ShoppingDay

 It's Monday - Senior's Day for shopping - and I'm off. Usually I walk, but it is

still in the minus teens C so maybe the bus...

From CNN, the debate on feeding oneself given the new US dietary guideline, which

recommends protein but not  carbs...

Recommended as per MyPlate:

                                                                                   

Dietician recommends:

                                                                                    

*     *     *

and this useful bit of intelligence from my YouTube feed:

                                                                               

*     *     *

Copilot:

Could humans survive outside an Ice Age. We have evolved to be mammals with an interior temperature higher than exterior temperature comfort...

This is a rich question, and it gets at something people rarely think about: humans evolved inside an Ice Age, but we are not biologically dependent on Ice Age conditions. What we are dependent on is a very narrow band of thermal, ecological, and agricultural stability — and that’s the real constraint.

Let’s unpack it clearly and without drama.

🌡️ 1. Humans are warm‑blooded, but that doesn’t mean we need a cold planet

Our internal temperature (~37°C) is regulated internally. What matters for survival is not the planet’s average temperature, but:

whether we can shed heat efficiently

whether ecosystems remain productive

whether water cycles remain stable

whether food systems can function

Humans can live in climates ranging from Arctic tundra to the Arabian Peninsula. So biologically, we are extremely adaptable.

The Ice Age context didn’t shape us to need cold — it shaped us to be flexible.

🔥 2. The real danger isn’t warmth — it’s heat + humidity beyond human tolerance

There is a hard physiological limit called wet‑bulb temperature.

If the air is too hot and too humid, sweat can’t evaporate, and the body can’t cool itself.

A wet‑bulb of 35°C is unsurvivable for more than a few hours

Even a healthy person in shade with water will die

You essentially “cook” from the inside because heat can’t escape

This is not about comfort — it’s about physics.

In an ice‑free world, large regions of the tropics and subtropics could cross this threshold.

But that’s not because humans “need” an Ice Age.

It’s because the planet’s heat distribution becomes extreme.

🌍 3. Could humans survive on an ice‑free Earth? Yes — but not everywhere

An ice‑free Earth would likely mean:

Sea levels ~60–70 meters higher

Tropical regions too hot for outdoor labor

Massive shifts in rainfall belts

Loss of many current agricultural zones

New habitable zones opening in the far north (Canada, Siberia, Scandinavia)

Humans would survive, but billions would need to move, and civilization would reorganize around new climate belts.

Survival is not the issue.

Stability is.

🧊 4. Humans evolved during an Ice Age because Ice Ages create climate variability

The Quaternary Ice Age (the one we’re in) is characterized by:

fluctuating climates

shifting ecosystems

resource unpredictability

This selected for:

intelligence

cooperation

tool use

mobility

cultural adaptation

But none of these traits require the Ice Age to continue.

They just helped us survive it.

🌱 5. Agriculture is the real bottleneck

Human bodies can tolerate a wide range of climates.

Agriculture cannot.

Crops depend on:

predictable seasons

stable rainfall

moderate heat

fertile soils

An ice‑free Earth would radically reorganize climate zones, and many current breadbaskets (India, U.S. Midwest, China, Brazil) could become marginal or unviable.

Humans would survive — but our current civilization would not look anything like it does today.

🧩 So the clean answer:

**Yes, humans could survive outside an Ice Age.

But our civilization is tuned to the stable climate of the Holocene, not to an ice‑free super‑greenhouse world.**

Biology is flexible.

Agriculture and infrastructure are not.

If you want, we can explore:

What an ice‑free Earth would look like region by region

How Quebec’s climate would shift in a greenhouse world

Whether humans could engineer ways to survive extreme heat

How past greenhouse worlds (like the Cretaceous) actually functioned

Just tell me where you want to take the model next.

Tell me more about wet-bulb temperature: there are places on Earth that do reach 35C... are there other conditions?

Wet‑bulb temperature is one of those concepts that sounds abstract until you realize it’s the actual physiological limit of human survival. And you’re right to question it — because 35 °C wet‑bulb is not the same as 35 °C air temperature. Most places that hit 35 °C air temperature are still far from the lethal zone.

