Съдържание

Условие I. Impact heating and partial melting II. Sinking of iron droplets III. Energy released by differentiation IV. Does the asteroid cool too quickly?Решение

Балкански 2026, Теория — Задача 2. How an Asteroid Forms Its Core

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от Балкански 2026

Concours Général de Physique «Minko Balkanski» · 31 юли 2026 г. · 10 т.

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Условие

We study a fictitious spherical asteroid called Balkania. It is initially homogeneous: iron is uniformly dispersed in silicate material. A collision heats it and produces partial melting. Iron droplets may then sink and form a metallic core. The model is deliberately simplified; any additional assumption must be stated clearly.

Subparts marked $\star$ are more difficult. They may be left for a second pass; no later question depends on them unless the required result is supplied again.

Three-stage schematic: a small impactor of radius b moving with speed u∞ towards a large sphere of radius R with dispersed iron dots („before collision: dispersed iron“); arrow „collision“ to a partially molten body with a wavy surface and arrows („partially molten body“, legend „• Fe ▬ silicates“, „schematic, not to scale“); arrow „settling“ to a differentiated body with a dark core of radius Rc, a mantle and a „cold crust“, outer radius R („after differentiation: metallic core“). — Figure II.1 – Collision, partial melting, settling, and core formation.

Data for Balkania.

Initial radius	$R = 2.0 \times 10^5\ \mathrm{m}$
Initial iron volume fraction	$\phi = 1/8$
Silicate density	$\rho_s = 3.0 \times 10^3\ \mathrm{kg\,m^{-3}}$
Iron density	$\rho_f = 7.0 \times 10^3\ \mathrm{kg\,m^{-3}}$
Mean specific heat capacity	$c = 1.0 \times 10^3\ \mathrm{J\,kg^{-1}\,K^{-1}}$
Initial temperature	$T_0 = 1250\ \mathrm{K}$
Onset-of-melting temperature	$T_s = 1400\ \mathrm{K}$
Effective latent heat of melting	$L = 2.5 \times 10^5\ \mathrm{J\,kg^{-1}}$
Thermal diffusivity	$\kappa = 1.0 \times 10^{-6}\ \mathrm{m^2\,s^{-1}}$
Surface emissivity	$\varepsilon = 0.80$

Collision. The impactor is spherical, has radius $b = 8.0 \times 10^4\ \mathrm{m}$ , and has the same composition and mean density as Balkania. Its speed at infinity is $u_\infty = 4.0 \times 10^3\ \mathrm{m\,s^{-1}}$ . Balkania is treated as fixed during the approach. The collision is perfectly inelastic: the bodies remain joined. A fraction $\chi = 0.50$ of the kinetic-energy loss during the collision is converted into uniform internal energy of the merged body; mass loss is neglected.

Iron droplets. Their radius is $a = 5.0 \times 10^{-3}\ \mathrm{m}$ . The dynamic viscosity of the molten silicate is $\eta = 20\ \mathrm{Pa\,s}$ .

I. Impact heating and partial melting

1. Determine Balkania's mean density $\rho$ . Then calculate the iron mass fraction.

2. Let $M$ be the mass of Balkania and $m$ the mass of the impactor.

2. a) Calculate $M$ , $m$ , and the ratio $\mu = m/M$ .

2. b) Using conservation of the impactor's mechanical energy during the approach, show that

v_{\mathrm{imp}}^2 = u_\infty^2 + \frac{2GM}{R + b},

where $v_{\mathrm{imp}}$ is its speed just before contact. Calculate $v_{\mathrm{imp}}$ and comment on the importance of gravitational acceleration.

3. Let $V$ be the common speed of the two bodies immediately after the collision.

3. a) Using conservation of momentum, express $V$ . Show that the kinetic-energy loss during the collision is

E_{\mathrm{loss}} = \frac{1}{2}\frac{Mm}{M + m}v_{\mathrm{imp}}^2.

3. b) Show that the deposited energy per unit mass of the merged body is

q_{\mathrm{imp}} = \frac{\chi}{2}\frac{\mu}{(1 + \mu)^2}v_{\mathrm{imp}}^2,

and calculate it.

3. c) The energy $c(T_s - T_0)$ first heats the body to $T_s$ . The remaining energy causes melting at approximately constant temperature. Determine the molten fraction $x$ . Iron droplets can move when $x \geq x_c = 0.30$ . Does the collision trigger differentiation?

3. d) $\star$ Neglecting gravitational acceleration during the approach, determine the minimum impactor radius $b_{\mathrm{min}} < R$ required to reach exactly $x = x_c$ .

II. Sinking of iron droplets

In Parts II–IV, to keep the model analytically simple, use Balkania's initial values $R$ and $M$ and neglect the mass increase caused by the impactor. The gravitational field is approximated by that of the initial homogeneous sphere of density $\rho$ . You may use the result that a homogeneous spherical shell produces no gravitational field at an interior point.

