CHAPTER 4
Results
The figures below are divided into two parts: a left-hand side where the density decreases with time from left to right, representing the expansion phase, and a right-hand side where the density increases with time from left to right, representing the collapse phase.
The dotted line in the right-hand part indicates when virialization occurs.
10−2 100
102 10−10 10−8 10−6 10−4 10−2 100 102
B [G]
10−1 101 103 105 107
10−10 10−8 10−6 10−4 10−2 100 102
nb [cm−3]
B0 = 0.01 nG 0.1 nG 0.5 nG 1 nG 3 nG 5 nG 10 nG 13 nG
Figure 4.1 – Without turbulence – Evolution of the physical magnetic field strength with baryonic density for several different initial comoving field strengths.
10−2 100
102 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3
B n b−α [G cm3α ]
B0 = 0.01 nG 0.1 nG 0.5 nG 1 nG 3 nG 5 nG 10 nG 13 nG
10−1 101 103 105 107
10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3
nb [cm−3]
Figure 4.2 – Without turbulence – Evolution of the magnetic field strength scaled with the baryonic density for several different initial comoving field strengths.
February 2013 4.1. MODEL WITHOUT TURBULENCE
The temperature evolution is shown in Figure 4.3. The thermal evolution for the B0 = 0.01 nG case is found to be nearly identical to the zero-field case. For B0= 0.1 nG, heat input from ambipolar diffusion during the expansion phase eventually increases the temperature above the temperature of the CMB, and for stronger fields the temperature is significantly increased, up to& 104K for B0 = 3 nG and above. At that temperature, the cooling is dominated by atomic hydrogen cooling (as opposed to molecular hydrogen cooling at lower temperatures) which is very effective. Another effect of this high temper-ature is that collisional ionization becomes efficient at increasing the ionization degree, which is shown in Figure 4.4; the ionization fraction increases with increasing magnetic field strength. This increased ionization renders ambipolar diffusion less efficient, which also prevents a further increase in temperature, as can be seen from the heating and cooling rates shown in Figure 4.6 for an initial magnetic field of B0 = 1 nG. The H2 fraction is shown in Figure 4.5. During the expansion phase, the higher ionization frac-tion results in an enhanced H2 fraction at turnaround, compared to the zero-field case.
However, the H2 fraction only increases with increasing magnetic field strength up to B0 = 1 nG; at B0 = 3 nG it is closer to the zero-field case again and then keeps increasing for increasing initial field strength once more. This is related to the high temperatures reached in those cases, which facilitates efficient collisional dissociation of H– and H2.
During the collapse phase, the thermal evolution becomes more complicated. After turnaround, the H2 fraction oscillates and evolves quite differently for different initial magnetic field strengths, as a result of the competition between collisional dissociation of H2 and increased H2 formation due to the enhanced electron fraction. This situation stabilizes approximately at a density of ∼1 cm−3; now the H2 fraction is larger for a larger initial magnetic field (except in the case where B0 = 13 nG), and increases only slightly further with increasing density. As a result of the enhanced H2 fraction, the stronger molecular hydrogen cooling now allows the halo with the stronger magnetic field to cool to a temperature below that of a less magnetized halo for a brief period, but then ambipolar diffusion heating begins to increase the temperature again, due to the gas becoming increasingly more neutral.
Note that for the zero-field case and for initial fields smaller than 1 nG the gas temperature increases adiabatically after turnaround, and when sufficient H2 is formed for efficient cooling and the temperature decreases again. The central temperature is smaller than the virial temperature of the host halo, which is ∼104K. This occurs because the innermost region starts to cool and collapse during the adiabatic compression and does not experience the virialization shock.
