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What is a dimensionless number, and why are the A. Berger equations said to produce these?

Here’s a clean, intuitive way to understand both ideas — dimensionless numbers and why André Berger’s astronomical‑forcing equations produce them — without drowning in jargon.

🌍 What is a dimensionless number?

A dimensionless number is a quantity that has no physical units — no meters, seconds, kilograms, watts, etc.

It’s just a pure ratio.

Why do scientists love them?

Because dimensionless numbers:

let you compare things that would otherwise have incompatible units

reveal underlying physical relationships

stay the same no matter what measurement system you use

often represent geometry, proportions, or normalized quantities

                                                                                 

                                                                              

🧠 Why does this matter for climate modeling?

Dimensionless orbital parameters are ideal because

they can be plugged into climate models regardless of units

they isolate the shape and orientation of Earth’s orbit

they separate geometry from absolute energy flux

they allow clean comparison across millions of years

Climate models then convert these dimensionless parameters into actual energy fluxes (W/m²) when computing insolation.

If you want, I can also:

derive one of Berger’s equations step‑by‑step

show how the dimensionless orbital parameters translate into real insolation

compare Berger (1978) to Laskar (2004)

map how these parameters drive glacial cycles

Just tell me where you want to go next.

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source: Wikipedia

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So where do the physical measurements come in; how are they done?

Great question, Louise — this is exactly the point where the abstract geometry of Berger’s orbital solutions meets the real, physical world. Let’s walk through it cleanly and concretely.

🌍 Where do the physical measurements come in?

Berger’s equations themselves are purely geometric and dynamical.

But the inputs to those equations — the things that describe Earth’s orbit — come from physical measurements made in several ways:

1. Astronomical observations (historical + modern)

These are direct measurements of:

positions of planets

angles in the sky

Earth’s orientation

timing of equinoxes and solstices

lunar and planetary perturbations

How they’re measured

Telescopes (optical, radio, infrared) track planetary positions with arc‑second precision.

Laser ranging to the Moon (Apollo retroreflectors) gives Earth–Moon distance to millimeter accuracy.

Radar ranging to planets (Venus, Mars) gives orbital distances.

Very Long Baseline Interferometry (VLBI) measures Earth’s rotation and wobble.

These measurements feed into the dynamical models that compute:

eccentricity

obliquity

precession

longitude of perihelion

orbital angular momentum

All of these are then converted into the dimensionless orbital parameters Berger uses.

2. Geophysical measurements of Earth’s rotation and tilt

Earth’s obliquity and precession rate are not guessed — they’re measured.

Tools used

VLBI: tracks Earth’s rotation axis relative to quasars

Satellite laser ranging: tracks Earth’s shape and wobble

GRACE and GRACE‑FO: measure mass distribution changes affecting rotation

Gyroscopes in space (Gravity Probe B historically)

These give the physical values of:

axial tilt (in degrees)

precession rate (arcseconds per year)

nutation amplitudes                                                                                 

Berger then expresses these in dimensionless trigonometric combinations like 

𝑒sin𝜛.

3. Planetary ephemerides (numerical integration)

This is where the heavy lifting happens.

Institutions like:

JPL (Jet Propulsion Laboratory)

IMCCE (France)

Observatoire de Paris

Laskar’s group (La2004, La2010 solutions)

integrate the equations of motion of the solar system using:

Newtonian gravity

relativistic corrections

measured masses of planets

measured initial positions and velocities

These integrations produce:

Earth–Sun distance as a function of time

orbital eccentricity

precession of perihelion

obliquity cycles

Berger’s 1976/1978 solutions were based on the best ephemerides of his time.

4. Solar constant and insolation measurements

When Berger converts orbital geometry into actual insolation (W/m²), he needs:

the solar constant (measured by satellites like SORCE, TIM, ACRIM)

Earth’s radius and geometry

atmospheric transmission (for some applications)

These are physical measurements with units.

But the orbital forcing terms remain dimensionless.

