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the ZAMS, the accretion rate is set to the previous non-zero value in order to get more realistic values for R∗I and R∗II. This allows us to avoid setting R∗ = RZAMS too early in the protostar’s evolution. If, however, the accretion slows after the sink has been in place for more than its KH time, we assume the star has reached its ZAMS radius, and we set Lacc = 0, R∗ = RZAMS, and L∗ = LZAMS. Note that typical KH times, where tKH = GM 2 ∗ /R∗L∗, range from 1000 yr for a large and rapidly accreting 10 M⊙ protostar (see e.g. Hosokawa et al. 2010) to ∼ 4 × 10 4 yr for a 15 M⊙ main sequence star. The typical KH luminosity for a 15 M⊙ star is LKH ∼ 5 × 10 4 L⊙ (see Fig. 3.1). Given our averaging scheme in which a minimum of one 0.015 M⊙ gas particle can be accreted over 100 years, this gives an effective minimum mea- surable accretion rate of 1.5 × 10 −4 M⊙ yr −1 . However, for M∗ > ∼ 10 M⊙, this minimum accretion rate still yields a value of R∗II that is smaller than RZAMS. In this case, we again set the protostellar luminosity and radius to its ZAMS values once the accretion rate has dropped to near-zero. In our case, the measured accretion rate drops very quickly after 500 yr. At this point the star has reached 15 M⊙, is still undergoing adiabatic expansion, and has tKH ∼ 1000 yr. The star then begins rapid KH contraction until the measured accretion rate becomes zero at 1000 yr. Though within the simulation we set L∗ = LZAMS as soon as the averaged accretion rate is zero, in reality the protostar is better described by a more gradual approach to RZAMS. In Figure 3.1 we show the protostellar values used in the simulation along with a more realistic ‘slow-contraction’ model which follows the same accretion history as the ‘with-feedback’ case until reaching an asymptotic growth rate of 10 −3 M⊙ yr −1 . The ‘slow-contraction’ model is then held at this rate, which is similar to the fiducial value used in Equation 11, and is also the typical accretion 68
ate found at late times in our ‘no-feedback case’ as well as the simulations of, e.g., Greif et al. (2011); Smith et al. (2011). This model well-matches the prescription used in the simulation, particularly in effective temperature and ionizing luminosity, and both predict that ionizing radiation will exceed the influx of neutral particles, and that break-out beyond the sink occurs at < ∼ 1000 yr. Though the protostar most likely does not reach the ZAMS within the time of our simulation, our simple model serves as a reasonable approximation for any unresolved accretion luminosity. We also note that Hosokawa et al. (2010) find that, for a given accretion rate, primordial protostars undergoing disk accretion will have considerably smaller radii than those accreting mass in a spherical geometry. The rapid contraction of the protostar between 500 and 1000 years therefore serves as an idealized representation of the sub- sink material evolving from a spherical to a disk geometry as it gains angular momentum. As discussed in Smith et al. (2011), the sink particle method requires several simplifying assumptions when constructing the protostellar model. By setting the protostellar mass equal to the sink mass, we are assuming that the small-scale disk which likely exists within the sink region has low mass compared to the protostar. We also assume that the accretion rate at the sink edge is the same as the accretion rate onto the star, when in reality after gas enters within racc it must likely be processed through a small-scale disk before being incorporated onto the star. However, our assumption may still be a good approximation of physical reality, given that the primordial protostellar disk study of Clark et al. (2011b) implies that the thin disk approximation would probably not be valid on sub-sink scales (e.g. Pringle 1981), and that strong gravitational torques can quickly drive mass onto the star. Furthermore, as 69
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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
Page 31 and 32: The size of the refinement levels h
Page 33 and 34: the sink a temperature and pressure
Page 35 and 36: same time, such as the fragments in
Page 37 and 38: Figure 2.2: Physical state of the c
Page 39 and 40: 2.4.2 Protostellar accretion 2.4.2.
Page 41 and 42: Despite some amount of rotational s
Page 43 and 44: Figure 2.6: Velocity field of the c
Page 45 and 46: Figure 2.7: Disk mass vs. time sinc
Page 47 and 48: Figure 2.8: Angular momentum struct
Page 49 and 50: sink tform [yr] Mfinal [M⊙] rinit
Page 51 and 52: sink (∼ 4 M⊙) that merged with
Page 53 and 54: actual final mass of the star is li
Page 55 and 56: the average temperature of all part
Page 57 and 58: disk accretion onto a primordial pr
Page 59 and 60: Figure 2.12: Impact of feedback on
Page 61 and 62: Figure 2.13: Comparison of specific
Page 63 and 64: the central core, eventually compri
Page 65 and 66: extending our simulation to later t
Page 67 and 68: that such a non-primordial origin t
Page 69 and 70: impose an upper mass limit to Pop I
Page 71 and 72: we use a ray-tracing scheme to foll
Page 73 and 74: 3.2.3 Sink Particle Method When an
Page 75 and 76: convenient way to directly measure
Page 77 and 78: where the sum now extends from the
Page 79 and 80: Figure 3.1: Evolution of various pr
Page 81: ased on the prescription of Omukai
Page 85 and 86: Figure 3.2: Evolution of various di
Page 87 and 88: Figure 3.3: Evolution of disk mass
Page 89 and 90: Figure 3.4: Temperature versus numb
Page 91 and 92: disk mass is able to level off in d
Page 93 and 94: Figure 3.6: Projected density and t
Page 95 and 96: onto the disk. The numerical experi
Page 97 and 98: that the I-front expands as an hour
Page 99 and 100: Figure 3.8: Evolution of various H
Page 101 and 102: ˙ M∗ = 3πΣν, (3.18) where ν
Page 103 and 104: the discrete nature of the gas part
Page 105 and 106: stars. We also note that, despite t
Page 107 and 108: current computational limits. If ra
Page 109 and 110: 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
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sink M∗(5000 yr) [M⊙] M∗(10 5
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present here, particularly if the s
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the stars grow to larger masses (se
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Here, Σ is the disk surface densit
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which has already detected GRBs at
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M⊙. They find that the specific a
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are initialized at high z with smal
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M⊙, where Nneigh 32 is the typic
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Figure 5.1: Top panels: Effective v
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Figure 5.2: Effect of relative stre
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Using cs ∝ T 1/2 results in ρ
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Figure 5.4: Evolution of gas proper
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a much smaller contribution from <
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abundances of HD, which also acts a
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first stars had very short lifetime
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of star-forming material. Here we t
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star formation, they found CR energ
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The GZK cutoff applies only to thos
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and virialization redshift zvir = 2
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and undergoes free-fall collapse. I
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β = 1 − −2 1/2 ɛ + 1 mHc2
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and where ΓCR(D) = 5 × 10 −29 e
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Figure 6.2: Thermal evolution of pr
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Figure 6.3: Minimum temperature rea
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Figure 6.4: Thermal evolution of pr
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The CR energy density can now be es
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2007). When considering realistic c
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6.3.5 Fragmentation scale As is evi
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2006). This decrease in fragmentati
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Chapter 7 Outlook Our understanding
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that Pop III stars likely experienc
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to be part of an area of scientific
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Barkana, R. and Loeb, A. (2001). In
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Bromm, V., Kudritzki, R. P., and Lo
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Daigne, F., Olive, K. A., Silk, J.,
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Gao, L., White, S. D. M., Jenkins,
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Heger, A., Woosley, S. E., and Spru
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Kratter, K. M. and Murray-Clay, R.
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Machida, M. N., Omukai, K., Matsumo
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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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