Order of magnitude: the physics

Estimate fast, estimate robustly, and the world starts to look constrained. Three estimates, from a city to the moon, and the one scaling law that runs through the whole course. CCBS · EST 5053 · Fall 2026 · lecture 1 · core page · 2026-09-01 (Tue) 10:40–12:15 · lecturer-taught This lecture opens the course and then teaches its first tool. The first fifteen minutes are the rules. The rest is three worked estimates, one for each move of the method: a city, an animal, a potato. Lecture 2 does the same for biology and chemistry. The claim being made for the method is stronger than it first sounds. Order-of-magnitude reasoning is not a way of getting approximate answers quickly. It is a way of finding the one constraint that decides a problem, before you have a model and often before you have data. The answer is a by-product. The constraint is the point. So this page does estimates rather than justifying them. There is a theorem underneath and it is worth knowing, but it is an account of why arguments like these work rather than a tool you need to run one. It is in the exposition, worked slowly, next to the blast-wave calculation that is its best demonstration.

The rules of the course live on their own page How the course runs, what you hand in, how you are graded, and where to get the AI tools is the course organization page. It is the first fifteen minutes of this lecture, and it is the reference you come back to all term.

Three pages, three depths This is the core: the spine of the lecture. Where this method comes from traces the history, from Galileo through Buckingham, Taylor, Fermi and Purcell to the Caltech course this one descends from. The full exposition is the tutorial read, with the derivations done rather than sketched, plus the Π theorem, the Trinity yield and ten questions to practise on. Both are for after the lecture, not during it.

Part 1

The course

What this is, in one section. The full rules live on the organization page.

1What this course is

Physics, system, industry: three schools asking three questions about life, and one course whose tools aim at all three. CCBS reads a cell as a machine you can reason about quantitatively, and three schools frame the reasoning. Physics asks what life is as an object: sizes, rates, energies and the constraints they impose, which is lectures 1, 2 and 7. System asks how life works as a machine: networks, dynamics, feedback, stochasticity, computation, which is lectures 3 to 6 and 12. Industry asks how life could be useful as a tool, which is lectures 13 to 16. You should leave able to reason about a biological problem with all three at once. Nothing needs prior background classes, only "an exuberant love for biology," which is mandatory. In 2026 an agent can digest a paper, write first-pass code and draft a figure on demand, so the emphasis moves to what an agent cannot do: understand, defend, and communicate. Order of magnitude comes first because it decides which of the other tools you need. Before writing a differential equation you should know whether the process takes a microsecond or a century, and whether the thing it acts on exists in ten copies or ten million. Half the time that knowledge dissolves the question.

Part 2

The method

Three moves, met in the order the three worked examples need them: name the bottleneck, write a balance and let the size cancel, calibrate against something you have touched.

2What the method is

Numbers can say a lot, but only when they are robust. Details are fragile. Orders of magnitude are not. The world is connected. Observations that seem unrelated can be deeply constraining, and Eratosthenes measured the size of the Earth from the length of a shadow at two latitudes. The instrument that sees such connections is order-of-magnitude reasoning: estimate the quantity, keep only the power of ten, and let the constraint do the arguing.¹

Order of magnitude An estimate right to within a factor. We keep

1

10

, not more. The claim is of the form "it is about

10^{6}

per second", never "approximately 1.3×10⁶".

Two properties make this the right first tool. It is fast, so you can do it mid-conversation, before deciding whether a project is worth a month. More importantly it is robust: the factors that dominate an estimate are few, so the answer survives being wrong about every detail, whereas a precisely written answer claims a precision it does not have. Details are fragile. Orders of magnitude are not.

It is also teachable, which is not obvious. Caltech ran a whole course on it for years, and its notes are still the best single document on the subject.²³ What follows is three moves and one worked example of each. Nothing here is a theorem. You will get your hands dirty and then, if you want the account of why it works, read the exposition.

