Supplement 1: Chapters1-4 - clarkrorschpublication

Supplementary Material for Chapters 1 to 4

The Excel workbooks used for the calculation of the lunar temperatures for Figure 2.2 and the solutions to the Lorenz equations given in Figures 3.1, 3.2 and 3.1 are described and can be downloaded using the links provided. The dataset of the solar flux values used in the calculations in the book is also provided. The values at 0.5 hour intervals for 365 days of the year at latitudes from 0 to 85° in 5° intervals are provided in a csv file.

A .pdf file of this Supplementary Material may be downloaded using the link

Download.SupplementaryMaterial.Chaps. 1-4

1) The Calculation of the Surface Temperature along the Lunar Equator

The calculation of the lunar surface temperature along the equator is given in the Excel workbook Lunar Calcs.xlsx. This has three worksheets, Lunar T Calc, Lunar Plot and Fresnel R. The first worksheet contains the calculations based on Equations 2.1 to 2.4, the second contains the plot data and the third contains the Fresnel Reflection calculation.

The workbook may be downloaded using the link.

Download Workbook LunarCalc.xlsx

(Note: the worksheet will be saved to the Downloads folder. If the file does not open, or there is a message box about not enough memory or disk space, right click to open the properties box and click on unblock)

The daytime temperature is calculated using simple blackbody theory. The emitted LWIR flux E_ir from the surface is equal to the absorbed solar flux:

E_ir = I_sunCosθ(1 - R_θ)A ......................................................Eqn. (2.1)

The surface temperature T_s (K) is calculated from the Stefan Boltzmann Law:

T_s = (E_ir/σε)^0.25....................................................................Eqn. (2.2)

where Isun is the incident solar flux, set to 1368 W m^-2, θ is the solar zenith angle, R_θ is the angle dependent surface reflection, here set to the Fresnel reflection for a refractive index of 1.5, A is the surface absorption, set to 0.9, σ is Stefan’s constant and ε is the surface emissivity, set to 0.95.

The night time cooling can be calculated iteratively using the blackbody emission coupled to a thermal reservoir with a heat capacity Cq.

ΔT_s = tσεT⁴/Cq ....................................................................Eqn. (2.3)

T_s(n+1) = T_s(n) - ΔT_s........................................................... Eqn. (2.4)

Here t is the time step interval, set here to 0.1 lunar hours and C_q is the heat capacity of the surface layer coupled to the blackbody emission. The units here are J m^-2. The depth of the coupled layer is not defined. Two different values of C_q were used, 3.84E4 J m^-2 for the initial surface cooling and 3.73E5 J m^-2 for the thermal conduction limited case. The contents of the three worksheets are as follows:

Worksheet: Lunar T Calc

Columns 2 and 3 rows 15 to 33 contain the solar angle and the Fresnel reflection calculated for a refractive index of n = 1.5. These data are taken from the worksheet Fresnel R, column 15.

The solar angle in degrees is in column 7 and this is converted to lunar time in column 6 using

Hour =RC[1]*6/90

(6 lunar hours = 90 degrees in solar angle)

The incident solar flux at normal incidence is set to 1368 W m^-2 in cell R8C8 and the surface absorption is set to 0.9 in cell R9C8. The incident solar flux is calculated in column 8 using Eqn. (2.1).

E_ir = R8*COS(RADIANS(RC[-1]))*(1-C3)*R9

The surface emissivity is set to 0.95 in cell R10C9. The surface temperature is calculated in column 9 using Eqn. (2.2).

T_s (K) = (RC[-1]/(0.00000005672*R10))^0.25

The night time cooling is calculated in columns 25 to 29.

The time interval is set to 0.1 lunar hours in cells R11C25 and R11C28. The lunar time is calculated in column 25 by adding the time interval to the previous time in the cell above.

t =R[-1]C+R11

The start temperature for the first part of the calculation is set to 150 K in cell R13C26. The start temperature for the second part of the calculation is set to 114 K in cell R22C26.

The surface emissivity is set to 0.95 in cell R11C27. The LWIR flux emitted by the surface is calculated in column 27.

E_ir = R11*0.00000005672*RC[-1]^4

The total heat emitted over 0.1 lunar hours is calculated in column 28. (1 lunar hour is 27.3 earth hours).

