-supernovae & Neutron Stars 3.1. (ALL) When an iron stellar core collapses in a
ID: 2268991 • Letter: #
Question
-supernovae & Neutron Stars 3.1. (ALL) When an iron stellar core collapses in a supernova event, it converts all o the iron into neutrons. However the resulting neutrons each individually represent a higher-energy state than the iron nuclei. How much energy does it take to convert 2 Mo of 5626Fe into neutrons, assuming that each neutron masses the same as it would in free space? 3.2. (ALL) How does the above value compare to the total amount of energy that was released in creating the 626Fe from hydrogen in the first place? 3.3. (ALL) How much-energy is liberated-by the gravitational collapse of a the aforemen- tioned 2 Mo core from a radius of 1 Re, down to a radius of 20 km? Assume constant density (not so bad of an assumption in this case).Explanation / Answer
(3.1) Mass of neutron = 1.674 × 10-27 kg
In Fe molecule there are 30 neutrons, total mass= 30×1.674×10-27 kg
Energy released for conversion into neutron for one molecule
E = mc2 = 30×1.674×10-27×( 2.99 ×108)2 kgm/s
= 30×1.674×10-27×9.8×( 2.99 ×108 )2 W
=4.39 ×10-8 W
Energy released due to two such molecules = 2×4.39×10-8
=8.78 ×10-8 W
(3.2)Number of electrons in one iron molecule = 26
Mass of electron = 9.1 ×10-31 kg
Energy required for one molecule E = mc2
= 26×9.31×10-31 ×9.8×( 2.99 ×108 )2 W
=2.12 ×10-11 W
For two molecules =4.24 ×10-11 W
(3.3) Energy liberated = (0.60 b)÷(1-b/2) where, b= GM/Rc2
Total mass of neutrons in two molecules
= 2×30×1.674×10-27 kg
=1×10-25 kg
G=6.674 ×10-11 m3kg-1s-2
R=20×103 m
b=(6.674×10-11×1×1025×9.8)/[20×103×(2.99 ×108)2]
=3.65×10-6
E= (0.6×3.65×10-6)/[1-(3.65×10-6)/2 ] W
= 2.19 ×10-6 W
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