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x = −2. Using x = −2 is therefore a more conservative choice that does not assume any modifications to standard Fermi acceleration theory. By equating the total CR energy density UCR with the integral of the differential CR spectrum over all energies, the normalizing density factor nnorm is estimated to be nnorm = UCR ɛmax ɛminln ɛmin ≈ 1 UCR 10 ɛmin , (6.2) where we get approximately 1/10 for the coefficient choosing ɛmax = 10 15 eV and ɛmin = 10 6 eV. The differential energy spectrum is therefore with dnCR dɛ = This can also be written as UCR ɛ2 minln ɛmax ɛmin ɛ ɛmin −2 , (6.3) UCR(z) ≈ pCRESNfPISNΨ∗(z)tH(z)(1 + z) 3 . (6.4) UCR(z) ≈ 2 × 10 −15 erg cm −3 pCR × fPISN 2 × 10 −3 M −1 ⊙ 0.1 ESN 10 52 erg 1 + z 21 Ψ∗ 2 × 10 −2 M⊙ yr −1 Mpc −3 3 2 , (6.5) where pCR is the fraction of SN explosion energy, ESN, that goes into CR energy, and fPISN is the number of PISNe that occur for every solar mass unit 148
of star-forming material. Here we take the above star-formation rate to be constant over a Hubble time tH, where tH(z) 2 × 10 8 1 + z yr 21 −3/2 , (6.6) evaluated at the relevant redshift. We assume that each star quickly dies as a PISN with ESN = 10 52 erg, appropriate for a 200 M⊙ star (Heger and Woosley 2002), ten percent of which is transformed into CR energy (e.g. Ruderman 1974). The value of ten percent is derived from Milky Way (MW) energetics, and here we have simply extrapolated this to PISNe. Very little is known about what value of pCR applies to PISNe specifically, so assuming their shock structure to be similar to local SNe appears to be a reasonable first guess. We choose 1/500 M⊙ for fPISN, so that there is one PISN for every 500 M⊙ of star-forming material. This implies that somewhat less than half of the star forming mass falls within the PISN range. Compared to PISNe, the usual core-collapse SNe (CCSNe) thought to accelerate CRs in the Milky Way have an explosion energy that is lower by about an order of magnitude. However, the masses of their progenitor stars are also much lower, ranging from ∼ 10 − 40 M⊙. Thus, if we have an IMF extending to this lower mass range, there will be approximately an order of magnitude more SNe per unit mass of star-forming material. However, because ESN for low-mass (∼ 10 M⊙) CCSNe progenitors will be lower by a similar amount, these two effects may cancel out and result in a total UCR that is comparable to the PISN case. Furthermore, the difference in shock velocities, ush, between a PISN and a CCSN should not be more than a factor of a few. Though CCSNe have a lower explosion energy, this energy is used to accelerate roughly ten times less ejected mass, Mej, than in a PISN. Simple 149
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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
Page 111 and 112: It is apparent that the angular mom
Page 113 and 114: h = 0.7. To accelerate structure fo
Page 115 and 116: acc = Lres 50 AU, where: Lres 0.5
Page 117 and 118: years, up to several dynamical time
Page 119 and 120: sp = GM∗ c2 , (4.1) s where cs is
Page 121 and 122: Figure 4.2: Left: Angular momentum
Page 123 and 124: through the disk, and the disk is e
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
Page 133 and 134: sink M∗(5000 yr) [M⊙] M∗(10 5
Page 135 and 136: present here, particularly if the s
Page 137 and 138: the stars grow to larger masses (se
Page 139 and 140: Here, Σ is the disk surface densit
Page 141 and 142: which has already detected GRBs at
Page 143 and 144: M⊙. They find that the specific a
Page 145 and 146: are initialized at high z with smal
Page 147 and 148: M⊙, where Nneigh 32 is the typic
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
Page 157 and 158: a much smaller contribution from <
Page 159 and 160: abundances of HD, which also acts a
Page 161: first stars had very short lifetime
Page 165 and 166: star formation, they found CR energ
Page 167 and 168: The GZK cutoff applies only to thos
Page 169 and 170: and virialization redshift zvir = 2
Page 171 and 172: and undergoes free-fall collapse. I
Page 173 and 174: β = 1 − −2 1/2 ɛ + 1 mHc2
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
Page 195 and 196: to be part of an area of scientific
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.
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Rollinde, E., Vangioni, E., and Oli
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Simon, M., Ghez, A. M., Leinert, C.
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Tanvir, N. R., Fox, D. B., Levan, A
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Rays, volume 576 of Lecture Notes i
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Zatsepin, G. T. and Kuz’min, V. A
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