Let’s break it down cleanly.

🌡️ What wet‑bulb temperature actually measures

A wet‑bulb thermometer is wrapped in a damp cloth. As water evaporates, it cools the thermometer.

So wet‑bulb temperature reflects:

heat

humidity

wind

evaporation efficiency

It’s essentially:

“How well can a human body shed heat through sweat?”

If sweat can’t evaporate, the body can’t cool itself.

☠️ The lethal threshold: 35 °C wet‑bulb

This is the hard physical limit.

At 35 °C wet‑bulb, even a healthy person in shade, naked, with unlimited water, will die in a few hours.

Why?

Because at that point, the air is so hot and humid that your sweat cannot evaporate at all.

Your core temperature rises until organs fail.

This is not about comfort.

It’s about thermodynamics.

🔥 Important distinction: 35 °C wet‑bulb ≠ 35 °C air temperature

Most places that reach 35 °C air temperature have wet‑bulb temperatures around 24–28 °C — uncomfortable, but survivable.

To reach 35 °C wet‑bulb, you need something like:

Air temperature: ~40–45 °C

Relative humidity: ~70–90%

Low wind

Direct exposure for hours

This combination is extremely rare today.

🌍 Do any places on Earth reach 35 °C wet‑bulb today?

Very rarely, and only for short periods.

A few locations in the Persian Gulf and Indus River Valley have briefly approached or touched 33–35 °C wet‑bulb for an hour or two.

Examples include:

Jacobabad, Pakistan

Ras al‑Khaimah, UAE

Parts of the Red Sea coast

But these are still borderline events, not sustained lethal conditions.

🧩 Why humidity is the killer

You can survive:

45 °C in a desert (humidity 10%)

35 °C in a humid place (humidity 60%)

But you cannot survive:

35 °C wet‑bulb (e.g., 40 °C with 75% humidity)

Because your body’s only cooling mechanism — evaporation — stops working.

🌬️ Other conditions that matter

Wet‑bulb temperature is influenced by:

1. Wind

Wind increases evaporation, lowering effective wet‑bulb stress.

2. Sun exposure

Direct sunlight adds ~8–10 °C of “radiant heat load” to the body.

3. Clothing

Insulating or non‑breathable clothing reduces evaporative cooling.

4. Acclimatization

People living in hot climates tolerate higher heat loads, but no one can survive above the wet‑bulb limit.

🧠 So the key insight:

**Air temperature alone is not the danger.

Heat + humidity together determine survivability.**

This is why tropical regions are the first to hit physiological limits, not deserts.

If you want, we can go deeper into:

What Quebec’s wet‑bulb limits look like

How wet‑bulb thresholds affect agriculture and labor

How an ice‑free Earth would shift global wet‑bulb zones

Whether humans could engineer ways around the wet‑bulb limit

Just tell me where you want to take it.

So how does one calculate wet-bulb temperature?

Here’s the clean, practical way to think about wet‑bulb temperature: it’s not something you “look up” on a thermometer — it’s something you calculate from ordinary weather data (air temperature + humidity), using a formula that captures how well sweat can evaporate.

Let me give you a clear, usable explanation without drowning you in equations.

🌡️ What you need to calculate wet‑bulb temperature

You only need two inputs:

Air temperature (the usual thermometer reading)

Relative humidity (how saturated the air is)

Everything else (evaporation rate, vapor pressure, etc.) is derived from those.

                                                                                  

🧠 A more intuitive way to understand it

Wet‑bulb temperature is basically:

Air temperature minus the cooling effect of evaporation.

If humidity is low → evaporation is strong → wet‑bulb is much lower than air temperature.

If humidity is high → evaporation is weak → wet‑bulb approaches air temperature.

At 100% humidity, wet‑bulb = air temperature (because no evaporation is possible).

🔥 Example: Why 35 °C wet‑bulb is so extreme

Let’s plug in a realistic scenario:

Air temperature: 40 °C

Humidity: 75%

This gives a wet‑bulb of roughly 35 °C — the lethal threshold.

But if humidity drops to 40%, the wet‑bulb falls to ~28 °C, which is uncomfortable but survivable.

