FormingWorlds
diff --git a/‎book/09_earth_venus/earth_venus.md‎
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diff --git a/‎book/10_mercury_mars/mercury_mars.md‎
Lines changed: 8 additions & 6 deletions b/‎book/10_mercury_mars/mercury_mars.md‎
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@@ -9,8 +9,7 @@ Earth and Venus are nature's best controlled experiment in comparative planetolo
 They formed from the same nebular reservoir within roughly $0.3$ AU of each other, contain similar bulk inventories of silicate, iron, and volatile elements, and have masses and radii within $20\%$ of each other.
 Yet their present-day surface conditions could not be more different.
 Earth is wet, geologically active, magnetically shielded, and inhabited; Venus is dry, encased in a $92$ bar carbon-dioxide atmosphere at a surface temperature of $735$ K, and the only one of the rocky planets without a global magnetic field.
-Why?
-What turned two nearly identical starting points into two radically different end states?
+Identifying what set these two nearly identical starting points on such radically different evolutionary paths is the central question of this lecture.
 This lecture works through that question in three parts.
 Part 1 reviews Earth as a reference planet, Part 2 surveys Venus as the alien twin, and Part 3 brings the comparison together to extract the underlying physics.
 The blackboard derivation in the middle introduces the Simpson-Nakajima runaway greenhouse limit, a thermodynamic boundary in phase space that, once crossed, makes the divergence essentially irreversible.
@@ -454,7 +453,7 @@ Reproduced from {cite:t}`Goldblatt2013`.
 
 In numbers, the Simpson-Nakajima limit comes out to about $280\text{--}310$ W/m$^2$, depending on details of the radiative transfer model and the assumed atmospheric composition {cite:p}`Goldblatt2013,Kasting1988`.
 Earth today absorbs about $240$ W/m$^2$ of stellar flux, comfortably below the limit.
-Venus, at $0.72$ AU, absorbs roughly $1.91 \times$ more stellar flux per unit area than Earth would at the top of its atmosphere; the absorbed solar flux at Venus is much higher than the Simpson-Nakajima limit, which is precisely why Venus cannot host liquid water at the surface today.
+Venus, at $0.723$ AU, absorbs roughly $1.91 \times$ more stellar flux per unit area than Earth would at the top of its atmosphere; the absorbed solar flux at Venus is much higher than the Simpson-Nakajima limit, which is precisely why Venus cannot host liquid water at the surface today.
 Once Venus crossed the limit some time in its history, the runaway became a one-way process: the ocean evaporated to space, water photolysed in the upper atmosphere, hydrogen escaped (preferentially over deuterium, leaving the residual D/H ratio strongly enriched), and the surface dried out permanently.
 What is left is the dry, $\mathrm{CO_2}$-dominated, $735$ K hothouse we observe today.
 
@@ -495,7 +494,7 @@ $$
 p_{\mathrm{sat}}(T) \approx p_{\mathrm{ref}}\, \exp\!\left[ -\frac{L}{R_v}\!\left(\frac{1}{T} - \frac{1}{T_{\mathrm{ref}}}\right) \right]
 $$ (eq:saturation-pressure)
 
-with $p_{\mathrm{ref}} = 611$ Pa and $T_{\mathrm{ref}} = 273$ K (the triple point of water).
+with $p_{\mathrm{ref}} = 611$ Pa and $T_{\mathrm{ref}} = 273.16$ K (the triple point of water, rounded to $273$ K below).
 
