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6.3.2 Thermal and chemical evolution In calculating the evolution of the primordial clouds, we solve the com- prehensive chemical reaction network for all the species included in Johnson and Bromm (2006), and consider cooling due to H, H2, and HD. The temperature of the CMB sets the lower limit to which the gas can cool radiatively (e.g. Larson 1998). Assuming that CRs with the above energy spectrum impinge upon the primordial gas cloud, we add the respective heating and ionization rates. Once a low-energy CR enters the high-density region of the cloud, each time it ionizes an H atom, an electron with average energy < E >= 35 eV is released (Spitzer and Tomasko 1968). Including the ionization energy of 13.6 eV implies that a CR proton loses approximately 50 eV of kinetic energy upon each scattering. This places a limit on the number of scatterings a CR can undergo in a cloud as well as a limit on the distance into the cloud that it can reach before it loses all its energy to ionization. This distance can be described by a penetration depth where (Schlickeiser 2002) and − Dp(ɛ) ≈ βcɛ − dɛ dt ion , (6.21) dɛ = 1.82 × 10 dt ion −7 eV s −1 nH0f(ɛ), (6.22) f(ɛ) = (1 + 0.0185 lnβ) 158 2β 2 β3 , (6.23) 0 + 2β3
β = 1 − −2 1/2 ɛ + 1 mHc2 . (6.24) Here, mH is the mass of a proton, β = v/c, and (dɛ/dt)ion is the rate of CR energy loss due to ionization. The cutoff value of β0 0.01 is appropriate for CRs traveling through a medium of atomic hydrogen, since β0c = 0.01c is the approximate orbital velocity of electrons in the ground state of atomic hydrogen. When the velocity of CRs falls below β0c, the interaction between the CRs and electrons will sharply decrease, as will the ionization rate (Schlickeiser 2002), but for our study all CRs are assumed to be above this critical velocity. Thus, for a given distance D into a cloud, a CR has an effective optical depth of D/Dp. Fig. 6.1 shows the dependence of Dp on CR energy for a neutral hydrogen density of nH 0 = 1 cm−3 , which is typical for densities in the minihalo case. Densities will of course greatly increase towards the end of the free-fall evolution, but the size of the collapsing gas cloud, and thus the distance CRs must travel, will decrease. Although the overall attenuation would be slightly greater if the time-dependent density evolution were accounted for instead of using a constant attenuation value, when comparing these two cases the dif- ference is not large enough to yield a significant variation in the minihalo’s temperature evolution. For simplicity, only the typical density was therefore used in calculating the attenuation. As can be seen in Fig. 6.1, the lowest-energy CRs do not get attenuated until they travel a distance of about a few hundred pc, so in gas clouds of this size or smaller the CR flux will not be significantly attenuated. This also shows that the low-energy CRs are the ones that will have the greatest ionization and heating contribution to the cloud, as they more readily release 159
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Copyright by Athena Ranice Stacy 20
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New Insights into Primordial Star F
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Acknowledgments I first want to giv
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angular momentum to rotate at nearl
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2.4.2.5 Thermodynamics of accretion
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Chapter 7. Outlook 177 Bibliography
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List of Figures 2.1 Density project
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1.1 Motivation Chapter 1 Introducti
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leaving behind a neutron star or bl
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describe studies that have attempte
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CRs may therefore provide a pathway
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expected that some of the gas withi
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our numerical methodology in Chapte
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scattering is the major source of o
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portance. The reaction rates for an
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The size of the refinement levels h
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the sink a temperature and pressure
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same time, such as the fragments in
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Figure 2.2: Physical state of the c
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2.4.2 Protostellar accretion 2.4.2.
