Question

Question 2 (20 points) Venus has mass my = 4.87 x 1024 kg and radius Ry = 6.05 x 103 km. Surface conditions are temperature To = 740 K, and pressure Po = 9.3 MPa. The lapse rate is approximately constant for most of the troposphere, up to 2 60 km where T1 263 K. Its atmosphere can be approximated as 100% carbon dioxide, CO2. You can assume that gravity is constant at its sea-level value. Using your result from Question 1 a) Plot the pressure, temperature, and density as a function of altitude for this segment of the Venusian atmosphere (0 <zs 60 km). Normalize your results by the sea-level value, i.e. plot P/P, T/T,, and p/po. Plot all three curves on the same set of axes, with altitude z on the vertical (y) axis. b) Compute the mass of this segment of the Venusian atmosphere. Tip: From Newton's law of gravitation 90 = my where G = 6.674 X 10-11 m3/(kg s?) is the universal gravitational constant.

Answer 1

Given: m = 4.87*1024 kg R1 = 6.05 10 km To = 740 K Po = 9.3 MPa Zmax = 60 km T = 263 K Solution: From Newton's law of gravitation, acceleration due to gravity on Venus is, 80 = Gm, _6.67x10-"x 4.87x1024 2 - =8.8798=8.88 m/s? R (6.05109) Since, lapse rate (a) is constant, the temperature variation can be written as: T=To-az -------> (1) T, ET, -azmax a= (T-TO) / zmax = (740-263)/60= 7.95 K/km Dividing To on both side of equation (1), we get, T = = 7,= -0.010742.- > (2) From Hydrostatic, we know that, Using ideal gas equation P=pRT and equation (1). P = RT RIT, - az) After rearranging, dPdz P R(To-az) Integrating the above equation, we get, Rearranging,

\/Ra -------> (3) Density ratio is found using ideal gas equation, PPRT P PORT р РІР, ----> (4) Po T/T (b) Mass of the stratosphere Mass of stratosphere between z and z+dz region. P dm = px 2(R, +z)dz = x2(R, + z)dz RT Total mass, <=60 260 P m= | dm= ! 2= 1 =0 RT 27(R+z)dz = 3.9815*10' kg (Ans) The value of P and T are substituted in above equation from equations (2) and (3), which will also contain z. Thus the integration is difficult. Hence, The above integration is found numerically in Matlab using trapezoidal rule.

Tho rho/rho z (km) 0 0.2 0.4 T/T or P/P 0.6 or rho/rho 0 0

Matlab code:-----------------------------

clear all; clc;

mv = 4.87E24; % mass of Venus
Rv = 6.05E6; % radius of Venus
P0 = 9.3E6; % Pressure in Pa
T0 = 740; % temperature in K

T1 = 263;
zm = 60E3; % max height
g0 = 8.88;   
alpha = (T0-T1)/zm;
MW = 44; % molecular mass of CO2
R = 8314/MW; % gas constant   

z=0:zm/100:zm;

TT0 = 1-(alpha/T0).*z; % temperature ratio

pp0 = (TT0).^(g0/(R*alpha)); % pressure ratio

rr0 = pp0./TT0; % density ratio

plot(TT0,z./1000,'Linewidth',2)
xlabel(' T/T_0 or P/P_0 or rho/rho_0')
ylabel('z (km)')
set(gca,'fontsize',16)
hold on;
plot(pp0,z./1000,'Linewidth',2)
hold on;
plot(rr0,z./1000,'Linewidth',2)
legend('T/T_0','P/P_0','rho/rho_0')

% finding mass
P = pp0*P0;
T = TT0*T0;

dm =2*pi*(P./(R.*T)).*(Rv+z);

mass = trapz(z,dm); % mass of stratosphere

%-------------------- End of code ------------