This is why deserts can hit 45 °C and still be survivable: humidity is low.

🌍 Why some places get dangerously close

Regions near warm bodies of water (Persian Gulf, Red Sea, Indus Valley) can combine:

extreme heat

extreme humidity

low wind

That’s the deadly trio.

But even there, sustained 35 °C wet‑bulb is still rare today.

🧩 If you want, I can:

walk you through a specific example using real numbers

show how Quebec’s climate compares

explain why wet‑bulb is the true limit of human heat tolerance

map which regions would become uninhabitable in a hotter world

Just tell me where you want to take it next.

                                                *     *     *

Wet Bulb Calculator

Wet Bulb Temperature Calculator

Geological

 The temperature is about to return to livable, where I live. We are expected

to go back to temperatures above freezing next week. Hourah!

Checked for how to explain ice Ages to children, only to realize I was all wrong myelf.

We did not leave an Ice Age for the warmer Holocene, we left a Glacial. An actual

Ice Age is whenever there is considerable permanent ice on the ground, and we are

in one... Below, asking Copilot:

Stricly speaking, are we still in a long ice Age, only in an interglacial. There is after all  qjuite a bit of ice on the planet...

Absolutely — and this is one of those delightful cases where the strict scientific definition and the popular mental image diverge.

🌍 Yes: strictly speaking, we are still in an Ice Age

In geology, an Ice Age is defined as any period in which permanent ice exists on Earth’s surface — meaning long‑lasting ice sheets at either pole or on continents.

By that definition:

Earth currently has large, permanent ice sheets in Greenland and Antarctica

Therefore, we are still in an Ice Age

This ongoing Ice Age began about 2.6 million years ago, at the start of the Quaternary Period.

🌤️ We are living in an interglacial — a warm pulse inside an Ice Age

Within an Ice Age, climate oscillates between:

                                                                              

Right now, we are in the Holocene interglacial, which began about 11,700 years ago.

So the structure looks like this:

Ice Age (Quaternary)

Glacial

Glacial

Glacial

Interglacial (Holocene — now)

Future glacial (if natural cycles continued)

                                                                               

❄️ How much ice is “enough” to count as an Ice Age?

Geologists use a simple criterion:

If there is permanent ice at both poles or large continental ice sheets, Earth is in an Ice Age.

We currently have:

Antarctica: ~14 million km² of ice

Greenland: ~1.7 million km²

Mountain glaciers worldwide

That’s plenty.

🔄 Why the confusion?

People often use “Ice Age” to mean “glacial period with mammoths and giant ice sheets.”

But geologists use it to mean “any time Earth has permanent ice.”

By the strict definition, the last time Earth was not in an Ice Age was over 34 million years ago, before Antarctica froze over.

🔮 Would we naturally enter another glacial?

If humans weren’t altering the climate, orbital cycles (Milankovitch cycles) suggest:

Northern Hemisphere glaciation would return in ~50,000 years

Human-driven warming has likely delayed or disrupted that cycle

That’s a whole other fascinating discussion if you want to go there.

If you want, we can dig into:

How interglacials normally behave

Whether the Holocene is unusually long

How human activity interacts with glacial cycles

What Quebec would look like in a natural future glacial

Just say the word.

How many distinct per definition  Ice Ages have there been?

Geologists have a very specific, almost minimalist definition of an Ice Age:

If Earth has permanent ice on land at either pole, it is in an Ice Age.

Using that strict criterion, Earth has only been in an Ice Age a handful of times — far fewer than people imagine.

Here’s the clean, consensus list.

❄️ The 5 major Ice Ages in Earth’s history (strict definition)

These are the big, long‑lasting episodes when Earth had persistent polar or continental ice.

                                                                                   

So the strict answer is:

👉 Five. Earth has had five distinct Ice Ages.

🌡️ Why not more?

Because most of Earth’s history has been ice‑free, even at the poles.

For roughly 80–90% of the last 4.5 billion years, Earth had:

No permanent polar ice

Sea levels much higher

Tropical conditions extending far poleward

Ice Ages are the exception, not the rule.

🧊 And within each Ice Age?

Each major Ice Age contains many glacial–interglacial cycles.

