_bary.append(e_bary)

    r_helio  = neptune.at(times).observe(sun).apparent().distance().km
    e_helio = (r_helio.max() - r_helio.min()) / (r_helio.max() + r_helio.min())
    e_neptune_helio.append(e_helio)

e_neptune_bary = np.array(e_neptune_bary)
e_neptune_helio = np.array(e_neptune_helio)

# Earth
years_earth = np.arange(2000, 17001, 50)
days = np.arange(367)
e_earth_bary, e_earth_helio = [], []
for year in years_earth:
    times = ts.utc(year, 1, days)

    x_bary = earth.at(times).position.km
    r_bary = np.sqrt((x_bary**2).sum(axis=0))
    e_bary = (r_bary.max() - r_bary.min()) / (r_bary.max() + r_bary.min())
    e_earth_bary.append(e_bary)

    r_helio  = earth.at(times).observe(sun).apparent().distance().km
    e_helio = (r_helio.max() - r_helio.min()) / (r_helio.max() + r_helio.min())
    e_earth_helio.append(e_helio)

e_earth_bary = np.array(e_earth_bary)
e_earth_helio = np.array(e_earth_helio)

if True:
    fig, (ax1, ax2, ax3) = plt.subplots(3, 1)
    ax1.plot(years_neptune, e_neptune_bary, label='barycentric')
    ax1.plot(years_neptune, e_neptune_helio, label='heliocentric')
    ax1.set_xlim(1950, 17050)
    ax1.set_ylim(0, None)
    ax1.legend()
    ax1.set_ylabel('eccentricity')
    ax2.plot(years_earth, e_earth_bary, label='barycentric')
    ax2.plot(years_earth, e_earth_helio, label='heliocentric')
    ax2.set_xlim(1950, 17050)
    ax2.set_ylim(0, None)
    ax2.legend()
    ax2.set_ylabel('eccentricity')
    ax3.plot(years_neptune, e_neptune_bary, label='Neptune barycentric')
    ax3.plot(years_earth, e_earth_helio, label='Earth heliocentric')
    ax3.set_xlim(1950, 17050)
    ax3.set_ylabel('eccentricity')
    ax3.set_xlabel('Year')
    ax3.set_ylim(0, None)
    ax3.legend()
    fig.suptitle('DE441, e = rmax-rmin / rmax+rmin', fontsize=14)
    plt.show()


---

### Answer by username_2 on 2021-10-20. Score: 2

Earth's Eccentricity varies between about 0.000055 and 0.0679. This is the first [Milankovich cycle.](https://en.wikipedia.org/wiki/Milankovitch_cycles#Orbital_eccentricity) When Earth' eccentricity is at its lowest, it is lower than that of Neptune.

> The Earth's orbit approximates an ellipse. Eccentricity measures the departure of this ellipse from circularity. The shape of the Earth's orbit varies between nearly circular (with the lowest eccentricity of 0.000055) and mildly elliptical (highest eccentricity of 0.0679).4 Its geometric or logarithmic mean is 0.0019. The major component of these variations occurs with a period of 413,000 years (eccentricity variation of ±0.012). Other components have 95,000-year and 125,000-year cycles (with a beat period of 400,000 years). They loosely combine into a 100,000-year cycle (variation of −0.03 to +0.02). The present eccentricity is 0.017 and decreasing.

---

4Laskar J, Fienga A, Gastineau M, Manche H (2011). "La2010: A New Orbital Solution for the Long-term Motion of the Earth" [PDF](http://www.aanda.org/articles/aa/pdf/2011/08/aa16836-11.pdf). Astronomy & Astrophysics. 532 (A889): A89. arXiv:[1103.1084](https://arxiv.org/abs/1103.1084)