Figure 1. The method, in three moves. Numbered in the order the lecture meets them, one worked example each. Move 1 is a judgement about the physics and can be wrong. Move 2 is algebra and cannot. Move 3 is an empirical claim, so it is the one that has to be checked against something independent, which is what §7 does.

3Hangzhou: the bottleneck decides

Evacuating a city by plane or by car. The same flux formula answers both, and the bottleneck, not the vehicle, decides.

Move 1, on something macroscopic and familiar. Evacuate all of Hangzhou, about $10^{7}$ people. The question sounds like a choice of transport. It is not.

By air

One airport. Under emergency loading a plane leaves every two minutes, and each carries about $10^{2}$ people:

(86400\,\mathrm{s}/\mathrm{day}) / (120\,\mathrm{s}/\text{ plane}) \approx 7\times 10^{2} \text{ planes }/\text{day } \times 10^{2} \text{ people } \approx 7\times 10^{4} \text{ people }/\text{day}

(1)

For $10^{7}$ people that is about 140 days. Too slow, and note where the failure is: the aircraft is the fastest vehicle available and it is irrelevant, because the constraint is the number of runways.

By road

Count lanes instead. Ten highways leave the city and in an emergency all lanes run outbound, so call it $10^{2}$ lanes. A lane is a pipe with three numbers: people per vehicle, speed, spacing.

\text{Flux } = N v / L = (4 \text{ people } \times 20\,\mathrm{m}/\mathrm{s}) / (40\,\mathrm{m}) = 2 \text{ people }/s \text{ per lane}

(2)

With $10^{2}$ lanes that is $1.7\times 10^{7}$ people per day, so the city clears in about half a day. Same formula, opposite verdict, and all that changed is the number of parallel pipes.

The move that matters

Now the obvious objection. Are there enough cars? What if people drive faster, or slower? Each is a fragile detail, and the way to kill all of them at once is to find what is fixed. The spacing $L$ is not a free choice: you leave a gap because you need time to react, so $L = v\tau$ with $\tau \approx 2\,\mathrm{s}$ set by human reflexes. Substitute:

\text{Flux } = N v / L = N v / (v\tau ) = \pf{N / \tau}

(3)

Figure 2. The vehicle cancels. Once the spacing is written as

v\tau

, the speed drops out of the flux entirely and only the occupancy and the reaction time remain. A crowd walking at 1 m/s two metres apart moves people at a quarter the rate of cars at 72 km/h, not a twentieth.

The lesson, stated generally

The speed cancelled, because going faster forces a proportionally bigger gap. So when an estimate depends on a quantity you are unsure of, look for the relation tying it to something fixed, and substitute. If the uncertain quantity cancels, you never needed it. If it does not, you have learned that it is one of the few things worth measuring. Either outcome is progress, which is why the move is worth making even when you expect it to fail.

Part 3

The scales

What stays constant when size changes, what does not, and the most useful scaling in the course.

4Jump height: what cancels

Move 2. Set the energy stored in muscle against the height gained, and the size cancels on both sides at once. What comes out is that a flea and a horse should reach the same height.

Muscle is a fixed volume fraction $f_{m}$ of the body, and it does a fixed work per unit volume $w$ , so the energy available for a jump is $E_{m} = f_{m} w L^{3}$ . The mass to be lifted is $m = \rho L^{3}$ . Set the one against the other:

\rho L^{3} g h = f_{m} w L^{3} \, \Rightarrow \, \pf{h = f_{m} w / (\rho g)}, \, \text{ independent of } L

(4)

The $L^{3}$ cancels on both sides, so jump height does not depend on size at all. Only the muscle fraction and the quality of the muscle survive. Putting in $f_{m} \approx 0.15$ and $w \approx 10^{5}\,\mathrm{J}/\mathrm{m}^{3}$ gives $h \approx 1.5\,\mathrm{m}$ , the same for a locust, a cat and a person.

Read that as a ceiling rather than a prediction, because not all of an animal's muscle points the right way. A standing human jump lifts the centre of mass about half a metre and a high jumper with a run-up gets most of the rest, so the estimate is high by a factor of three, which is what an order-of-magnitude claim promises.

