tokens: this is 0.3 ≲ f h ≲ 0.4 for halo masses M 12 ∼ (0.5−2) at redshift z ≳ 2 (Roca-Fàbrega et al. 2019 ), while observations of M 12 ∼ (1−10) haloes at z ≳ 0.1 suggest f h ∼ 0.2−0.3 (Singh et al. 2018 ). We define f h,0.3 = f h /0.3. Δ( R v ) is the ratio of the density at R v to the mean density in the halo, . For a singular isothermal sphere Δ( R v ) = 1/3, while for an NFW halo with concentration c = (5, 10, 20), Δ( R v ) ∼ (0.24, 0.19, 0.14), respectively. We write Δ 1/6 = Δ( R v )/(1/6). Assuming cosmological parameters Ω m = 0.3 and , we obtain for the stream density at R v where δ 100 = δ/100 and . For a hydrogen mass fraction of X = 0.76, this yields a hydrogen number density in the stream at R v of We hereafter write . The stream radius at R v can be constrained using the mass flux through the stream, where is the total mass accretion rate on to the halo virial radius, and f s is the fraction of baryonic accretion along the gas stream. Cosmological simulations suggest f s ∼ (0.2−0.5) with a typical value of f s = 1/3 (i.e. three significant streams; Danovich et al. 2012 ). In the Einstein de Sitter (EdS) regime (valid at z > 1), the accretion on to the virial radius is well approximated by (Dekel et al. 2013 ) with halo-to-halo variance encapsulated by the normalization s ∼ (0.5−2). Combining equations ( 15 ), ( 23 ), ( 25 ), and ( 26 ), we obtain with . Together with equation ( 16 ), we have Note that this is larger by a factor of ∼2 than the corresponding equation (68) in Padnos et al. ( 2018 ), which seems to be due to a typo, as it is consistent with their equation (67). Using equation ( 21 ) for δ, we see that and is independent of redshift. More massive haloes are thus fed by relatively narrower streams compared to their virial radii. We hereafter substitute . The hydrogen column density perpendicular to the stream axis is N H, 0 = n H, 0 R s . Using equations ( 24 ) and ( 27 ), this yields For haloes at z ∼ 2, this is large enough for the streams to be largely self-shielded against the UV background if the neutral fraction is x H i ≳ 10 −3 . In collisional ionization equilibrium (CIE), this is true for (e.g. Goerdt et al. 2010 ), i.e. Θ s ≲ 2. Of course, streams may still be susceptible to local sources of UV radiation, such as star formation in satellite galaxies located along the stream, or starbursts/AGN activity in the central galaxy. To summarize, given the halo mass and redshift, the precise temperatures of the stream and halo gas, Θ s and Θ h , respectively, determine the density contrast, δ. The stream Mach number, M b , is then interchangeable with η, the ratio of stream-to-virial velocity. The stream density is then determined by , the normalization of the hot halo gas density near R v . Finally, the ratio of stream radius to virial radius is set by , the normalization of the gas accretion rate along the stream. For η ∼ 1, , and , a stream at z = 2 has , (0.05−0.28), and (0.03−0.16) for δ = 30, 100, and 300, respectively.
ner_tags: O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula B-Citation I-Citation I-Citation I-Citation O O O O B-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula B-Citation I-Citation I-Citation I-Citation O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O B-TextGarbage I-TextGarbage O O B-Model I-Model I-Model B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Model I-Model O O B-Formula I-Formula I-Formula I-Formula I-Formula B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O B-Formula I-Formula I-Formula I-Formula O O O O B-Citation I-Citation I-Citation I-Citation O O O B-Model I-Model I-Model B-Model B-Model O O B-Formula I-Formula I-Formula O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O B-Wavelength O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O B-Wavelength O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula B-TextGarbage O O O O O B-Formula I-Formula I-Formula O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O