5. Geological validation (indirect measurements)

To check that the orbital solutions are correct, scientists compare them to:

ice cores

sediment cores

coral growth bands

speleothems

varves

magnetic reversals

These contain climate cycles that match the predicted:

41,000‑year obliquity cycle

23,000‑year precession cycle

100,000‑year eccentricity modulation

This doesn’t produce the orbital parameters, but it validates them.

🧩 Putting it all together

Here’s the workflow:

Astronomers measure planetary positions, Earth’s tilt, rotation, and distances.

Ephemeris models integrate these measurements to produce orbital elements over millions of years.

Berger takes those orbital elements and expresses them as dimensionless geometric ratios.

Climate scientists then convert those dimensionless parameters into actual insolation using measured solar constants.

So the physical measurements are the inputs; the dimensionless numbers are the outputs of the orbital geometry.

If you want, I can also map:

exactly which measurements feed which Berger parameter

how Laskar’s modern solutions differ from Berger’s

how uncertainties propagate over millions of years

how these orbital parameters translate into glacial cycles

Just tell me which direction you want to explore.

What was Milankovitch himself working from to derive his cycles?

Milankovitch wasn’t doing anything mystical or speculative — he was working from the best astronomical measurements and mathematical tools available between ~1900 and 1940, and he built his climate‑cycle theory by combining classical celestial mechanics, observational astronomy, and hand‑calculated integrals of solar radiation.

What’s fascinating is that he did all of this before computers, before modern ephemerides, and even before precise satellite measurements. Yet his results were astonishingly close to what Berger and Laskar later computed with supercomputers.

Let me lay out exactly what he used.

⭐ 1. Classical celestial mechanics (Laplace, Lagrange, Poisson)

Milankovitch relied heavily on 18th–19th century analytical solutions for the motions of planets:

perturbation theory

secular variations of orbital elements

precession of the equinoxes

gravitational interactions among planets

These gave him time‑dependent formulas for:

eccentricity

longitude of perihelion

obliquity

precession rate

These formulas were not as precise as modern numerical integrations, but they were good enough to capture the major 23k, 41k, and 100k‑year cycles.

⭐ 2. Astronomical observations available in his era

Milankovitch used the observational data compiled by:

Newcomb (1895)

Le Verrier

Hansen

Oppolzer

Poincaré’s refinements

These provided:

measured orbital elements of Earth and planets

measured precession constants

measured obliquity

measured solar constant (rough early estimates)

These were the physical measurements that anchored his equations.

⭐ 3. Newcomb’s planetary tables (his main numerical foundation)

This is the big one.

Milankovitch explicitly based his orbital calculations on Simon Newcomb’s 1895 Tables of the Sun, which were the international standard until the 1980s.

Newcomb’s tables gave:

Earth’s orbital eccentricity as a function of time

longitude of perihelion

mean anomaly

precession constants

obliquity values

These were derived from centuries of telescopic observations and mathematical smoothing.

Milankovitch used these as the initial conditions for his long‑term orbital integrations.

                                                                            

every latitude

every day of the year

across tens of thousands of years

He used:

Earth–Sun distance from orbital geometry

solar declination from obliquity and precession

hour angle geometry

trigonometric expansions

These integrals produced the famous summer insolation curves that drive glacial cycles.

⭐ 5. Geological evidence available at the time

Milankovitch cross‑checked his astronomical predictions against:

Alpine glacial deposits

Scandinavian glacial sequences

marine terraces

early deep‑sea sediment interpretations

This wasn’t as precise as modern δ¹⁸O records, but it gave him confidence that the cycles were real.

⭐ 6. No computers — everything was analytic or tabulated

Milankovitch worked with:

logarithm tables

trigonometric tables

slide rules

hand‑drawn graphs

multi‑page integrals

He produced thousands of pages of calculations, all by hand, to generate what Berger later automated.

🧩 So what did Milankovitch actually have?

Here’s the short version:

                                                                                        

🧠 Why Berger and Laskar later replaced Milankovitch’s numbers

Milankovitch’s method was correct.

His inputs (Newcomb’s tables) were outdated.

