Astroparticle Physics

17.1 Chapter 1 3394. By definition (see the Glossary) the following relation holds between the ratio of intensitiesI 1 ,I 2 and the difference of two star magnitudes m 1 ,m 2 :so thatm 1 − m 2 =−2.5log 10 (I 1 /I 2 ),I 1 /I 2 = 10 −0.4(m 1−m 2 ) ,and from m = 1 one gets I 1 /I 2 ≈ 0.398 or I 2 ≈ 2.512 I 1 .5. Let N be the number of atoms making up the celestial body, µ the mass of the nucleon,and A the average atomic number of the elements constituting the celestial object. Gravitationalbinding dominates ifGM 2R>Nε,where ε is a typical binding energy for solid material (≈ 1 eV per atom). Here numericalfactors of order unity are neglected; for a uniform mass distribution the numerical factorwould be 6/5, see Problem 13.5. The mass of the object is M = NµA, so that thecondition above can be written asGM 2R = GM R NµA>Nε or MR > εµAG .M/R can be rewritten asMR = 4 3 πR2 ϱ = (4 3 π) 1/3 (43 π) 2/3 R 2 ϱ 2/3} {{ }M 2/3 ϱ 1/3 = (4 3 π) 1/3 M 2/3 ϱ 1/3 ,which leads to( 43 π) 1/3 M 2/3 ϱ 1/3 > εµAGε ) 3/2 1or M>( √ .µAG 43 πϱThe average density can be estimated to be (again neglecting numerical factors of orderunity; for spherical molecules the optimal arrangement of spheres without overlappingleads to a packing fraction of 74%)ϱ = µ ,43 πr3 Bwhere r B = r e /α 2 is the Bohr radius (r B = 0.529 × 10 −10 m), r e the classical electronradius, and α the fine-structure constant. This leads toM> ( ε ) 3/2 1√= ( ε ) 3/2 r 3/2B√ = 1 (εr B ) 3/2 .µAG 43 πµ/(4 3 πr3 B ) µAG µ µ 2 AG