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corresponding threshold, which is ɛGZK = 5 × 10 19 eV in today’s Universe. This cutoff has recently been observed by the HiRes experiment (Abbasi et al. 2008). In the high-redshift Universe, however, the GZK cutoff will be some- what lower, as can be seen as follows: While in today’s Universe the average energy of a CMB photon is ɛCMB = 2.7kBTCMB = 6 × 10 −4 eV, at higher redshifts this energy will be larger by a factor of (1 + z). In the CR rest frame, the CMB photon energy is ɛ ′ CMB ≈ γ 6 × 10 −4 eV (1 + z) , (6.10) where γ is the Lorentz factor of a CR proton. Equating ɛ ′ CMB with ɛt, we find γ ≈ 2 × 1011 (1 + z) . (6.11) We can now calculate the CR energy for which the threshold for photo-pion production is reached: ɛGZK(z) = γmHc 2 ≈ 3 × 1020 eV , (6.12) (1 + z) where mH is the mass of a proton. A more precise calculation, carrying out an integration over the entire Planck spectrum and over all angles, yields ɛGZK(z) = 5 × 1019 eV 1 + z ≈ 2 × 10 18 −1 1 + z eV . (6.13) 21 Thus, at redshifts of 10 or 20, the GZK cutoff is around an order of magnitude smaller than in today’s Universe, giving a robust upper limit to the CR energy. 152
The GZK cutoff applies only to those CRs that travel large distances through the CMB. A CR can undergo a maximum of approximately 10 inter- actions with CMB photons before falling below the GZK limit (e.g. Longair 1994). Given an interaction cross section of σ = 2.5×10 −28 cm 2 for photo-pion production and a CMB density of nγ = 5 × 10 2 cm −3 (1 + z) 3 , the interaction mean-free-path is λ = (σnγ) −1 1025 cm (1 + z) 3. (6.14) The maximum distance from which ultra-high energy CRs (UHECRs) with ɛCR ≥ 5 × 10 19 eV/(1 + z) could have originated is therefore dmax 10λ 1026 cm 3 kpc (1 + z) 3 1 + z 21 −3 . (6.15) Thus, at z = 20 a UHECR impinging upon a primordial gas cloud must have been accelerated in a source no more than a proper distance of ∼ 3 kpc away. The origin of UHECRs remains unknown even in today’s Universe, though possible sources range from pulsar winds to active galactic nuclei (AGN) and gamma-ray bursts (GRBs) (e.g. de Gouveia Dal Pino and Lazarian 2001; Waxman 2001; Torres et al. 2002). Though structures at z = 15 and z = 20 are not yet massive enough to form AGN, pulsars and GRBs may be plausible UHECR sources at these redshifts (e.g. Bromm and Loeb 2006). 6.2.4 Magnetic fields Magnetic fields are an important component of CR studies due to their effects on both CR acceleration and propagation through the Universe. For instance, the maximum energy a CR can reach with the Fermi acceleration 153
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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
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
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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 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 163 and 164: of star-forming material. Here we t
Page 165: star formation, they found CR energ
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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
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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.
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Page 215 and 216: 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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