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Radiative Time Constant

The volumetric heating rate, Q
partial Q/partial T ident 1/taur
Qir = sigma · T4 · partial epsilon/partial z
taur = (rho Cp)/(4 sigma T3 partial epsilon/partial z)
Harshvardan and Cess (1976, Tellus 28 p.1-10) give a value for Earth CO2 of partial epsilon/partial z = -0.0081 per Km and Cess and Khetan (1973, JQSRT 13 p.995-1009) give a value for Jupiter, Saturn, Neptune, and Uranus of partial epsilon/partial z = -0.0068, -0.0063, -0.0061, and -0.0058 per Km, respectively.
taur = (rho H Cp)/(4 epsilon sigma T3)
The phase shift, Phi, is the seasonal lag due to the radiative time constant. It arises from the 1st Fourier moment of the thermal response. A value of Phi=90° indicates a full seasonal shift ( e.g., the warm maximum expected at the summer solstice would occur at the autumnal equinox).
Phi = tan-1[ 2 pi · taur/tauo ]
where tauo is the orbital period. For epsilon = 0.3 the values are:

 
Teff
K		Peff
mBar		rho
gm/cm3		H
Km		taurad
years		phase
°
Venus229.020.0004.6230e-054.8620.0073.94
Earth255.0500.0006.8297e-047.4710.13139.47
Mars210.06.0001.5124e-0510.6050.0061.17
Jupiter124.0400.0008.6132e-0519.1464.53467.39
Saturn95.0300.0008.1128e-0536.97520.79477.27
Uranus59.0430.0002.0161e-0424.234131.31184.15
Neptune59.0500.0002.3443e-0419.206121.00877.67
Titan85.01.0004.0568e-0618.1520.05382.54