 **Step 1: The IR optical depth and the photosphere temperature.**
 The outgoing longwave radiation emerges from a thermal photosphere, defined as the level where the IR optical depth, integrated downward from the top of the atmosphere, reaches order unity.
@@ -552,7 +551,7 @@ $$
 Earth is comfortably below the runaway greenhouse limit, by about $40$--$70$ W/m$^2$.
 At the orbit of Venus, the same calculation gives $F_{\mathrm{abs}}^{\mathrm{Venus}} \approx 460$ W/m$^2$ (assuming an Earth-like albedo, which is wrong for present Venus but is the relevant comparison for an early ocean-bearing Venus), well above the limit.
 Setting $F_{\mathrm{abs}} = F_{\mathrm{OLR}}^{\max}$ and solving for the orbital distance gives the **inner edge of the classical habitable zone** at roughly $0.95$--$0.99$ AU for present-day solar luminosity {cite:p}`Kasting1993,Kopparapu2013,Goldblatt2013`.
-Venus, at $0.72$ AU, sits well inside the inner edge.
+Venus, at $0.723$ AU, sits well inside the inner edge.
 
 **Key insight.**
 The runaway greenhouse is not an artefact of one model or one parameter choice; it is a thermodynamic feature of the water phase diagram combined with the elementary physics of radiative transfer through a saturated atmosphere.
@@ -786,12 +785,11 @@ The rates of these reactions are uncertain by orders of magnitude, and pinning t
 
 ### Why did Earth and Venus diverge?
 
-We are now in a position to answer the question that opened this lecture.
-What turned two nearly identical starting points into two radically different end states?
+We are now in a position to answer the question that opened this lecture, namely what set two nearly identical starting points onto such radically different evolutionary paths.
 The answer involves four physical inputs that differed (or differed slightly) between Earth and Venus, plus the nonlinear couplings that turned modest input differences into runaway outcomes.
 
 The first input is the **solar flux**.
-At $0.72$ AU, Venus receives $1.91 \times$ the solar flux per unit area that Earth does at $1$ AU.
+At $0.723$ AU, Venus receives $1.91 \times$ the solar flux per unit area that Earth does at $1$ AU.
 This is the single biggest lever, and it is the only input where the two planets are very different at the start.
 With the present-day Sun, Venus is comfortably above the Simpson-Nakajima limit and Earth is comfortably below.
 With a young, faint Sun, the picture is more nuanced: Venus may have been just above or just below the threshold, depending on cloud cover and atmospheric composition, and the question of whether early Venus was habitable becomes a question of how the solar flux interacted with the cloud feedback and the carbonate-silicate cycle.
@@ -955,7 +953,7 @@ The central piece of physics is the **Simpson-Nakajima runaway greenhouse limit*
 This limit, in the range $280\text{--}310$ W/m$^2$, defines the inner edge of the habitable zone and is essentially independent of the surface temperature once the atmosphere becomes optically thick in the IR.
 A planet whose absorbed solar flux exceeds the limit cannot host a steady state with liquid surface water; the runaway greenhouse drives the ocean into the upper atmosphere, photolyses the water, and loses the hydrogen to space.
 The process is one-way: there is no thermodynamic path back to the wet state.
-Earth, at $1$ AU, sits below the limit; Venus, at $0.72$ AU, sits well above it.
+Earth, at $1$ AU, sits below the limit; Venus, at $0.723$ AU, sits well above it.
 
 The **carbonate-silicate cycle** is the negative feedback that has stabilised Earth's climate within the liquid-water window for $4$ Gyr, but it cannot operate on present Venus because the cycle requires both liquid surface water (for the weathering sink) and active plate tectonics (for the subduction return leg).
 Once Venus lost its water, the sink was destroyed; once the subduction stopped, the return leg was destroyed.
 
@@ -9,7 +9,9 @@
 
 In {ref}`lecture09` we treated Earth and Venus as a near-twin pair: very similar in mass and bulk composition, yet wildly divergent in surface conditions. This lecture takes the opposite approach. Mercury and Mars are nature's *limiting cases* for rocky planets in our solar system. Mercury is the smallest and the densest, sitting closest to the Sun and stripped of almost all volatiles. Mars is roughly half the diameter of Earth, sits at the outer edge of the classical habitable zone, and preserves a sedimentary and atmospheric record that points to a wetter, warmer past. Together they bracket Earth and Venus on every axis that matters for terrestrial-planet evolution: size, heliocentric distance, volatile inventory, and dynamo longevity.
 
