Lecture Notes for Astronomy 321, W 2004 1 Stellar Energy ...

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subject to a spherically symmetric gravitational field, for which E = 0 represents the dividing line between m being gravitationally bound or unbound. So we have: • K > 0 ⇒ E < 0 ⇒ “bound” (“closed”), R returns to 0. • K < 0 ⇒ E > 0 ⇒ “unbound” (“open”), R goes to infinity. • K = 0 (“flat”) ⇒ E = 0 ⇒ R → ∞ and Ṙ → 0 as t → ∞. These solutions for R(t) as a function of t are summarized in Figure 12. The last solution, for a flat universe, corresponds in our analog to the solution for the classical escape velocity, where the velocity of the test mass goes to zero at infinity. We note that K = 0 is quite strongly favored by current observations. We also note that in this case, the condition E = 0 means that two large numbers K and U, the total kinetic and potential energies of the universe, have to cancel to large precision. We will discuss this seemingly unlikely situation when we discuss “inflation.” It is interesting to evaluate the time evolution of the three cases above. For K = 0, it is easy to show that the Friedmann equation with ρ = M/(4πR 3 /3) becomes R 1/2 dR = √ 2GM dt, which gives R ∝ t 2/3 and the age of the universe as 2/(3H o ). Furthermore, the critical density which enables this E = 0 solution (see homework) can straightforwardly be shown to be ρ c = 3H2 o 8πG . (28) For K > 0, one can show using a similar analysis that a “big crunch” occurs at a time 2πGM/c 3 ∼ 10 Gyr after the big bang. 34
Figure 12: Time evolution of the scale parameter for the three possible types of spacetime curvature, K, assuming non-relativistic matter dominates the energy density. 35
Page 1 and 2: Lecture Notes for Astronomy 321, W
Page 3 and 4: 1.3 Four forces of nature The elect
Page 5 and 6: Figure 1: Binding energy per nucleo
Page 7 and 8: 2 Stellar Energy Generation - PP-ch
Page 9 and 10: 2.2 The pp interaction: Coulomb bar
Page 11 and 12: compared to the quantum barrier pen
Page 13 and 14: 0.01 0.009 0.008 0.007 0.006 I (MeV
Page 15 and 16: Figure 8: Yet another high-temperat
Page 17 and 18: part due to the solar neutrinos dis
Page 19 and 20: came up with a viable detector cons
Page 21 and 22: neutrino species have small, but fi
Page 23 and 24: urning in the next sub-shell outwar
Page 25 and 26: solar luminosity, which is comparab
Page 27 and 28: As it turns out, the neutral curren
Page 29 and 30: 4.6 Further collapse and black hole
Page 31 and 32: approaches zero. This distance R s
Page 33: 5.3 The Friedmann equation and a us
Page 37 and 38: Table 1: Properties of cosmic fluid
Page 39 and 40: 7 Vacuum energy and Inflation In Se
Page 41 and 42: 7.2 Inflationary scenario We saw ab
Page 43 and 44: 8 Fluids and 2nd Friedmann equation
Page 45 and 46: Figure 15: Energy (kT ) per particl
Page 47 and 48: decay modes, and hence N ν . The c
Page 49 and 50: Figure 18 gives this critical densi

Lecture Notes for Astronomy 321, W 2004 1 Stellar Energy ...

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