Q =27.3*3600*R11*RC[-1]

The change in temperature is calculated in column 29 by dividing by the heat capacity C_q. Two different heat capacities are used. For the first part of the calculation, C_q1 is set to 3.84E+04 in cell R9C29 and for the rest of the calculation, C_q2 is set to 3.73E+5 in cell R11C29. For rows 13 to 21, the change in temperature, ΔT is given by

ΔT = RC[-1]/R9

For Rows 22 to 133 the change in temperature,

ΔT is given by ΔT = RC[-1]/R11

The new temperature in column 26 is then calculated by subtracting the ΔT from Column 29 from the temperature in the same row in Column 26.

T=R[-1]C-R[-1]C[3]

The temperature data from C9 and C26 are copied to Column 13 for plotting. The start and end points for the surface cooling calculations are added to Column 14.

The results for the night time cooling are copied to column 33 and potted twice. The first plot provides a visual check of the data. The second plot is superimposed on the measured data.

To further check the calculation, the LIWR flux is calculated from the temperature in Column 55 and the absorbed solar flux is copied to Column 56. These data are then plotted to verify that the absorbed solar and emitted LWIR flux match. The data are also plotted on an enlarged scale.

C55: E_ir =(0.00000005672*R8)*RC[-42]^4

C56: Absorbed Solar =RC[-48]

Worksheet: Lunar Plot

The temperature data from Vasavada et al is in Column 13 and the calculated data from the worksheet Lunar T Calc is in Column 14. The ranges for the three parts of the calculation, 0.0 to 6.0, 6.0 to 6.9 and 6.9 to 18.0 lunar hours is indicated by plotting the points in column 15.

Reference

Vasavada, A. R., J. L. Bandfield, B. T. Greenhagen, P. O. Hayne, M. A. Siegler, J-P Williams and D. A. Paige (2012), “Lunar equatorial surface temperatures and regolith properties from the Diviner Lunar Radiometer Experiment” J. Geophys. Res. 117 E00H18 pp. 1-12. [https://doi.org/10.1029/2011JE003987]

Worksheet: Fresnel R

The Fresnel reflections are calculated based on the equations from Jenkins and White (1976), Fundamentals of Optics, 4th Edition, 1976 McGraw Hill, p.524

R_s/E_s = -Sin(ϕ - ϕ´)/Sin(ϕ + ϕ´) ........................................................ Eqn. S1a

R_p/E_p = Tan(ϕ - ϕ´)/Tan(ϕ + ϕ´) ....................................................... Eqn. S1b

E´_s/E_s = 2Sinϕ´Cosϕ/Sin(ϕ + ϕ´) .......................................................Eqn. S1c

E´_p/E_p = 2Sinϕ´Cosϕ/[Sin(ϕ + ϕ´)Cos(ϕ - ϕ´)] ................................. Eqn. S1d

Where E, R and E´ are the amplitudes of the electric vectors for the incident, reflected and refracted light, the subscripts s and p denote the two orthogonal polarizations and ϕ, ϕ´ are the angles of incidence and refraction.

The intensities are given by the square of the amplitudes and the total intensity is the average of the intensities of the two polarizations.

The refractive index is set to 1.5 in cell R4C2. The refractive index for air is set to 1 in cell R3C2 and the angle increment is set to 1 degree in cell R5C2.

The angle of incidence is given in Column 1 and the angle of refraction is given in Column 2.

ϕ´ = 180*ASIN(SIN(PI()*C1/180)*R3C2/R4C2)/PI()

The quantities Rs/Es, Rp/Ep, E´s/Es, E´p/Ep are calculated in columns 4 through 7.

R_s/E_s =-SIN(PI()*(C1-C2)/180)/SIN(PI()*(C1+C2)/180)

R_p/E_p =TAN(PI()*(C1-C2)/180)/TAN(PI()*(C1+C2)/180)

E´_s/E_s =2*SIN(PI()*(C2)/180)*COS(PI()*(C1)/180)/SIN(PI()*(C1+C2)/180)

E´_p/Ep = 2*SIN(PI()*(C2)/180)*COS(PI()*(C1)/180)/(SIN(PI()*(C1+C2)/180) *COS(PI()*(C1-C2)/180))

These are converted to intensities in columns 9 through 12.