-The structure of the lecture follows that ambition. Part 1 takes Mercury as a metal-world case study, threading the needle from orbital dynamics through interior structure, surface morphology, polar volatiles, exosphere, and exploration history. Part 2 turns to Mars and walks through its interior, geological epochs, surface highlights, evidence for past water, the early-climate puzzle, modern atmospheric loss, and exploration. The blackboard derivation, on the Jeans escape flux, sits naturally in the Mars half because thermal escape is the textbook entry point for understanding Mars' atmospheric history. Part 3 then steps back and uses the comparison between the two extremes (and their contrast with Earth and Venus) to extract the general lessons that the rest of the course will need: size and distance set the trajectory; timing and dynamo longevity modulate the outcome; and habitability is a multidimensional question with at least four largely independent ingredients.
+The structure of the lecture follows that ambition. Part 1 takes Mercury as a metal-world case study, threading the needle from orbital dynamics through interior structure, surface morphology, polar volatiles, exosphere, and exploration history. Part 2 turns to Mars and walks through its interior, geological epochs, surface highlights, evidence for past water, the early-climate puzzle, modern atmospheric loss, and exploration. The blackboard derivation, on the Jeans escape flux, sits naturally in the Mars half because thermal escape is the textbook entry point for understanding Mars' atmospheric history.
+
+Part 3 then steps back and uses the comparison between the two extremes (and their contrast with Earth and Venus) to extract the general lessons that the rest of the course will need: size and distance set the trajectory; timing and dynamo longevity modulate the outcome; and habitability is a multidimensional question with at least four largely independent ingredients.
 
 A note on the data underlying this lecture: of all the rocky bodies in the solar system, Mercury and Mars have changed the most dramatically in our understanding over the last fifteen years. NASA's *MESSENGER* mission revolutionised Mercury science between 2011 and 2015; ESA/JAXA's *BepiColombo* is now en route for orbit insertion in 2026. NASA's *InSight* lander placed a seismometer on Mars in 2018 and produced the first reliable internal-structure constraints for the planet by 2021. *Curiosity* and *Perseverance* are still operating on the surface as of this writing. Where I quote numbers, they reflect the post-*MESSENGER*, post-*InSight* state of the art; where the numbers remain contested, I say so explicitly.
 
@@ -202,16 +204,16 @@ The decisive transformation came with **MESSENGER** (MErcury Surface, Space ENvi
 
 ### Mars overview: half Earth, one tenth the mass
 
-Mars is the second-smallest planet, with a mass of $0.107\,\Mearth$ and a radius of $0.532\,\Rearth = 3389\ \mathrm{km}$. Its mean density is $3.93\ \mathrm{g\ cm^{-3}}$, the lowest of the four terrestrial planets. The orbit is well within the habitable zone of {ref}`lecture05`, with semi-major axis $a = 1.524\ \mathrm{AU}$ and eccentricity $e = 0.0934$. Crucially, the obliquity is currently $25.19^\circ$, almost identical to Earth's $23.4^\circ$, so Mars has Earth-like seasons. The atmosphere is thin ($\sim 6\ \mathrm{mbar}$ surface pressure) and dominated by $\mathrm{CO_2}$, with a global mean surface temperature near $210\ \mathrm{K}$.
+Mars is the second-smallest planet, with a mass of $0.107\,\Mearth$ and a radius of $0.532\,\Rearth = 3389\ \mathrm{km}$. Its mean density is $3.93\ \mathrm{g\ cm^{-3}}$, the lowest of the four terrestrial planets. The orbit sits near the outer edge of the conservative habitable zone of {ref}`lecture05`, with semi-major axis $a = 1.524\ \mathrm{AU}$ and eccentricity $e = 0.0934$. Crucially, the obliquity is currently $25.19^\circ$, almost identical to Earth's $23.4^\circ$, so Mars has Earth-like seasons. The atmosphere is thin ($\sim 6\ \mathrm{mbar}$ surface pressure) and dominated by $\mathrm{CO_2}$, with a global mean surface temperature near $210\ \mathrm{K}$.
 