Left: a large circle with centre O („centre of Balkania“), a radius vector r to a small droplet near the surface moving with velocity v inward. Arrow „enlargement“ to the right: a grey droplet of radius a („moving iron droplet“) with velocity v pointing „towards the centre“, force F_eff pointing left and drag F_D pointing right; labels |F_eff| = (ρf − ρs)Vg(r) and |F_D| = 6πηav. Note below: „Effective weight drives the droplet inward; Stokes drag opposes the motion.“ — Figure II.2 – Position, velocity, and force balance of an iron droplet.

4. At a distance $r \leq R$ from the centre, show that the gravitational field strength is

g(r) = \frac{4\pi G\rho}{3}r.

Calculate the surface acceleration $g(R)$ .

5. After differentiation, all the iron forms a pure spherical core of radius $R_c$ . By conserving the total iron volume, determine $R_c/R$ .

6. An iron droplet of volume $\mathcal{V}$ is at distance $r$ from the centre.

6. a) Taking account of weight and buoyancy, show that the inward resultant force has magnitude

F = (\rho_f - \rho_s)\mathcal{V}g(r).

6. b) At low Reynolds number, the Stokes drag force has magnitude $F_D = 6\pi\eta a v$ . Assuming terminal speed, show that

v_t(r) = \frac{2(\rho_f - \rho_s)a^2}{9\eta}g(r).

6. c) Calculate $v_t(R)$ . Check the consistency of the Stokes regime using

\mathrm{Re} = \frac{2\rho_s a v_t}{\eta}.

7. $\star$ Over a very short time interval, take the position $r$ and hence $v_t$ as constant. Establish the equation of motion of a droplet released from rest in the form

\frac{dv}{dt} = \frac{v_t - v}{\tau}.

Express and calculate $\tau$ . Estimate the distance $v_t(R)\tau$ and justify the terminal-speed assumption used in Question 6.

8. Consider a droplet initially at the surface. Since $g(r) \propto r$ , write $v_t(r) = kr$ .

8. a) Obtain an expression for $k$ , then calculate the settling time $t_{\mathrm{set}}$ from $R$ to $R_c$ .

8. b) Deduce the dependence of $t_{\mathrm{set}}$ on $a$ and $\eta$ . Determine the minimum droplet radius $a_{100}$ that reaches the core in less than 100 yr, with all other data unchanged.

9. $\star$ At the end of differentiation, calculate the field $g_f(R_c)$ just outside the core, neglecting the field of the outer mantle. Compare it with the initial homogeneous field $g_i(R_c)$ . Does the calculation of $t_{\mathrm{set}}$ tend to overestimate or underestimate the actual final settling time?

III. Energy released by differentiation

Assume radial ordering is preserved: the iron initially contained inside a sphere of radius $r$ is found, after differentiation, inside a pure iron sphere of radius $r_f$ .

Left: a sphere with a grey shell of radius r („initial state, dispersed iron shell“); arrow labelled $r_f = \phi^{1/3} r = r/2$ to the right: a sphere with a small dark shell of radius r_f inside the core („final state, shell inside the core“). Below: dV_f = φ4πr²dr and the note „Conservation of iron volume imposes the same radial scale factor on every shell.“ — Figure II.3 – Radial mapping of iron shells during differentiation.

10. Show that $r_f = r/2$ .

11. For a small iron volume initially at distance $r$ , approximate its displacement as a motion from $r$ to $r/2$ in the initial field.

11. a) Show that the gravitational work released per unit iron volume is

w(r) = (\rho_f - \rho_s)\int_{r/2}^{r} g(s)\,ds = \frac{\pi}{2}G\rho(\rho_f - \rho_s)r^2.

11. b) The iron volume initially contained in the shell $[r, r + dr]$ is $dV_f = \phi 4\pi r^2 dr$ . Hence obtain the total released energy,

E_{\mathrm{diff}} = \frac{\pi^2}{20}G\rho(\rho_f - \rho_s)R^5.

12. Assume that this energy is fully converted into heat within the asteroid.

12. a) Calculate the mean temperature rise $\Delta T_{\mathrm{diff}} = E_{\mathrm{diff}}/(Mc)$ . Compare it with the impact heating.

12. b) Determine the radius $R_\star$ for which differentiation alone would produce a mean temperature rise of 100 K, with all other data unchanged. Interpret the result.

IV. Does the asteroid cool too quickly?

A sphere with a thin dark outer layer („cold crust, thickness ℓ“) and a grey inner sphere („warm interior forming core“); arrows pointing outward all around labelled „thermal radiation“. Note below: „Two limiting models are compared: an isothermal interior and purely diffusive transport.“ — Figure II.4 – Thermal structure, surface radiation, and diffusion depth.

13. At temperature $T_s$ , radiation received from space is neglected.

13. a) Calculate the initial radiated power

P_{\mathrm{rad}} = 4\pi R^2\varepsilon\sigma T_s^4.