Figure 3. Flat over three decades of body length. Jump height against body length, both logarithmic. Four animals spanning a thousandfold range of size sit on one flat line, the geometric mean of their heights, which is equation (4) with the

L^{3}

already cancelled. Had height scaled with size instead, these points would fall on a diagonal across three decades. The flea is the exception, a factor of 23 low, and §5 is why.

This argument is a hundred years old, and Haldane credited it to Galileo

J. B. S. Haldane's 1926 essay On Being the Right Size puts it exactly: "the height to which an animal can jump is more nearly independent of its size than proportional to it."⁴ His figures are a flea at about two feet and a man at about five, and he credits the argument to Galileo's square-cube law of 1638. Note the hedge, "more nearly independent than proportional". His own two numbers differ by a factor of 2.5, and he is telling you what an order-of-magnitude claim actually claims.

5The wall at τ = L/v

Height came out flat because the energy and the mass both scale as $L^{3}$ . The time available to spend that energy does not, and that is a wall a small animal cannot climb.

Equation (4) says what energy is available. It says nothing about whether an animal can get it out in time, and that is a separate question with a different answer. A flat height means a flat takeoff speed, $v = \sqrt{2gh} \approx 3\,\mathrm{m}/\mathrm{s}$ for everyone. But the animal has only its own body length in which to reach that speed, so the time available is

\tau \approx L / v

(5)

and unlike $h$ , this depends on size, linearly. A human has 0.34 s, a cat 0.07 s, a flea 0.7 ms. Fast muscle needs roughly 0.1 s to shorten by half its length, so somewhere between the cat and the flea the animal runs out of time.

Figure 4. The time runs out before the energy does. The time available to deliver the jump,

\tau = L/v

, against the 0.1 s that one muscle contraction takes. Unlike the height in Figure 3 this tracks body length directly, and the flea sits a factor of 150 below the line. That is why it also sits below the flat height line, and why it needs a spring.

Equation (5) is right, and the film says so. Sutton and Burrows recorded hedgehog fleas $1.8\,\mathrm{mm}$ long leaving the ground in $1.4\,\mathrm{ms}$ at $1.3\,\mathrm{m}/\mathrm{s}$ , and $1.8\,\mathrm{mm} / 1.3\,\mathrm{m}/\mathrm{s} = 1.4\,\mathrm{ms}$ to two figures.¹¹ A millisecond is genuinely what the flea has, and one muscle contraction is a hundred times longer.

In power terms the same statement is sharper. Charge the flea the 11 per cent of body mass a locust devotes to jumping, and delivering the energy in 1.4 ms demands $6000\,\mathrm{W}/\mathrm{kg}$ against the $190$ striated muscle can produce, a factor of thirty. So whatever the flea does, it does not power the jump with muscle.

What it does instead is use muscle as a winch rather than as an engine. It contracts slowly against an elastic pad of resilin over hundreds of milliseconds, holds the load with a latch, and releases it in under a millisecond.¹³ The spring sets the power, not the muscle. Energy is conserved and power is amplified, because power is energy over time and the spring shortens the time.

The flea, measured

Ten hedgehog fleas, Archaeopsyllus erinacei, filmed at 5000 frames per second.¹¹ Mass $0.7\,\mathrm{mg}$ , length $1.8\,\mathrm{mm}$ , take-off in $1.4\,\mathrm{ms}$ at $1.3\,\mathrm{m}/\mathrm{s}$ at $39^\circ$ , about $98\,g$ of acceleration. Every flea number on this page is one of those.

The 11 per cent is the softest input in the argument, measured on locusts and applied to a flea. Bennet-Clark and Lucey, who found the spring, charged the flea 20 per cent, which drops the requirement to about seventeen times muscle.¹³ Doubling the softest input moves the margin and not the conclusion. Knowing which input you could double without minding is most of what an estimate is for.