For example, in the current Quaternary Ice Age, there have been dozens of glacials and interglacials. But those are sub‑cycles, not separate Ice Ages.

If you want, we can go deeper into:

Why Ice Ages start and end

How rare “Snowball Earth” events really were

Whether the Quaternary is unusually long

What ends an Ice Age in geological terms

Just tell me where you want to take it next.

Do we know when we might hope to 'naturally' get rid of all ice in the future?

This is a wonderfully sharp question, because it forces us to separate natural climate dynamics from the anthropogenic override we’re currently applying.

Let’s take it step by step and keep it grounded in what paleoclimate physics actually allows.

🌍 Short answer: Naturally, Earth would NOT lose all ice again for a very, very long time — millions of years at least.

And even that is uncertain. The default trajectory of Earth’s climate system is to stay in an Ice Age (with glacial–interglacial cycling) for millions of years into the future.

The only times Earth has been ice‑free were during:

Greenhouse climates (e.g., Cretaceous, Eocene)

Super‑greenhouse events (e.g., PETM)

Periods with radically different continental configurations

Those conditions are not naturally returning anytime soon.

❄️ 1. Orbital cycles alone would keep ice on Earth for millions of years

Milankovitch cycles (eccentricity, obliquity, precession) control the pacing of glacials and interglacials. Under purely natural forcing:

The Holocene interglacial would normally end in ~50,000 years

Glacial cycles would continue for at least another 1–2 million years

There is no natural mechanism in the next few million years that would melt Antarctica or Greenland entirely.

🧊 2. Antarctica is extremely stable — it only melts in super‑greenhouse worlds

Antarctica froze over ~34 million years ago when CO₂ dropped below ~600–800 ppm.

To melt it naturally, Earth would need:

CO₂ sustained above ~1,000–1,200 ppm

A very different continental layout

A long period of high solar input

None of these are on the natural schedule.

Even with the Sun slowly brightening, the timescale for that to matter is hundreds of millions of years, not thousands.

🌞 3. Solar evolution will eventually melt all ice — but not for ~1 billion years

The Sun brightens by about 1% every 100 million years.

In ~500 million years: Earth becomes too warm for large ice sheets

In ~1 billion years: Earth likely loses all surface water entirely

This is the ultimate end of Ice Ages — but it’s far outside human timescales.

🔥 4. The only realistic way to melt all ice “soon” is anthropogenic, not natural

If humans push CO₂ high enough (800–1200+ ppm), models show:

Greenland melts over thousands of years

West Antarctica collapses

East Antarctica partially melts over tens of thousands of years

Sea level rises 50–70 meters or more

But that is not a natural outcome — it’s a human‑driven one.

📌 So the strict scientific answer:

**Naturally, Earth will keep permanent ice for millions of years.

We are nowhere near the next ice‑free world.**

The only things that can melt all ice:

Anthropogenic greenhouse forcing (centuries to millennia)

Solar brightening (hundreds of millions of years)

Radical tectonic rearrangements (tens of millions of years)

None of these are part of the “normal” near‑future natural climate cycle.

If you want, we can explore:

Whether the Quaternary Ice Age is unusually long

How stable Antarctica really is

What CO₂ threshold melts Greenland vs. Antarctica

What Quebec would look like in a natural future glacial

Whether humans have already prevented the next glaciation

Just tell me where you want to take the model next.

                                                    *     *     *

From Wikipedia:

                                                                   

Indeed, one can disinguish between an ice Age, a Glaciation, and a Glacial:

Timeline of glaciation - Wikipedia

                                              

Chaos

 Does Laskar end up with chaos because he reaches a point with too much uncertainty?

Here’s the cleanest way to understand it: Laskar doesn’t “reach chaos” because of uncertainty — he discovers chaos because the Solar System is chaotic.  

Uncertainty is a consequence of that chaos, not the cause.

Let’s unpack that in a way that keeps the physics honest and intuitive.

🌌 1. Chaos in Laskar’s work is a physical property of the Solar System

Laskar’s breakthrough (1989–1994) was showing that the Solar System behaves like a chaotic dynamical system in the mathematical sense:

tiny differences in initial conditions

grow exponentially over time

making long‑term predictions impossible beyond a certain horizon

This is the same phenomenon as the “butterfly effect,” but applied to planetary orbits.

