Earth's temperature profile is a coupled problem: sunlight supplies energy, radiation carries energy to space, and radiation plus convection shape the atmosphere below. A lapse rate gives temperature differences; the radiation budget helps anchor the absolute temperatures.

1. Start above the atmosphere

Earth intercepts sunlight over pi*R^2, but has surface area 4*pi*R^2. Global-average incoming sunlight is therefore S/4. If A is planetary albedo and OLR is outgoing thermal infrared radiation, conservation of energy gives, neglecting the tiny geothermal contribution: [1]

N = S*(1-A)/4 - OLR
N = (1/(4*pi*R^2))*dE/dt

E is energy stored in the Earth system. Positive N means heat accumulates; equilibrium means N = 0. With S approximately 1361 W/m^2 and A approximately 0.30:

Incoming sunlight: 340 W/m^2
Reflected sunlight: 102 W/m^2
Absorbed sunlight: 238 W/m^2
Required equilibrium OLR: 238 W/m^2

Convection and evaporation redistribute energy inside this boundary. Cooling to space occurs through radiation. Today's Earth has a positive energy imbalance; equilibrium is a teaching reference. [2]

2. Convert outgoing flux into an effective temperature

The Stefan-Boltzmann law is F = sigma*T^4. Define:

T_e = (OLR/sigma)^(1/4)
sigma = 5.6704*10^-8 W/(m^2*K^4)

For OLR approximately 238 W/m^2, T_e is approximately 255 K (-18 C). This is a blackbody-equivalent emission temperature, not the air temperature at the literal top of the atmosphere. Actual infrared escapes from many heights, depending on wavelength, clouds and location; some escapes directly from the surface. [3,4]

3. Connect that temperature to the surface

Let z increase upward and define the environmental lapse rate:

Gamma(z) = -dT/dz
T(0) = T(z_e) + integral from 0 to z_e of Gamma(z) dz

That integration is exact for a specified temperature profile. A simple effective-emission-height approximation takes T(z_e) approximately equal to T_e. If the ground and near-surface air are closely coupled:

T_surface approximately T_e + integral from 0 to z_e of Gamma(z) dz

For an illustrative constant Gamma = 6.5 K/km and z_e = 5 km:

T_surface approximately 255 + 6.5*5
= 287.5 K, approximately 14 C

Those chosen values illustrate the connection; they do not independently predict global surface temperature. Radiative transfer, convection, humidity and clouds determine the actual emission heights and profile. Ground skin temperature can differ from nearby air. [5]

For a dry adiabatic parcel, the first law and hydrostatic balance give: [6]

c_p*dT - dp/rho = 0
dp = -rho*g*dz
Gamma_dry = g/c_p, approximately 9.8 K/km

Here c_p is specific heat at constant pressure, rho is air density, and g is gravitational acceleration.

Moist convection releases latent heat; the environmental lapse rate varies. The example's 6.5 K/km is not the dry adiabatic rate. [5,6]

4. Where greenhouse gases enter

Infrared opacity determines which parts of the profile contribute radiation escaping to space. Greater opacity can shift emission toward higher, colder tropospheric air in affected spectral bands. At an unchanged temperature profile this reduces outgoing energy; warming and other adjustments then work toward restoring balance. [2,4]

Pressure and gravity help determine a temperature gradient. That gradient alone cannot set the surface's absolute temperature. Radiation, opacity and the lapse rate work together. Nor can one constant tropospheric lapse rate be extended through the stratosphere, where temperature can increase with height. [3,5]

Sources

[1] NASA: Climate and Earth's Energy Budget
[2] NASA CERES: measured radiation budget and energy imbalance
[3] Kerry Emanuel, MIT: radiation and atmospheric structure
[4] NASA GISS: infrared spectra and the greenhouse effect
[5] Brian Rose, University at Albany: radiative-convective equilibrium
[6] MIT: adiabatic lapse-rate derivation, slides 19-20