The spring buys the jump, not the height

$1.3\,\mathrm{m}/\mathrm{s}$ at $39^\circ$ lifts a flea about $3.4\,\mathrm{cm}$ , and Bennet-Clark and Lucey measured $3.5\,\mathrm{cm}$ on a different species by a different route in 1967.¹³ So the latch gets the flea off the ground, and it still lands a factor of twenty below the flat line. Haldane's “about two feet” is an estimate, not a measurement, and so is the 20 cm that circulates in textbooks: that one is Bennet-Clark and Lucey's own figure from the volume of resilin, printed four lines from the 3.5 cm they filmed.

Shrinking stops helping before the flea, too. Below a threshold set by the spring material the smallest jumpers cannot load a spring without breaking it, so take-off speed peaks in the tens of milligrams.¹² Two constraints, energy and time, and the small animal is caught by the second one.

6Heat: deriving t ≈ x²/κ

Fourier's law, conservation, one equation, and the scaling that survives: double the size, quadruple the time. This is the most useful line in the course.

Heat flux is proportional to the local temperature gradient, and energy is conserved. Those two facts give the heat equation in four lines, and the picture they are read off is three points on a rod.

Figure 5. The whole derivation is read off this picture. Flux enters the middle segment from the hot side and leaves on the cold side. What does not leave is stored, and that sentence is the heat equation. Every textbook draws it, because there is no simpler way to see conservation.

J = -K \nabla T \,\,(\text{Fourier}) \,\,\, \Delta E = C_{v} \delta T \,\,(storage)

(6)

\delta x C_{v} \delta T = [J(x) - J(x+\delta x)] \delta t \, \Rightarrow \, \pf{\partial T/\partial t = \kappa \nabla^{2}T}, \, \kappa = K/C_{v}

(7)

Two material constants have collapsed into one. Now take the order-of-magnitude step: replace each derivative by the variation over one characteristic scale, $\partial /\partial t \to 1/t$ and $\partial^{2}/\partial x^{2} \to 1/x^{2}$ . The temperature cancels and

T/t \approx \kappa T/x^{2} \, \Rightarrow \, \pf{t \approx x^{2}/\kappa}

(8)

Only one functional form survives, and it is quadratic in size. Every diffusive process in this course obeys it, with a different letter in place of $\kappa$ : heat, molecules, momentum, even the spread of a mutation. Learn this one line.

7From a potato to the moon

Move 3. The scaling needs one material constant, and you can get it out of something you have cooked. Then it reaches sixteen orders of magnitude past the kitchen.

Equation (8) has exactly one constant in it, and move 3 says to fix that constant from something you have personally experienced rather than from a table. So bake a potato. It takes more than an hour and not a whole afternoon, so call it two hours. A potato is about 10 cm across and the heat has to reach the middle, so the distance is a few centimetres: call it 3 cm. That is the entire experiment, and equation (8) run backwards turns it into a number:

\kappa \approx x^{2}/t = (0.03\,\mathrm{m})^{2} / (7200\,\mathrm{s}) = \pf{1.2\times 10^{-7}\,\mathrm{m}^{2}/\mathrm{s}}

(9)

Now check it against something independent, because move 3 is the one move in the method that is an empirical claim rather than algebra. A potato is mostly water, and water's conductivity, density and heat capacity are all textbook:

\kappa_{\text{water}} = K/(\rho c_{p}) = 0.60 / (1000 \times 4180) = \pf{1.4\times 10^{-7}\,\mathrm{m}^{2}/\mathrm{s}}

(10)

The kitchen was right to fifteen per cent, which is better than the estimate deserves and much better than an order of magnitude promises. What matters more is that it is robust. Take the corners of anything a person might claim about a potato, 3 to 5 cm and one to three hours, and $\kappa$ runs from $8\times 10^{-8}$ to $7\times 10^{-7}$ : a box a factor of eight wide, with the true value inside it.¹⁰ Two things you already knew have fixed a material constant to within a decade.