For initial fields in the range 3 − 12 nG, something interesting happens at high den-sities; the heating from ambipolar diffusion is so strong that at a certain point an insta-bility forms. The molecular hydrogen cooling cannot compensate for the strong heating anymore, and the temperature suddenly increases. Then atomic hydrogen cooling takes over and stabilizes the temperature at ∼8000 K. At the same time, collisional dissoci-ation of H2 becomes dominant over H2 formation processes and the H2 fraction drops steeply. However, when the temperature increases, also the ionization fraction increases strongly and this again aids the formation of H2. Some molecules reform, but the high temperature and density prevent it from becoming an important coolant again, and the gas temperature stays high. The smallest B0 for which an instability as described above occurs, in this case 3 nG, will be referred to as Bcrit,inst0 , the critical field strength for
forming an instability. For an even stronger initial magnetic field, B0 = 13 nG, H2 never becomes an important coolant as it cannot form fast enough. Ambipolar diffusion heat-ing, which is the main heating process, is balanced by atomic hydrogen cooling at all times during the collapse, and thus the gas stays hot. The smallest B0 for which this occurs will be referred to as B0crit,H2, the critical field strength for which H2cooling never becomes efficient. The problem with these large fields, however, is that the magnetic Jeans mass becomes large compared to the halo mass, and thus the halo will likely not collapse at all.
10−2 100
102 101 102 103 104
T [K]
10−1 101 103 105 107
101 102 103 104
nb [cm−3]
B0 = 0.01 nG 0.1 nG 0.5 nG 1 nG 3 nG 5 nG 10 nG 13 nG
Figure 4.3 – Without turbulence – Evolution of the gas temperature with baryonic density for several different initial comoving field strengths.
February 2013 4.1. MODEL WITHOUT TURBULENCE
10−2 100
102 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1
x e
B0 = 0.01 nG 0.1 nG 0.5 nG 1 nG 3 nG 5 nG 10 nG 13 nG
10−1 101 103 105 107
10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1
nb [cm−3]
Figure 4.4 – Without turbulence – Evolution of the electron fraction with baryonic density for several different initial comoving field strengths.
10−2 100
102 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1
x H 2
B0 = 0.01 nG 0.1 nG 0.5 nG 1 nG 3 nG 5 nG 10 nG 13 nG
10−1 101 103 105 107
10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1
nb [cm−3]
Figure 4.5 – Without turbulence – Evolution of the H2 fraction with baryonic density for several different initial comoving field strengths.
10−2 100
102 100 102 104 106 108 1010
dT/dz
Adiabatic h/c AD heating HI−cooling H2−cooling Compton h/c
10−1 101 103 105 107
100 102 104 106 108 1010
B0 = 1 nG
nb [cm−3]
Figure 4.6 – Without turbulence – Evolution of the heating and cooling rates from various processes (as labeled) for an initial comoving field strength of 1 nG.
February 2013 4.1. MODEL WITHOUT TURBULENCE
10−2 100
102 10−1 100 101 102 103 104 105
Scale [pc]
2R 2π/k
max(n
b) λJ λJB
Integral scale
10−1 101 103 105 107
10−1 100 101 102 103 104 105 B0 = 1 nG
nb [cm−3]
Figure 4.7 – Without turbulence – Evolution of various scales in the cloud for an initial comoving field strength of 1 nG. The top black line represents the diameter of the spherical cloud, the blue dotted line shows the scale associated with kmaxas it evolves with density, the red and green dash-dot lines represent the thermal and magnetic Jeans length, respectively, and the purple solid line is the integral scale.
10−2 100
102 100 102 104 106 108 1010
dT/dz
Adiabatic h/c AD heating HI−cooling H2−cooling Compton h/c
10−1 101 103 105 107
100 102 104 106 108 1010
B0 = 13 nG
nb [cm−3]
Figure 4.8 – Without turbulence – Evolution of the heating and cooling rates from various processes (as labeled) for an initial comoving field strength of 13 nG.
10−2 100
102 10−1 100 101 102 103 104 105
Scale [pc]
2R 2π/kmax(n
b) λJ λJB
Integral scale
10−1 101 103 105 107
10−1 100 101 102 103 104 105 B0 = 13 nG
nb [cm−3]
Figure 4.9 – Without turbulence – Evolution of various scales in the cloud (as labeled) for an initial comoving field strength of 13 nG.