tokens: of our problem. Since a series of earlier works have studied this topic (Rees 1988 ; Kochanek 1994 ; Khokhlov Melia 1996 ; Strubbe Quataert 2009 ; Kasen Ramirez-Ruiz 2010 ), we only summarize here the main results that are most relevant to our work, including the kinetic energy, internal velocity dispersion and the opening angle with respect to the central SMBH. An UDS originates from the part of a star which at the time of tidal disruption has a positive energy. This unbound part contains about half of the mass of the star if the star initially approaches the SMBH along a parabolic orbit with a pericentre distance R p comparable to R t – an probable case in real galaxies (Rees 1988 ; Magorrian Tremaine 1999 ). In this case, the unbound material distribute in a range of specific energy between and (Guillochon Ramirez-Ruiz 2013 ; Stone, Sari Loeb 2013 ). As a result, the total kinetic energy E k of the UDS is where m * ≡ M * /M ⊙ , r * ≡ R * / R ⊙ . Here we have assumed an even distribution of unbound material in specific energy, i.e. , which was proposed by Rees ( 1988 ) and later confirmed by smooth-particle hydrodynamics simulations (Evans Kochanek 1989 ). Although we made a couple simplifications to derive equa-tion ( 2 ), the result agrees well with the median value computed from a statistically representative ensemble of UDSs (Guillochon et al. 2015 ). Using this equation and the fact that for main-sequence stars at m * > 1 (Demircan Kahraman 1991 ; Gorda Svechnikov 1998 ), we further derive . This scaling relation indicates that more massive stars generally give rise to more energetic UDSs. It is worth noting that equation ( 2 ) does not include the additional gain of kinetic energy during the compression and bouncing of the star at the orbital periapsis (Brassart Luminet 2008 , and references therein). We neglected it because the bouncing energy , about β 2 GM * / R * per unit mass (Guillochon et al. 2009 ), is usually much smaller than , where β ≡ R t / R p is the so-called penetration factor. Take tidal disruption of a solar-type star ( M * = M ⊙ and R * = R ⊙ ) for example. When M • = 4.3 × 10 6 M ⊙ which is the mass of the SMBH in the Milky Way (MW; Genzel, Eisenhauer Gillessen 2010 ), the penetration factor β cannot be greater than 4.3; otherwise, the star will enter a radius smaller than the ‘innermost bound circular orbit’ (twice the Schwarzschild radius for a non-rotating black hole) and eventually be swallowed as a whole by the SMBH. Because of such a limit to β, the ratio is small, in fact in the range (0.0063, 0.12) for the SMBHs in the mass range of 10 6 M • /M ⊙ 10 8 . We notice that our E k differs by a factor of β 2 from that derived by Khokhlov Melia ( 1996 ), Strubbe Quataert ( 2009 ) and Kasen Ramirez-Ruiz ( 2010 ). This is because those authors calculated using the formula , i.e. they assumed that the energy spread is determined at R p instead of at R t . However, recent studies of the tidal-disruption
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