The moon

Now spend it somewhere nobody has been. Rock conducts about seven times better than water, $\kappa \approx 1.0\times 10^{-6}\,\mathrm{m}^{2}/\mathrm{s}$ , and the moon's radius is 1737 km. Nothing else about the moon is needed. Equation (8) gives the time for its centre to feel its surface:

t \approx R^{2}/\kappa = (1.74\times 10^{6})^{2} / (1.0\times 10^{-6}) \approx 3\times 10^{18}\,\mathrm{s} \approx \pf{10^{11}\,\text{yr}}

(11)

Twenty times the age of the solar system. So the question was the wrong question. Conduction has not cooled the moon and is nowhere near finishing, which means the answerable question is not how long it takes but how far it has got. Turn equation (8) around and ask what thickness has had time to cool in the 4.57 billion years available:

d = \sqrt{\kappa t} = \sqrt{(1.0\times 10^{-6})(1.44\times 10^{17})} \approx \pf{3.8\times 10^{5}\,\mathrm{m}}

(12)

Call it 380 km, the outer 22 per cent of the radius. The interior has not heard about the surface at all. That is a checkable prediction about a body nobody in this room has touched, it took two lines of arithmetic, and the constant in it came from dinner.

Figure 6. Sixteen decades on one line. Left:

t = x^{2}/\kappa

from a protein crossing a bacterium to the moon cooling, with a band spanning κ from water to rock. The band is a factor of 7 wide and invisible at this scale, which is why one line really does carry all of them. Right: the calibration above. The dashed box is the range of everything anyone would claim about baking a potato, and water's true line runs through it.

Two habits, both cheap

Check the constant, then quote it. Move 2 is algebra and cannot betray you. Move 1 is a judgement about the physics and can. Move 3 is an empirical claim, so an anchor you have never checked against the material property it is supposed to be is a number you invented, and a reader who cannot see which value you used cannot check anything downstream of it.

An absurd answer is a result. Equation (11) came back at twenty times the age of the solar system, and the useful reading is not that the arithmetic failed. It is that conduction is not the mechanism at that size, which is exactly why the next question was how far it got rather than how long it took. An estimate that returns nonsense has told you which physics to stop assuming.

8Below ten microns, inertia is gone

The rest of the course lives at a cell's size, so the question is which forces run that scale. The answer is not the one your own body has taught you.

From here the course works where cells work, and the first thing to establish about a new scale is which forces dominate it. Two forces act on anything moving through a fluid, and they scale differently with size, so their ratio is a property of the scale rather than of the object. A world in which one dominates behaves nothing like a world in which the other does, and our intuition was trained entirely inside one of them.

Estimate both, with the move from §6: replace every derivative by its variation over one characteristic scale. Take a body of size $L$ moving at speed $v$ through a fluid of density $\rho$ and viscosity $\eta$ . Inertia is mass times acceleration. The mass is $\rho L^{3}$ , and the body changes its velocity by about $v$ in the time $L/v$ it takes to move its own length, so

F_{\text{inertial}} \approx \rho L^{3} \cdot v / (L/v) = \pf{\rho v^{2} L^{2}}

(13)

Viscosity is a stress acting on a surface. The stress is $\eta$ times a velocity gradient, the gradient is about $v/L$ , and the surface is about $L^{2}$ , so

F_{\text{viscous}} \approx \eta (v/L) \cdot L^{2} = \pf{\eta v L}

(14)

Divide. Everything about the fluid and the body collapses into a single dimensionless number, which is the whole reason such a number is worth having:

F_{\text{inertial}} / F_{\text{viscous}} \approx \rho v^{2}L^{2} / (\eta v L) = \pf{Re = \rho v L/\eta}

(15)

Two force estimates and a division. Note that $Re$ carries one power of $L$ and one of $v$ , and a small animal is small in both, so the ratio collapses fast as you shrink. For a bacterium, with $\rho = 10^{3}\,\mathrm{kg}/\mathrm{m}^{3}$ , $v \approx 30\,\mu\mathrm{m}/\mathrm{s}$ , $L \approx 1\,\mu\mathrm{m}$ and $\eta = 10^{-3}\,\mathrm{Pa}\cdot \mathrm{s}$ :

