Manuscript Hawk01-A@1 · Hawk01-A.tex · section · 11979653

Quasistatic mass evolution and its limitations

The four-dimensional semiclassical source › Quasistatic mass evolution and its limitations · lines 440–481

Hawk01-A@1#eq:power, Hawk01-A@1#eq:massledger, Hawk01-A@1#eq:alpha, Hawk01-A@1#eq:nu

The outgoing source power in natural units is

P_{em} (M) = \int_{0}^{\infty} ω \frac{d ^{2} N}{d u d ω} d ω .

For a selected species list, a quasistatic mass ledger is

\frac{d M}{d u} = - P_{em} (M) + P_{abs} (u, M) .

In SI units the right-hand side is divided by $c^{2}$ when the powers are in watts and $M$ is in kilograms.

For the isolated massless scalar, the radial equation depends on $M ω$ . Define

α ν = \frac{1}{2 π} ℓ = 0 \sum^{\infty} (2 ℓ + 1) \int_{0}^{\infty} \frac{z Γ _{ℓ} ( z ; 1 )}{e ^{8 π z} - 1} d z, = \frac{1}{2 π} ℓ = 0 \sum^{\infty} (2 ℓ + 1) \int_{0}^{\infty} \frac{Γ _{ℓ} ( z ; 1 )}{e ^{8 π z} - 1} d z .

Here $Γ_{ℓ} (z; 1)$ denotes the dimensionless scattering function at dimensionless frequency $z$ .

Displayed formulas in this passage

Hawk01-A@1#eq:power · line 443
$P_{em} (M) = \int_{0}^{\infty} ω \frac{d ^{2} N}{d u d ω} d ω .$
Hawk01-A@1#eq:massledger · line 451
$\frac{d M}{d u} = - P_{em} (M) + P_{abs} (u, M) .$
Hawk01-A@1#eq:alpha · line 461
$α ν = \frac{1}{2 π} ℓ = 0 \sum^{\infty} (2 ℓ + 1) \int_{0}^{\infty} \frac{z Γ _{ℓ} ( z ; 1 )}{e ^{8 π z} - 1} d z, = \frac{1}{2 π} ℓ = 0 \sum^{\infty} (2 ℓ + 1) \int_{0}^{\infty} \frac{Γ _{ℓ} ( z ; 1 )}{e ^{8 π z} - 1} d z .$
Hawk01-A@1#eq:nu · line 461
$α ν = \frac{1}{2 π} ℓ = 0 \sum^{\infty} (2 ℓ + 1) \int_{0}^{\infty} \frac{z Γ _{ℓ} ( z ; 1 )}{e ^{8 π z} - 1} d z, = \frac{1}{2 π} ℓ = 0 \sum^{\infty} (2 ℓ + 1) \int_{0}^{\infty} \frac{Γ _{ℓ} ( z ; 1 )}{e ^{8 π z} - 1} d z .$