13. b) In an isothermal model, the molten fraction $x$ first solidifies at $T \simeq T_s$ . Taking the radiated power as constant and equal to its value in part a), estimate the time $t_{\mathrm{sol}}$ required to remove the latent heat $MxL$ .

13. c) $\star$ After solidification, write the differential energy balance governing $T(t)$ when the radiated power varies as $T^4$ . Using the mathematical hints, calculate the time $t_{\mathrm{sens}}$ required to cool from 1400 K to 1300 K. Estimate $t_{\mathrm{iso}} = t_{\mathrm{sol}} + t_{\mathrm{sens}}$ and compare it with $t_{\mathrm{set}}$ .

14. In a solid or a liquid without efficient mixing, heat diffuses over a distance of order

\ell \simeq \sqrt{\kappa t}.

14. a) Calculate $\ell$ during $t_{\mathrm{set}}$ and compare it with $R$ . Also estimate the diffusion time $R^2/\kappa$ across the entire asteroid.

14. b) The exact volume fraction in the surface layer of thickness $\ell$ is

f_{\mathrm{layer}} = 1 - \left(1 - \frac{\ell}{R}\right)^3.

Estimate it, then explain how a cold crust can coexist with an interior that remains partially molten. Conclude whether the core can form within the diffusive model.

14. c) $\star$ How would very efficient internal convection modify this conclusion? A qualitative discussion is sufficient.

Решение

Внимание: Непълно решение

В архива няма официално решение на тази задача.

Annexe : constantes et relations utiles — Reference sheet : constants and useful relations

Constantes communes / Common constants

G = 6.67 \times 10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}},\qquad \sigma = 5.67 \times 10^{-8}\ \mathrm{W\,m^{-2}\,K^{-4}},

1\ \mathrm{an} = 1\ \mathrm{yr} = 3.16 \times 10^{7}\ \mathrm{s},\qquad \ln 2 = 0.693,\qquad 1\ \mathrm{rad} = 57.3^{\circ}.

Français

Les relations ci-dessous peuvent être utilisées directement. Elles ne remplacent pas le choix et la justification du modèle physique.

Problème I – petites variations et optique

Pour $|\theta| \ll 1\ \mathrm{rad}$ :

\sin\theta \simeq \tan\theta \simeq \theta.

Au premier ordre :

F(z + \Delta z) - F(z) \simeq F'(z)\Delta z,

\Delta\left(\frac{1}{z}\right) \simeq -\frac{\Delta z}{z^2}.

Problème II – seuil, équations et intégrales

Pour la question 3d) :

q_c = c(T_s - T_0) + x_c L,

y = \frac{2q_c}{\chi u_\infty^2},\qquad \frac{\mu}{(1+\mu)^2} = y,

y\mu^2 + (2y - 1)\mu + y = 0.

La racine physique vérifie $0 < \mu < 1$ .

Si $dv/dt = (v_t - v)/\tau$ et $v(0) = 0$ :

v(t) = v_t\left(1 - e^{-t/\tau}\right).

Si $dr/dt = -kr$ :

\ln(r_2/r_1) = -k(t_2 - t_1).

Intégrales utiles :

\int_{r/2}^{r} s\,ds = \frac{3r^2}{8},\qquad \int_0^R r^4\,dr = \frac{R^5}{5},

\int T^{-4}\,dT = -\frac{1}{3T^3}.

Pour $|z| \ll 1$ : $(1 - z)^3 \simeq 1 - 3z$ .

English

The relations below may be used directly. They do not replace the choice and justification of a physical model.

Problem I – small changes and optics

For $|\theta| \ll 1\ \mathrm{rad}$ :

\sin\theta \simeq \tan\theta \simeq \theta.

To first order:

F(z + \Delta z) - F(z) \simeq F'(z)\Delta z,

\Delta\left(\frac{1}{z}\right) \simeq -\frac{\Delta z}{z^2}.

Problem II – threshold, equations, and integrals

For Question 3d):

q_c = c(T_s - T_0) + x_c L,

y = \frac{2q_c}{\chi u_\infty^2},\qquad \frac{\mu}{(1+\mu)^2} = y,

y\mu^2 + (2y - 1)\mu + y = 0.

The physical root satisfies $0 < \mu < 1$ .

If $dv/dt = (v_t - v)/\tau$ and $v(0) = 0$ :

v(t) = v_t\left(1 - e^{-t/\tau}\right).

If $dr/dt = -kr$ :

\ln(r_2/r_1) = -k(t_2 - t_1).

Useful integrals:

\int_{r/2}^{r} s\,ds = \frac{3r^2}{8},\qquad \int_0^R r^4\,dr = \frac{R^5}{5},

\int T^{-4}\,dT = -\frac{1}{3T^3}.

For $|z| \ll 1$ : $(1 - z)^3 \simeq 1 - 3z$ .

Оригинал в Архива: CGP_2026.pdf