Re \approx (10^{3})(3\times 10^{-5})(10^{-6}) / 10^{-3} = \pf{3\times 10^{-5}}

(16)

Eleven orders of magnitude below a human swimming, at roughly $10^{6}$ . That is not a change of degree, it is a change of which term in the equation of motion you are allowed to drop. Purcell put it as "about $10^{-4}$ or $10^{-5}$ . For these animals inertia is totally irrelevant. We know that F = ma, but they could scarcely care less."⁵

The consequence he draws is the one to remember. Stop pushing a bacterium and ask how far it coasts. Momentum against Stokes drag gives about $0.03$ Å, and Purcell's own figure is 0.1 Å, in 0.6 microseconds. Either way it is a fraction of an atom, so a cell cannot glide, cannot drift, and stops the instant it stops working.

Figure 7. Why bacteria have propellers and not paddles. Purcell's scallop theorem: at zero Reynolds number a swimmer whose shape changes reciprocally, retracing the same sequence of configurations backwards, has zero net displacement, however fast or slow it moves. Escaping the theorem requires a stroke that traverses a loop in shape space.

One more of his estimates carries straight into lecture 2, and it is the same competition again with diffusion in place of viscosity. Compare moving something a distance $l$ by stirring, $l/v$ , against moving it by diffusion, $l^{2}/D$ , which is §6's scaling with $D$ in place of $\kappa$ . The ratio is $lv/D$ , which Purcell introduces with "I'm sure this ratio has someone's name but I don't know the literature" and calls the stirring number. It is the Péclet number. A micron-sized cell with $D \approx 100$ µm²/s and any plausible internal flow has $lv/D < 1$ , so stirring a bacterium is pointless. Diffusion has already done the job.

Part 4

Using it

What the method is for once you leave physics, and what you do this week.

9The gap is where the biology is

In biology an estimate is a null hypothesis. It says what plain physics would do with no regulation at all, and the gap is where design lives.

Everything so far was physics, where the estimate is the answer. In biology the estimate is rarely the answer, and that is what makes it valuable. Compute what the system would do with nothing regulating it, then compare.⁷ Three outcomes, all three useful.

Table 1. What to conclude from the gap between the estimate and the observation.
Outcome	What it means	Example
Physics is far too slow	Some mechanism must exist that you have not accounted for. Go find it.	A protein diffusing down a 1 m axon needs $3\times 10^{2}$ years, so motors exist.
Physics is fast enough	No regulation is needed for this. Stop looking, and do not build a model of it.	Mixing a bacterium takes 10 ms, so a bacterium is well stirred by doing nothing at all.
The system sits at the bound	Nothing left to explain by design; physics already forbids doing better. This is the strongest kind of result.	The Berg–Purcell limit on how precisely a cell can read a concentration.⁶

The third row is the one people forget, and it undercuts the framing of the other two. Sometimes biology sits at the physical limit, and then there is no gap and no design to find. Berg and Purcell showed a chemotactic bacterium measures concentrations about as accurately as any device of its size could, given that molecules arrive by diffusion. An estimate equal to the observation is not a failed null model. It is the discovery that the organism is up against physics.

The ten-questions game

Ten questions, three minutes each, answers to within a factor of ten. The mass of all the RNA in a human cell. The number of ATP molecules in a bacterium at any instant. The length of one transcription cycle. The method is always the same: name the scale, name the bottleneck, then answer, and check afterwards against BioNumbers.⁸ We do one live and the room rates it 0 to 3 on the course rubric, which converts the rubric from a table into a shared standard.

10This week

Two deadlines, and the first is a test of the plumbing rather than of you.

Thursday 09-03, 23:59: sign up to teach. One file, one pull request, to the submissions repo. It merges itself if you named the file correctly and refuses with a reason if you did not. A lecture may have one to three students. Everyone teaches at least once, and completing an additional lecture as a volunteer earns extra credit. Lectures 3 and 4 teach on 09-08 and 09-10, so those sign-ups go first.
Monday 09-07, 23:59: the first extension mini-essay. Anything from lectures 1 or 2 you want to push on. The worked example shows the shape and the length. Eight run over the term and your best six count, so two free skips are built in.