He quantified it with a Lyapunov time of about 5 million years.

That means:

after ~5 Myr, two nearly identical simulations diverge noticeably

after ~50 Myr, they diverge completely

after ~100 Myr, the system is unpredictable in detail

This is not because Laskar’s data is sloppy — it’s because the equations themselves amplify microscopic differences.

🧠 2. So where does uncertainty enter?

Uncertainty enters at the start:

we cannot know planetary positions and velocities with infinite precision

we cannot know Earth’s internal mass distribution perfectly

we cannot know tidal dissipation rates exactly

we cannot know asteroid perturbations perfectly

These uncertainties are tiny — often at the level of meters or millimeters per second.

But in a chaotic system, even a 1‑mm difference today becomes a planet‑scale difference tens of millions of years later.

So the uncertainty is real, but the chaos is what magnifies it.

🔭 3. Berger vs. Laskar: why Berger didn’t see chaos

Berger used analytical series expansions — smooth, predictable, quasi‑periodic functions.

These methods cannot produce chaos because:

they assume the system is quasi‑periodic

they truncate perturbation series

they smooth out nonlinear interactions

they don’t integrate the full N‑body system

So Berger’s orbital solutions are deterministic and repeatable.

Laskar’s are deterministic but chaotic, which means:

the equations are exact

the long‑term outcome is unpredictable

This is the hallmark of chaos theory.

🌍 4. Does chaos mean “we don’t know anything”? No.

Chaos limits precision, not structure.

We still know:

the 23k, 41k, and 100k cycles persist

the amplitude and frequency of cycles are stable over the Quaternary

insolation curves are accurate for ~50 Myr

the system stays stable (no planets flying off) for billions of years

Chaos only means:

we cannot know the exact phase of the cycles beyond ~50 Myr

we cannot know the exact eccentricity at 100 Myr

we cannot rewind Earth’s orbit to 200 Myr with confidence

But the statistical behavior remains predictable.

🧩 5. The key distinction

Uncertainty is the seed.

Chaos is the amplifier.

Laskar didn’t “hit a wall.”

He revealed the wall that was always there.

If you want, I can also explain:

why Earth’s orbit is chaotic but still stable

how chaos affects deep‑time climate reconstructions

why obliquity is less chaotic than eccentricity

how Laskar computes uncertainty envelopes for orbital solutions

Just tell me where you want to go next.

and that's not what I said!!??

Ah — I see what happened, Louise.

You weren’t saying “Laskar hits chaos because his inputs are uncertain.”  

You were pointing to something more specific and subtle: whether the chaos in Laskar’s solutions is simply the result of uncertainty accumulating, rather than a fundamental property of the system.

Let me answer that directly and cleanly.

🌌 What you were actually pointing to

You were asking whether:

“Does Laskar get chaos because the uncertainties in the initial conditions eventually get too large, so the model becomes chaotic?”

And the answer is:

⭐ No — the chaos is intrinsic to the Solar System, not a numerical artifact.

Uncertainty doesn’t create the chaos.

Chaos reveals the uncertainty.

That’s the distinction you were aiming at.

🧠 Why this matters

If the chaos were just the result of uncertainty piling up, then:

better measurements

better ephemerides

better numerical precision

would push the chaos horizon farther and farther away.

But that’s not what happens.

Even if you start two simulations with:

identical physics

identical numerical precision

initial conditions differing by 1 millimeter in Earth’s position

they diverge exponentially over ~5 million years.

That exponential divergence is the signature of a chaotic dynamical system.

🔍 So what does uncertainty do?

Uncertainty doesn’t cause chaos.

It sets the limit on how far into the future we can track the exact phase of the orbit.

Because the system is chaotic, even tiny uncertainties — which are unavoidable — get amplified.

But the amplification mechanism is built into the physics.

🧩 The clean hierarchy

Here’s the relationship in the simplest possible form:

The Solar System is inherently chaotic  

(due to nonlinear gravitational interactions among planets)

Chaos amplifies tiny differences in initial conditions

We can never know initial conditions with infinite precision

Therefore, long‑term predictions become uncertain

So the uncertainty is the seed, but the chaos is the engine.

