A distance measurement for blazar TXS 0506+056 using its radio variability and very long baseline interferometry images

Open Access

Table 1.

Physical quantity (Q) transformation by Doppler effect and redshift.

Q^so to Q^em	Q^rec to Q^so	Q^rec to Q^em
ν^so = δν^em	ν^rec = ν^so/(1 + z)	ν^rec = δν^em/(1 + z)
dt^so = dt^em/δ	dt^rec = (1 + z)dt^so	dt^rec = (1 + z)dt^em/δ
dA^so = dA^em/δ²	dA^rec = dA^so	dA^rec = dA^em/δ²
dθ^so = dθ^em	dθ^rec = dθ^so	dθ^rec = dθ^em
$S_{ν}^{so} = δ^{3} S_{ν}^{em}$ $S_{\nu}^{\mathrm{so}}=\delta^{3}S_{\nu}^{\mathrm{em}}$	$S_{ν}^{rec} = S_{ν}^{so} / {(1 + z)}^{3}$ $S_{\nu}^{\mathrm{rec}}=S_{\nu}^{\mathrm{so}}/(1+z)^{3}$	$S_{ν}^{rec} = δ^{3} S_{ν}^{em} / {(1 + z)}^{3}$ $S_{\nu}^{\mathrm{rec}}=\delta^{3}S_{\nu}^{\mathrm{em}}/(1+z)^{3}$

Notes. The quantity on the emission frame (Q^em) is affected by the Doppler effect, which in the source frame (Q^so) is affected by redshift and in the receiver frame (Q^rec) is affected by the Doppler effect and redshift. ν is the frequency, A denotes the radiated surface area, θ is the angular size of the emission region, and S is the flux density. δ is the Doppler factor, and z is the redshift.

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