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### Spatial and Temporal Averages for Elliptical Orbits

posted Dec 16, 2013, 5:45 AM by Abel Mendez   [ updated Jan 4, 2014, 8:50 AM ]

Mean orbital values for distance, stellar flux, and equilibrium temperature can be computed with respect to spatial or temporal coordinates. Spatial averages are usually calculated with respect to the mean anomaly (E) or true anomaly (θ). Temporal averages are calculated with respect to time (t) or the eccentric anomaly (M). It is often assumed that the mean equilibrium temperature of a planet in an elliptic orbit can be simply calculated from its mean distance or flux, but this is not correct.

We calculated analytic solutions for both the spatial and temporal averages of distance, flux, and equilibrium temperature (Méndez et al., 2014, in preparation). Here a is the semi-major axis, e is the eccentricity, L is the stellar luminosity, A is the planet's bond albedo, and To = 278.5 K (i.e. the equilibrium temperature of Earth for zero albedo). A factor f is related to the effectiveness of atmospheric circulation and how the energy absorbed is transferred from the planet’s day to night sides (e.g. f = 1 for fast rotators and f = 2 for tidally locked planets without atmospheres)For convenience, the formulas derived here use 'exoplanet units' where distances are in AU, flux in solar units, and temperature in kelvins.

#### Mean Spatial Distance, Flux, and Equilibrium Temperature

 $\large&space;\bar{r}=a\sqrt{1-e^2}$ (1)

 $\large&space;\overline{F}=\frac{L(2+e^2)}{2a^2\(1-e^2)^2}$ (2)

 $\large&space;\overline{T}_{eq}=T_o\left (\frac{fL(1-A)}{a^2}\right )^ \frac{1}{4} \frac{2}{\pi \sqrt{1-e}} \; \mathbf{E}\left ( \frac{2e}{1+e} \right )$ (3)

Here r̅ , F̅, and T̅eq are the mean spatial values (i.e. with respect to the true anomaly) for planet distance, stellar flux, and planet equilibrium temperature. E is the complete elliptic integral of the second kind of the argument within parenthesis.

#### Mean Temporal Distance, Flux, and Equilibrium Temperature

 $\large&space;\left \langle r \right \rangle=a\left ( 1+\frac{e^2}{2} \right )$ (4)

 $\large&space;\left \langle F \right \rangle=\frac{L}{a^2\sqrt{1-e^2}}$ (5)

 $\large&space;\left \langle T_{eq} \right \rangle=T_o\left (\frac{fL(1-A)}{a^2}\right )^ \frac{1}{4}\frac{2\sqrt{1+e}}{\pi} \; \mathbf{E}\left ( \frac{2e}{1+e} \right )$ (6)

Here <r>, <F>, and <Teq> are the mean temporal values (i.e. with respect to time) for planet distance, stellar flux, and planet equilibrium temperature. E is the complete elliptic integral of the second kind of the argument within parenthesis.

All the previous equations become the well known expressions for circular orbits for e = 0, where there is no difference between spatial or temporal averages. For most applications the temporal averages are the ones necessary. Eq. 1, 4, and 5 are well known expressions (William and Pollard, 2002Perryman, 2011). Eq. 2, 3, and 6 are new derivations, but only Eq. 6 has practical implications. Eq. 4, 5, and 6 were also verified with a numerical simulation.

#### Using the Mean Temporal Equilibrium Temperature

The time-average stellar flux (Eq. 5) can't be used to calculate the time-average equilibrium temperature (Eq. 6) as if often assumed. The stellar flux increases to infinity as the eccentricity of the planet approaches one. However, the equilibrium temperature does not change accordingly, actually it decreases with eccentricity approaching a minimum of about 90% the value for a circular orbit. This seems contradictory but the stellar flux increases with 1/r2 while the equilibrium temperature with 1/√r. Therefore, the following expressions should not be used to calculate the average equilibrium temperature from the average distance or stellar flux since they produces large errors (>>10%) for highly eccentric orbits:

 $\large&space;\left \langle T_{eq} \right \rangle \neq T_o\left (\frac{fL(1-A)}{\left \langle r \right \rangle ^2}\right )^ \frac{1}{4} \neq T_o\left (\left \langle F \right \rangle f(1-A)\right )^ \frac{1}{4}$ (7)

where <r> is given from Eq. 4 and <F> is given from Eq. 5. These expression suggests that the equilibrium temperature increases with an increase in stellar flux due to eccentricity but it is quite the opposite.

Eq. 6 is the analytic solution to the equilibrium temperatures for eccentric orbits. Errors from assuming a circular orbits can be up to 10% for highly eccentric orbits. Still, this is not a large error compared with larger uncertainties associated with f and A. Summarizing, given the equilibrium temperature for circular orbits, the equilibrium temperature for elliptical orbits is:

 $\large&space;\left \langle T_{eq} \right \rangle=T_{eqc}\frac{2\sqrt{1+e}}{\pi} \; \mathbf{E}\left ( \frac{2e}{1+e} \right )$ (8)

where the equilibrium temperature for circular orbits Teqc is given by:

 $\large&space;T_{eqc} = T_o\left (\frac{fL(1-A)}{a^2}\right )^ \frac{1}{4} = T_\star \sqrt{ \frac{R_\star }{2a}}\left [ f\left ( 1-A \right ) \right ]^{1/4}$ (9)

and T* and R* are the effective temperature and radius of the star.

#### Notes

• There are many math libraries which include calculations for complete elliptic integral functions (e.g. Mathematica and GSL). The following expression could be used to simplify the calculation of Eq. 6 or 8. Errors from this approximation are less than 1% (less than 0.5% for e < 0.5).
 $\large&space;\frac{2\sqrt{1+e}}{\pi} \; \mathbf{E}\left ( \frac{2e}{1+e} \right )\approx \sqrt{1+\left (\frac{8}{\pi^2} -1 \right )e^{5/2}}$ (10)