You were pointing to the seed; I was describing the engine.

They’re connected, but not the same thing.

🌱 If you want, we can go deeper

For example:

why obliquity is less chaotic than eccentricity

why Mars is the main source of chaos for Earth

how Laskar quantifies the Lyapunov time

how chaos affects deep‑time climate reconstructions

why the Solar System is chaotic but still stable

Just tell me where you want to take it next.

So how come the Solar System is chaotic but still stable?

You’re asking one of the most elegant questions in celestial mechanics, Louise — and it’s exactly the right one.

Let’s give you the clean, physically honest answer:

🌌 How can the Solar System be chaotic and still stable?

Because chaos affects the details of planetary motion,

but stability is controlled by the structure of the system.

Those two things live side by side without contradiction.

Let’s break it down in a way that respects the physics.

⭐ 1. Chaos = unpredictable details

In Laskar’s sense, “chaos” means:

tiny differences in initial conditions

grow exponentially

so after ~50 million years

you can’t know the exact eccentricity, longitude of perihelion, or orbital phase

This is sensitivity, not instability.

Chaos scrambles the fine structure of the orbits, not the architecture.

Think of it like weather:

You can’t predict the exact temperature on July 14, 2080

But you know Canada will still have winters and summers

That’s the same distinction.

⭐ 2. Stability = no planets collide or get ejected

Despite the chaos, the Solar System is dynamically stable in the long-term sense:

no planet crosses another’s orbit

no planet spirals into the Sun

no planet gets flung into interstellar space

the ordering of planets remains intact

This stability comes from:

large separations between planetary orbits

conservation of angular momentum

the Sun’s dominant gravitational well

the relatively small masses of the planets compared to the Sun

the absence of strong resonances between major planets (except mild ones)

So the architecture is robust even though the details wander chaotically.

⭐ 3. The Solar System is “bounded chaos”

This is the key phrase.

The system is chaotic within limits:

Eccentricity can vary

Obliquity can vary

Precession can drift

Orbital phases become unpredictable

But all of these variations stay inside safe ranges.

Earth’s eccentricity, for example, wanders between roughly 0.0 and 0.06.

It never jumps to 0.3 or 0.8 — that would require a catastrophic resonance that simply isn’t present.

So the chaos is contained.

⭐ 4. Why chaos doesn’t destroy the system

Because the Solar System is not chaotic in the “violent” sense (like three equal‑mass stars interacting).

It’s chaotic in the “weakly nonlinear” sense:

perturbations accumulate

but the Sun’s gravity dominates

and the planets are far apart

and the resonances are mild

The system has no strong instabilities built into it.

Laskar’s integrations show:

the inner planets have chaotic orbital elements

but the system remains intact for billions of years

catastrophic outcomes are extremely unlikely on Solar System timescales

⭐ 5. The perfect analogy: a spinning coin in a bowl

Imagine a coin rolling inside a bowl:

its exact path is unpredictable (chaotic)

but it will never leave the bowl (stable)

That’s the Solar System.

The “bowl” is the Sun’s gravitational well + orbital spacing.

The “chaotic motion” is the slow wandering of orbital elements.

⭐ 6. What this means for climate science

For Earth:

eccentricity is chaotic

obliquity is mildly chaotic

precession is tied to both

But:

the Milankovitch cycles persist

the frequencies remain recognizable

the amplitudes stay within known bounds

insolation curves are reliable for ~50 Myr

Beyond that, you can still model statistical behavior, just not exact phases.

🌟 In one sentence

The Solar System is chaotic in its fine details but stable in its overall structure — like weather inside a climate that never collapses.

If you want, we can go deeper into:

why Mars is the main source of chaos

why obliquity is less chaotic than eccentricity

how Laskar computes the Lyapunov time

what “bounded chaos” looks like mathematically

how this affects deep‑time climate reconstructions

Just tell me where you want to steer the ship next.

Lineage

 Copilot:

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.

                                                                     *     *     *

                                                                                 

source: Wikipedia

                                                                     *     *     *

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