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Despite some amount of rotational s
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Figure 2.6: Velocity field of the c
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Figure 2.7: Disk mass vs. time sinc
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Figure 2.8: Angular momentum struct
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sink tform [yr] Mfinal [M⊙] rinit
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sink (∼ 4 M⊙) that merged with
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actual final mass of the star is li
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the average temperature of all part
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disk accretion onto a primordial pr
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Figure 2.12: Impact of feedback on
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Figure 2.13: Comparison of specific
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the central core, eventually compri
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extending our simulation to later t
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that such a non-primordial origin t
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impose an upper mass limit to Pop I
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we use a ray-tracing scheme to foll
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3.2.3 Sink Particle Method When an
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convenient way to directly measure
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where the sum now extends from the
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Figure 3.1: Evolution of various pr
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ased on the prescription of Omukai
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ate found at late times in our ‘n
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Figure 3.2: Evolution of various di
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Figure 3.3: Evolution of disk mass
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Figure 3.4: Temperature versus numb
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disk mass is able to level off in d
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Figure 3.6: Projected density and t
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onto the disk. The numerical experi
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that the I-front expands as an hour
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Figure 3.8: Evolution of various H
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˙ M∗ = 3πΣν, (3.18) where ν
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the discrete nature of the gas part
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stars. We also note that, despite t
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current computational limits. If ra
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the deaths of massive stars (see Wo
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It is apparent that the angular mom
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h = 0.7. To accelerate structure fo
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acc = Lres 50 AU, where: Lres 0.5
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years, up to several dynamical time
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sp = GM∗ c2 , (4.1) s where cs is
Page 121 and 122: Figure 4.2: Left: Angular momentum
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Page 125 and 126: steadily grows for the rest of the
Page 127 and 128: Fig. 4.5 shows these timescales for
Page 129 and 130: sink B. Once on the MS, the sink A
Page 131 and 132: Figure 4.6: Evolution of stellar ro
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Page 139 and 140: Here, Σ is the disk surface densit
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Page 143 and 144: M⊙. They find that the specific a
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Page 149 and 150: Figure 5.1: Top panels: Effective v
Page 151 and 152: Figure 5.2: Effect of relative stre
Page 153 and 154: Using cs ∝ T 1/2 results in ρ
Page 155 and 156: Figure 5.4: Evolution of gas proper
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Page 159 and 160: abundances of HD, which also acts a
Page 161 and 162: first stars had very short lifetime
Page 163 and 164: of star-forming material. Here we t
Page 165 and 166: star formation, they found CR energ
Page 167 and 168: The GZK cutoff applies only to thos
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Page 171: and undergoes free-fall collapse. I
Page 175 and 176: and where ΓCR(D) = 5 × 10 −29 e
Page 177 and 178: Figure 6.2: Thermal evolution of pr
Page 179 and 180: Figure 6.3: Minimum temperature rea
Page 181 and 182: Figure 6.4: Thermal evolution of pr
Page 183 and 184: The CR energy density can now be es
Page 185 and 186: 2007). When considering realistic c
Page 187 and 188: 6.3.5 Fragmentation scale As is evi
Page 189 and 190: 2006). This decrease in fragmentati
Page 191 and 192: Chapter 7 Outlook Our understanding
Page 193 and 194: that Pop III stars likely experienc
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Page 197 and 198: Barkana, R. and Loeb, A. (2001). In
Page 199 and 200: Bromm, V., Kudritzki, R. P., and Lo
Page 201 and 202: Daigne, F., Olive, K. A., Silk, J.,
Page 203 and 204: Gao, L., White, S. D. M., Jenkins,
Page 205 and 206: Heger, A., Woosley, S. E., and Spru
Page 207 and 208: Kratter, K. M. and Murray-Clay, R.
Page 209 and 210: Machida, M. N., Omukai, K., Matsumo
Page 211 and 212: Navarro, J. F. and White, S. D. M.
Page 213 and 214: Rollinde, E., Vangioni, E., and Oli
Page 215 and 216: Simon, M., Ghez, A. M., Leinert, C.
Page 217 and 218: Tanvir, N. R., Fox, D. B., Levan, A
Page 219 and 220: Rays, volume 576 of Lecture Notes i
Page 221 and 222: Zatsepin, G. T. and Kuz’min, V. A
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