IAO 2011, Теория, Група α — Задача 5. Jupiter disappeared

Orbital velocities

Let us find the most important parameters that we need for the solution. The orbital velocities of Jupiter and its moons are,

Jupiter: $T=2\pi\cdot778600000/4332.59/86400=13.069\,\mathrm{km/s}$,

and analogously:

After disappearance of Jupiter the velocity of each of its former moons relative to the Sun will remain and will be equal to sum of vectors of the velocity of Jupiter relative to the Sun and the velocity of the moons relative to Jupiter.

Transcription note: the symbol $T$ labels the speed calculation in the source.

5.1.

A body can leave the Solar system if its total energy in the field of gravitation of the Sun would exceed zero:

But also we know that for a circular orbit

Therefore, the body can leave the Solar system if its full speed at the distance of the orbit of Jupiter is at least $2^{1/2}$ times larger than the orbital velocity of Jupiter, i.e. $2^{1/2}\cdot13.07\,\mathrm{km/s}=18.48\,\mathrm{km/s}$. As we see, for any moon such a velocity is possible. At least, it is reached in a configuration of local opposition of the …

[The continuation is cut off between the bottom of page 4 and the top of page 5. The next surviving word is reproduced below.]

… disappears). The necessary drawing to this question should contain these sectors. Student have to find boundaries of the sectors geometrically from the diagram of geometrical summation of velocities or calculate the angles algebraically.

Answer: Io, Europa, Ganimede, Callisto, every moon in the case being in the sector shown at the drawing.

5.2.

A body can fall into the Sun in the opposite situation, if the velocity relative to the Sun is strongly reduced, namely reduced to zero the orbital component of the velocity, then it will move to the Sun or from the Sun by a thin ellipse and finally fall into the Sun. Or, as an alternative we may keep the velocity large, but its vector should be directed strictly to the Sun, so that the point of perihelion of the (not closed) orbit appear in the Sun. By the same diagram of geometrical summation of velocities we may find that falling into the Sun is possible only for Io and Europa, and for both of them it will take place only in two points, sometime before the local conjunctions with Sun (analog to new moon at Earth) and sometimes after. The necessary drawing to this question should contain these points. The student has to find the points geometrically from the diagram of geometrical summation of velocities or calculate their positions algebraically.

Answer: Io and Europa in the case being in the points of orbit shown at the drawing.

Note: the drawings will be presented for jury sometime later.

Source gap: those promised sector/point diagrams are absent from the supplied solution PDF and the inspected year inventory. They are not invented here.