> **Comments:**
> **username_1**: Without saying something about what Neptune's orbit will do in the long term, this is not yet an answer. The title of the question is "**Neptune's eccentricity stability**" and in the body the OP asks "**Will Neptune's orbital eccentricity** remain lower than earth's forever?" **But this only talk about Earth's orbit** likely because a Wikipedia page for Earth's orbit was easy to find. This is a helpful comment only. `-1`  
> **username_0**: @username_1 indeed. This doesn't answer my question, only gives a lovely background of the question if you ask it about Earth  
> **username_3**: The very first sentence here does answer the question, though.  
> **username_1**: @RoryAlsop the question is about the *time variation* of Neptune's eccentricity; "Will Neptune's orbital eccentricity remain lower than earth's forever?" The first sentence in this answer takes Neptune's eccentricity as a constant, exactly the opposite of what's asked.  
> **username_3**: No, it doesn't say anything of the sort @username_1. You have interpreted it to say it's a constant, but that is just your assumption.  
> **username_1**: @RoryAlsop there are four lights  

---

### Answer by username_1 on 2021-10-24. Score: 0

Answer supported with sources to

> Will Neptune's orbital eccentricity remain lower than earth's forever?

**tl;dr:** No, in about 10,000 years Earths' will go below Neptune's, which stays above it's current value for the next ~400,000 years.

---

It is hard to consider "forever" for two reasons:

1. "forever" is a long time
2. While numerical integration can be arbitrarily accurate, the inputs needed to model the solar system (e.g. exact positions and velocities at $t\_0$ and masses) can't be ascertained, and the calculation integrating from now until "forever" (or at least until the Sun expands and eats the Earth) is extremely long.

[@JamesK's answer](https://astronomy.stackexchange.com/a/47138/7982) cites a source that suggests Earth's eccentricity oscillates at several frequencies and will range between about 5.5×10-5 and 6.8×10-2.

[The orbital evolution of the Sun–Jupiter–Saturn–Uranus–Neptune system on long time scales](https://link.springer.com/article/10.1007/s10509-020-03855-w) (also [here](https://www.researchgate.net/publication/343911430_The_orbital_evolution_of_the_Sun-Jupiter-Saturn-Uranus-Neptune_system_on_long_time_scales)) show Neptune staying *fairly low* in eccentricity for 2 million years, **but never going anywhere near as low as Earth is expected to.**

### No. Not "forever" and not even for very long!

So for the next two million years or so, Neptune's eccentricity will remain in the same ballpark, whereas Earth's will repeatedly dip way below it.

---

From [here](https://link.springer.com/article/10.1007/s10509-020-03855-w) showing that Neptune will stay above it's current 0.008 eccentricity for at least 400,000 years.

[![enter image description here](https://i.sstatic.net/b5wgT.png)](https://i.sstatic.net/b5wgT.png)

**below:** Cropped from [Milankovitch Cycles Orbit and Cores](https://en.wikipedia.org/wiki/File:MilankovitchCyclesOrbitandCores.png) and annotated to show Neptune's minimum for the next 400 kiloyears.

It looks like in very roughly only 10,000 years Earth's eccentricity will dip below that of Neptune's.

[![enter image description here](https://i.sstatic.net/34iRn.png)](https://i.sstatic.net/34iRn.png)

> **Comments:**
> **username_0**: Thank you for the informative answer! If Neptune's eccentricity won't stay as it is forever, what's its lowest and highest?  
> **username_1**: @איתימרלוב my answer explains why "forever" can't be addressed. What you're asking can't be answered.  
> **username_0**: If this cannot be answered, can you at least tell me which planet in our solar system excluding Mercury through Mars has the eccentricity that varies the least overtime?  
> **username_1**: @איתימרלוב That's a **different question than what you've asked here**, so it's better to post that a new question. But "over time" is too vague just as "forever" is impossible. The DE411 ephemeris that I used in [this answer](https://astronomy.stackexchange.com/a/47189/7982) spans the years -13,200 to 17,191 so we have reliable information there. Outside of that one has to do a lot of poking around looking at various papers that might not agree with each other. That's probably more work than it's worth for me, but if you post as a new question then others may want to give it a try.  

---