 Mars has two small natural satellites, Phobos and Deimos, which we will discuss as a separate topic below. We will also see that the Martian obliquity, although similar to Earth's today, has been chaotically variable on $\sim 100$ Myr timescales, oscillating between near-zero and over $60^\circ$ as a consequence of resonance with planetary perturbations. This is essential to keep in mind when interpreting evidence for past water at the poles or for periodic ice ages on Mars.
 
-Compared to Mercury and Venus, Mars is at first glance more obviously Earth-like: a similar day length (24h 37min), similar obliquity, polar ice caps, dust storms, and a thin but meteorologically active atmosphere. Yet the surface is cold, dry, and uninhabitable today, and the most exciting Mars science of the last two decades has been the discovery that this was not always so. Reading the geological record, we now know that early Mars hosted abundant surface and near-surface liquid water for at least $\sim 100$ Myr, perhaps episodically for a billion years. Mars therefore offers the best preserved record we have of what an Earth-like planet looks like once it loses its atmosphere and dries out. That makes it both an irreplaceable target for astrobiology and a cautionary tale for assessments of habitability around other stars.
+Compared to Mercury and Venus, Mars is at first glance more obviously Earth-like: a similar day length ($24$ h $37$ min), similar obliquity, polar ice caps, dust storms, and a thin but meteorologically active atmosphere. Yet the surface is cold, dry, and uninhabitable today, and the most exciting Mars science of the last two decades has been the discovery that this was not always so. Reading the geological record, we now know that early Mars hosted abundant surface and near-surface liquid water for at least $\sim 100$ Myr, perhaps episodically for a billion years. Mars therefore offers the best preserved record we have of what an Earth-like planet looks like once it loses its atmosphere and dries out. That makes it both an irreplaceable target for astrobiology and a cautionary tale for assessments of habitability around other stars.
 
 
 ### Phobos and Deimos: the twin moons
 
-Phobos is the larger and inner of Mars' two moons, with a mean radius of $11.27\ \mathrm{km}$ and an orbital period of only $7$ hours and $39$ minutes, considerably shorter than the Martian rotation period of $24$ hours and $37$ minutes. Because Phobos orbits faster than the planet rotates beneath it, the tidal bulge it raises on Mars lags behind, and the resulting torque pulls Phobos *inward* rather than outward. The orbital radius is decreasing by approximately $1.8\ \mathrm{cm/yr}$, and the moon will be tidally disrupted or impact the planet within $30$ to $50$ Myr.
+Phobos is the larger and inner of Mars' two moons, with a mean radius of $11.27\ \mathrm{km}$ and an orbital period of only $7$ h $39$ min, considerably shorter than the Martian rotation period of $24$ h $37$ min. Because Phobos orbits faster than the planet rotates beneath it, the tidal bulge it raises on Mars lags behind, and the resulting torque pulls Phobos *inward* rather than outward. The orbital radius is decreasing by approximately $1.8\ \mathrm{cm/yr}$, and the moon will be tidally disrupted or impact the planet within $30$ to $50$ Myr.
 
 Deimos is smaller (mean radius $6.2\ \mathrm{km}$), has a longer orbital period of about $30$ hours, and is slowly receding from Mars on a much longer timescale. Both bodies are irregular in shape, very dark (reflectance under $7\%$), and spectroscopically similar to D-type asteroids, the most primitive class of small bodies in the outer solar system. Their bulk densities are low, $\sim 1.86\ \mathrm{g\ cm^{-3}}$ for Phobos, implying high porosity (perhaps $25\%$ to $35\%$), consistent with a rubble-pile internal structure {cite:p}`Kuramoto2022`.
 