Intensity Rs/Is etc. =RC[-5]*RC[-5]

The intensities of the reflected and transmitted light are given in columns 15 and 16

Reflected intensity R = (RC[-6]+RC[-5])/2

Transmitted intensity T = (RC[-5]+RC[-4])/2

2) The Solution to the Lorenz Equations

In 1963, Edward Lorenz developed a simple mathematical model for convection now known as the Lorenz equations

dX/dt = p.(Y - X) .......................................................................Eqn. (3.1a)

dY/dt = -X.Z + r.X - Y ..............................................................Eqn. (3.1b)

dZ/dt = X.Y - b.Z ......................................................................Eqn. (3.1c)

These describe the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. X is proportional to the rate of convection, Y and Z are proportional to the horizontal and vertical temperature variation, p is the Prandtl number and r is the Rayleigh number. Lorenz discovered that the solutions to these equations were sensitive to small changes in the starting values. He restarted one of his calculations using slightly rounded numbers from the printout of a previous calculation and obtained a very different answer. The equations may be solved numerically by using the equations in the form:

ΔX = p(Y - X) Δt ........................................................................Eqn. (3.2a)

ΔY = (-X.Z + r.X - Y) Δt ............................................................Eqn. (3.2b)

ΔZ = (X.Y - b.Z) Δt ....................................................................Eqn. (3.2c)

The values obtained for each time step are added to the values from the previous step.

The calculation of the solutions to the Lorenz equations using p = 10, r = 8/3 and b = 28 are given in the workbook LorenzEquations.xlx. These are the same values that were used by Lorenz in 1963. He used the notation σ, b and r for p r and b. The workbook may be downloaded using the link

Download LorenzEquations.xlx

It may be necessary to unblock the file in order to open it.

The workbook contains four worksheets, Calculation, Plots1, Plots 2 and Plots 3.

The first worksheet contains the calculation, x-z, z-y plots and the y time series plot.

The values for p, r, b and dt are given in R5C13 to R8C13

The fixed points, ±b(r-1)^0.5 are calculated in cells R12C16 and R13C16.

x is calculated in column 14 using

x =(R5C13*(R[-1]C[2]-R[-1]C))*C13+R[-1]C

y is calculated in column 15 using

y =(R[-1]C[-1]*R[-1]C[1]-R6C13*R[-1]C)*C13+R[-1]C

z is calculated in column 16 using

z =(R[-1]C[-2]*(R7C13-R[-1]C[-1])-R[-1]C)*C13+R[-1]C

The y values are copied over to column 21and the number of iterations is input into column 20. These values are used to plot the time series.

The results for the plots used in the book are given in the three worksheets Plots1, Plots 2 and Plots 3.

The calculation values are copied over into columns 12 through 16. To show the trajectory of the plot, the data are separated into positive and negative values in columns 19, 20 and if needed, 21.

The data for the x, y, z time series plots are contained in columns 26 to 29.

Reference

Lorenz, E. N. (1963), “Deterministic nonperiodic flow” Journal of the Atmospheric Sciences 20(2) pp. 130-141. [https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO semicolon 2]

(this link contains a semicolon that causes formatting problems with the text editor)

3) Solar Flux Data

The solar flux data was calculated using the ‘clean air’ algorithm in IEEE Standard 738-1993, IEEE Standard for calculating the current temperature relationship of bare overhead conductors. [https://ieeexplore.ieee.org/document/347648] (This has now been revised to IEEE Std 738-2012. [https://ieeexplore.ieee.org/document/6692858] The 738-1993 clear atmosphere solar flux algorithm was used in the calculations for this book). https://ieeexplore.ieee.org/document/6692858

The calculation uses a polynomial expansion of the solar elevation angle θ_e.

ΔQ_sun = (A + Bθ_e + Cθ_e2 + Dθ_e3 ……….) ...........................................Eqn. (4.2)

θ_e is calculated from the hour angle, h, the solar declination δ and the local latitude φ.

sinθ_e = coshcosδcosφ + sin δ sinφ .........................................................Eqn. (4.3)

The values of the coefficients for a clear atmosphere are

A = -3.92441

B = 5.92762

C = -0.17856

D = 0.003223

E = -3.35E-05

F = 1.81E-07

G = -3.79E-10

The fraction of the solar flux absorbed at the land surface depends on the angle of incidence and the surface absorbance. For the land surface, the absorbed solar flux is given by

Q_sunabs = A_surf(1 - R_θ)Q_sunCosθ ............................................................Eqn. (4.4)

where A_surf is the surface absorption and R_θ is the angle dependent surface reflection and Cosθ is the Lambert cosine term.

The results of these calculations for 5° latitude intervals from 0 to 85° are given in the file SolarFluxDataIEEE738-1993.csv. This contains the calculation with and without the Fresnel reflection term for A = 1 for 365 days of the year at half hour intervals.

The file may be downloaded using the link:

Download File SolarFluxDataIEEE738-1993.csv

The file is in csv format. It may be necessary to unblock the file in order to open it.