Every number on this page is checkable, and some of them will not survive being checked. Finding one that does not would make an excellent first essay, and the exposition lists eight places worth pushing on. So does the problem set that goes with this lecture, in particular its blast-wave problem, which reaches a conclusion about the world from a scaling exponent alone.⁹

†References & sources

Every claim and number on this page traces to one of these. Items marked PDF have their full text in the course's literature/ archive.

F. Xiao, CCBS 2025 Lecture 1: Order of Magnitude reasoning (handwritten notes), with the 2025 scribe notes by Qinguo Liu and Yihang Ding. notes scribe The source of the Hangzhou, jump-height and potato examples.
P. Goldreich, S. Mahajan and S. Phinney, Order-of-Magnitude Physics: Understanding the World with Dimensional Analysis, Educated Guesswork, and White Lies, draft of 1 August 1999. PDF The Caltech course this lecture descends from. See the lineage page.
S. Mahajan, The Art of Approximation in Science and Engineering, MIT 6.055 draft (2009), published as Street-Fighting Mathematics, MIT Press (2010). PDF The method taught as a discipline rather than a knack.
J. B. S. Haldane, "On Being the Right Size", in Possible Worlds and Other Papers (1926). PDF The jump-height argument in full, a century early, with the release-time caveat attached and the mechanism misdiagnosed.
E. M. Purcell, "Life at low Reynolds number", Am. J. Phys. 45, 3–11 (1977). PDF Re for a bacterium, the 0.1 Å coasting distance, the scallop theorem, and the stirring number.
H. C. Berg and E. M. Purcell, "Physics of chemoreception", Biophys. J. 20, 193–219 (1977). PDF The case where the estimate and the organism agree, which is the third row of Table 1.
R. Phillips and R. Milo, "A feeling for the numbers in biology", PNAS 106, 21465 (2009). PDF The argument for a numerical spine in cell biology.
R. Milo et al., "BioNumbers: the database of key numbers in molecular and cell biology", Nucleic Acids Res. 38, D750 (2010). PDF · Cell Biology by the Numbers Where to look up any number in Part 4, and lecture 2's main reference.
F. Xiao, CCBS 2025 problem set 1. PDF Carries the blast wave as a student exercise, along with the terminal-velocity and bomb-area estimates worked in the exposition.
lectures/research/lecture01/analysis/oom_numbers.py and make_figures.py. Every number on this page. Runs with no arguments and prints all 92 of them, with each input marked as either a value read from a full text or a standard constant.
G. P. Sutton and M. Burrows, "Biomechanics of jumping in the flea", J. Exp. Biol. 214, 836–847 (2011). PDF Table 2, page 838, is every measured flea number on this page: $0.7\,\mathrm{mg}$ , $1.8\,\mathrm{mm}$ , $1.4\,\mathrm{ms}$ , $1.3\,\mathrm{m}/\mathrm{s}$ , $39^\circ$ , $6000\,\mathrm{W}/\mathrm{kg}$ . Ten Archaeopsyllus erinacei at 5000 frames per second. The paper's own subject is where the flea pushes from, and the answer is the tibia and tarsus, not the trochanter.
M. Ilton et al., "The principles of cascading power limits in small, fast biological and engineered systems", Science 360, eaao1082 (2018). PDF The spring-failure wall behind the muscle-power wall, and the observation that take-off speed peaks in the tens of milligrams. The froghopper and pygmy mole cricket figures are theirs.
H. C. Bennet-Clark and E. C. A. Lucey, "The jump of the flea", J. Exp. Biol. 47, 59–76 (1967). PDF The 3.5 cm jump height, the resilin spring, and the 20% muscle fraction. The lineage page works through their estimate, which is this lecture's method used to find a mechanism rather than to bound one.