@@ -539,7 +541,7 @@ This is the **Jeans escape flux**, named after James Jeans who derived it in 192
 
 When $\lambda \to 0$ (the planet's gravity is irrelevant), the factor $(1+\lambda)e^{-\lambda} \to 1$ and the formula reduces to $n\sqrt{\kB T/(2\pi m)}$, which is exactly one quarter of $n\langle v\rangle$, the standard kinetic-theory result for the flux of molecules through an aperture in a gas. So the formula behaves correctly in the no-gravity limit.
 
-When $\lambda \gg 1$, the exponential dominates and the flux falls off as $e^{-\lambda}$. The dependence on $\lambda$ is steep: a change in $\lambda$ from $5$ to $10$ reduces the escape flux by a factor of $\sim 150$.
+When $\lambda \gg 1$, the exponential dominates and the flux falls off as $e^{-\lambda}$. The dependence on $\lambda$ is steep: a change in $\lambda$ from $5$ to $10$ reduces the escape flux by a factor of $\sim 80$ (the ratio $(1+\lambda_1)e^{-\lambda_1}/[(1+\lambda_2)e^{-\lambda_2}]$ for $\lambda_1 = 5$, $\lambda_2 = 10$ gives $(6/11)\,e^{5} \approx 81$).
 
 **Application to Mars.**
 
@@ -572,7 +574,7 @@ The exponential is then $e^{-230} \approx 10^{-100}$, which is *zero* for any pr
 Jeans escape is selective. Light species (H, He, $\mathrm{H_2}$, possibly $\mathrm{D}$) are stripped by thermal escape on Gyr timescales; heavy species (C, N, O, Ar, $\mathrm{CO_2}$) are essentially immune. If Mars has lost a substantial inventory of $\mathrm{CO_2}$ over its history, the loss must have happened by **non-thermal** processes that bypass the Maxwell-Boltzmann velocity distribution entirely. We turn to those processes next.
 ```
 
-A few comments on the derivation. The factor $(1+\lambda)$ in equation {eq}`eq:jeans` reflects the fact that the escaping molecules carry away not only their kinetic energy (the raw Maxwellian tail) but also the work done against gravity as they climb out, which biases the escaping population toward higher initial velocities. In the limit $\lambda \to \infty$ the formula reduces to $\Phi_J \sim n v_{\mathrm{th}} \lambda e^{-\lambda}$, which is the standard high-$\lambda$ asymptotic form often quoted in textbooks.
+A few comments on the derivation. The factor $(1+\lambda)$ in equation {eq}`eq:jeans` reflects the fact that the escaping molecules carry away not only their kinetic energy (the raw Maxwellian tail) but also the work done against gravity as they climb out, which biases the escaping population toward higher initial velocities. In the limit $\lambda \to \infty$ the formula reduces to $\Phi_J \sim n\,v_{\mathrm{th}}\,\lambda\,e^{-\lambda}$, the standard high-$\lambda$ asymptotic form often quoted in textbooks. Note that the "$\sim$" hides a numerical prefactor: written exactly, $(1+\lambda)e^{-\lambda} \to \lambda e^{-\lambda}$ for $\lambda \gg 1$, and $\sqrt{\kB T/(2\pi m)} = v_{\mathrm{th}}/(2\sqrt{\pi})$ with $v_{\mathrm{th}}$ the most-probable speed defined above, so the textbook scaling carries an implicit $1/(2\sqrt{\pi})$ that we have absorbed into the proportionality.
 
 The exobase concept is crucial: collisions below the exobase scramble velocities and prevent the high-velocity tail from accumulating, so escape effectively only happens at and above the exobase. The exobase altitude itself is set by where the mean free path equals the scale height, which depends on temperature, composition, and gravity. For Mars the modern exobase sits about $200\ \mathrm{km}$ above the surface; for Earth it is around $500$--$700\ \mathrm{km}$.