Modern ephemerides (DE431, La2004, La2010) use:

radar ranging

lunar laser ranging

spacecraft tracking

relativistic corrections

numerical integration of N‑body equations

So the cycles are the same, but the phase and amplitude are now more accurate.

If you want, I can map Milankovitch → Berger → Laskar in a clean lineage, or show exactly where their orbital solutions differ.

How is Laskar different from Berger?

Here’s a clear, structured way to see how Laskar and Berger differ — not just in results, but in philosophy, mathematics, data sources, and computational power. This is one of those places where the lineage of orbital‑forcing science becomes beautifully clear.

🌍 The short version

Berger = analytical, semi‑numerical orbital solutions + detailed insolation formulas  

Laskar = full N‑body numerical integration of the Solar System with chaos analysis

They are complementary, but Laskar’s work replaces Berger’s orbital elements with far more precise ones.

🔭 1. Their goals were different

Berger (1976–1991)

Wanted accurate insolation curves for paleoclimate models

Needed orbital elements (eccentricity, obliquity, precession) as inputs

Built analytic series expansions for orbital parameters

Focused on climate forcing, not planetary dynamics

Laskar (1986–present)

Wanted high‑precision long‑term planetary motion

Studied chaos in the Solar System

Built full numerical N‑body integrations

Focused on dynamics, not climate

Berger used the best orbital data available.

Laskar generated the best orbital data available.

🧮 2. Their mathematical methods differ radically

Berger’s method

Analytical expansions of orbital elements

Based on classical perturbation theory

Uses trigonometric series with coefficients from astronomical observations

Computationally light (1970s computers)

This produces smooth, predictable orbital cycles.

Laskar’s method

Direct numerical integration of the full Solar System

Includes:

all planets

Moon

relativistic corrections

tidal dissipation

chaotic divergence

Uses symplectic integrators designed for long‑term stability

Requires supercomputers

This produces orbital solutions that are:

more accurate

more detailed

chaotic beyond ~50 Myr

🧭 3. Their data sources differ

Berger

Used:

Newcomb’s 1895 tables

Bretagnon’s semi‑analytical planetary theories

Observational astronomy of the mid‑20th century

These were excellent for their time but limited.

Laskar

Used:

spacecraft tracking

radar ranging

lunar laser ranging

modern planetary masses

relativistic corrections

high‑precision ephemerides (DE series)

This is why Laskar’s orbital elements are now the standard.

📈 4. Their outputs differ

Berger outputs

Eccentricity 

𝑒(𝑡)

Obliquity 

𝜀(𝑡)

Precession parameter 

𝑒sin𝜛

Insolation at any latitude/day

Climate forcing curves

These are dimensionless orbital parameters + insolation in W/m².

Laskar outputs

Full orbital elements for all planets

Chaotic divergence estimates

High‑precision Earth orbital solutions (La1988 → La2010)

Obliquity and precession evolution

No insolation calculations (others compute that using his data)

Laskar gives the raw orbital mechanics; Berger gives the climate‑ready forcing.

🧠 5. Accuracy and timescale

Berger

Accurate for ~1–3 million years

Good for Quaternary climate studies

Limited by analytical approximations

Laskar

Accurate for ~50 million years

Beyond that, chaos dominates

Provides uncertainty envelopes

Now the gold standard for deep‑time orbital forcing

🧩 6. How they fit together in modern climate science

Today’s workflow is:

Use Laskar’s orbital solutions (La2004, La2010)

Feed them into Berger‑style insolation equations

Produce high‑precision insolation curves for paleoclimate models

So the field now uses:

Laskar for orbital mechanics

Berger for insolation geometry

They are not competitors — they are sequential steps in the same pipeline.

🌟 In one sentence

Berger built the climate‑forcing formulas; Laskar built the most accurate orbital inputs for those formulas.

If you want, I can also map:

exactly which Berger equations are replaced by Laskar inputs

how Laskar’s chaos findings limit deep‑time climate reconstructions

a side‑by‑side table of Berger vs. Laskar orbital elements

how Milankovitch → Berger → Laskar form a continuous scientific lineage