In Class Problems, November 2, 2009

Review

In consultation with the other members of your group, solve the problems below.

1. Formaldhyde (H₂CO) has been discovered in interstellar space. Calculate its mean molecular speed for T = 280 K. Would our Moon retain this gas for billions of years? What about Titan (R=2600 km, M=1.34 x 10²³ kg)?

V_rms = (3 k T /m)^0.5
T = 280 K
m = 30 x 1.67 x 10^-27 kg

V_rms = (3 x 1.67 x 10^-23 x 280 / 5.01 x 10^-26)^0.5
V_rms = 481 m/s

V_esc = (2 G M / R)^0.5

V_esc(moon) = (2 x 6.67 x 10^-11 x 7.3 x 10²² / 1.738 x 10⁶)^0.5
V_esc(moon) = 2.4 km/s

V_esc(Titan) = (2 x 6.67 x 10^-11 x 1.34 x 10²³ / 2.6 x 10⁶)^0.5
V_esc(Titan) = 2.6 km/s

For both the Moon and Titan, V_esc/V_rms is approximately 5, so formaldhyde has a lifetime of a few hundreds of millions of years (slightly longer for Titan).

2. Venus has an average density of 5200 kg/m³. Assuming a mantle density of 4500 kg/m³ and a core radius of 0.4xR_V, what is the core density of Venus?

M = M_core + M_mantle
rho_a = 5200 kg m^-3
rho_m = 4500 kg m^-3
R_c/R_V = 0.4

4/3 pi R_V³ rho_a = 4/3 pi R_c³ rho_c + 4/3 pi R_V³ rho_m - 4/3 pi R_c³ rho_m
R_V³ rho_a = (0.4 R_V)³ rho_c + R_V³ rho_m - (0.4 R_V)³ rho_m
rho_a - rho_m + (0.4)³ rho_m= (0.4)³ rho_c
rho_c = 15400 kg m^-3

3. The mean albedo of Mars is 0.16. Mars' semimajor axis is 1.524 AU and the orbital eccentricity is 0.093. What is the noontime temperature of Mars at perihelion? At aphelion? What is the typical peak wavelength at which the surface emits? What wavelength region would be the best for detecting a planet such as Mars (optical, radio, infrared, x-ray, gamma-ray, sub-mm)?

T_ss = 394 (1-A)^0.25(r_p)^-0.5
A = 0.16
R_peri = 1.524 (1 - 0.093) = 1.382 AU
R_ap = 1.524 (1 + 0.093) = 1.666 AU

T_peri = 394 ( 1- 0.16)^0.25(1.382)^-0.5 = 321 K
T_ap = 394 ( 1- 0.16)^0.25(1.666)^-0.5 = 292 K

lambda_max= 2898/ T (micron) = 2898/305 = 9.5 micron, which is in the infrared region.

4. Assume the sun radiates like a black body with a temperature of 5800 K and a radius of 6.96 x 10⁵ km. What is the flux from the sun at the solar surface? What is the flux (in W/m²) from the sun at a distance of 1.524 AU (Mars' semimajor axis)? What is the flux from the sun at a distance of 39.44 AU (Pluto's semimajor axis)?

F = sigma T⁴
T= 5800 K
sigma = 5.67 x 10^-8 W m^-2 K^-4

F = 6.42 x 10⁷ W m^-2

F(mars) = L/(4 pi R_mars²) = (F x 4 x pi x R_sun²) / (4 pi R_mars²)
= F(surface) x (R_sun/R_mars)²
= 598 W m^-2

F(pluto) = F(surface) x (R_sun/R_pluto)²
= 0.893 W m^-2

Use the exam formula sheet for constants and formulae.