Using de Broglie’s approach, ?nd the allowed energy levels and the electron radiifor a hydrogen–like atom, i.e. one in which a single electron orbits a nucleus ofcharge +Ze, where Z is the atomic number. Z = 1 then gives the hydrogen atomitself.1)Find the lowest (n = 1) energy level and orbital radius for the case Z = 92.See if you can ?nd an estimate for the radius of the U nucleus, and comparewith this orbital radius.2)Angular momentum in the stellar systems: as discussed in the context of thevirial theorem, a protostellar system must lose energy in order to coalesce,condense, and heat up. Furthermore, it must also shed angular momentum.A gas cloud of stellar mass will initially possess enormous angular momen-tum even if it is rotating very slowly, from random motion of its constituentmaterial. Some protostellar systems solve, or partially solve, this problem by evolving into close binary stars. Others probably do so by “growing” a plan-etary system. We are accustomed to think of the planets in our own solarsystem as inconsequential in relation to the Sun — negligible in terms of massand in terms of power production. However, the orbital motion of the planetsaccounts for well over 90% of the total angular momentum of the solar system,as the following calculations will show.i: Assume a ball of H gas of uniform density, 500 atoms/cm3, one light–year in radius, rotating uniformly about its centre so that its equator ismoving at the very moderate speed of 1 m s- 1. Compute its mass andits angular momentum.It is an oversimpli?cation to suppose that a cloud of interstellar gas will ever ro-tate uniformly, but this calculation gives the order of magnitude of the amountof angular momentum which such a cloud can possess.ii: Assuming that the Sun is a Standard Model star rotating uniformly, com-pute its angular momentum. Take its period to be 30 days.In fact the Sun rotates somewhat more rapidly at its equator (period 24.6 days)than near its poles (period 34 days).iii: Calculate the total orbital angular momentum of Jupiter, Saturn, Uranusand Neptune around the Sun.iv: Compare the three angular momenta from i, ii, and iii.v: One of the largest angular velocities known among stars is exhibitedby Altair (Aquilæ, the 12th brightest star in the sky), with a periodof approximately 6.5 hours. How would that period have been deter-mined? Assuming that Altair can be described by the Standard Model,with M = 1.6Mand R = 1.77R, calculate its rotational angular mo-mentum.Altair is of spectral type A7 V; stars earlier than about F5 rotate, usually, muchfaster than the later stars. This has been interpreted to mean that early MainSequence stars, for whatever reason, are less prone to “growing” planetary sys-tems than late MS stars.The moment of inertia of a uniform ball of radius R and mass M about its centre is( 2=5) M R2 while the moment of inertial of a Standard Model star is 0.0754 M R2 . Dataon planetary masses and orbital parameters can be found in Carroll and Ostlie, or inWikipedia.
Jitender Pal , 11 Years ago
Grade 9
