Mars has an albedo of 15. Given that it doesn't have much of an atmosphere, estimate the mean surface temperature on Mars. Note: the semimajor axis of the orbit of Mars is 1.52 AU. (Please enter your answer in units of degrees Kelvin)
To estimate the mean surface temperature on Mars, we can use the Stefan-Boltzmann Law, which relates the temperature of a planet to its albedo and distance from the Sun.
1. Convert the albedo from a percentage to a decimal: Albedo = 15 / 100 = 0.15.
2. Calculate the solar constant, which is the amount of solar radiation reaching Mars. The solar constant for Earth is approximately 1361 watts per square meter (W/m^2). Since Mars is farther away from the Sun, we need to adjust this value based on its distance.
The semi-major axis of Mars' orbit is given as 1.52 AU. 1 AU (astronomical unit) is the average distance between the Earth and the Sun, which is about 150 million kilometers (or 93 million miles). Therefore, the distance of Mars from the Sun is 1.52 * 150 million km = 228 million km.
The solar constant at Mars can be calculated as follows:
Solar Constant at Mars = (Solar Constant at Earth) * (1 AU / Mars-Sun distance)^2
= 1361 * (1 / (228 million / 150 million))^2
3. Now, we can use Stefan-Boltzmann Law to estimate the mean surface temperature on Mars. The law states that:
Solar constant = σ * Temperature^4
Rearranging the formula:
Temperature = (Solar Constant / σ)^(1/4)
where σ is the Stefan-Boltzmann constant, which is approximately 5.67 x 10^-8 W/m^2K^4.
4. Calculate the mean surface temperature on Mars:
Temperature = (Solar Constant at Mars / σ)^(1/4)
Let's calculate the mean surface temperature on Mars:
To estimate the mean surface temperature on Mars, we can use the concept of planetary equilibrium temperature. This temperature represents the temperature a planet would have if it were a perfect blackbody and absorbed all the sunlight falling on it.
The equilibrium temperature can be calculated using the formula:
Teq = (1 - α)^(1/4) * (L / (16 * σ * π * d^2))^0.25
Where:
Teq = equilibrium temperature
α = albedo (reflectivity) of the planet
L = luminosity of the Sun
σ = Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/(m^2*K^4))
d = distance from the planet to the Sun
Given that Mars has an albedo (α) of 15 (expressed as a percentage), we need to convert it to a decimal fraction (divide by 100). So α = 0.15.
The semimajor axis of the orbit of Mars (d) is given as 1.52 AU. We need to convert this to meters using the conversion factor that 1 AU equals about 149.6 million kilometers.
Now, we need to determine the luminosity of the Sun (L). The Sun's luminosity is approximately 3.828 x 10^26 Watts.
Plugging in the values to the formula, we can calculate the equilibrium temperature:
Teq = (1 - 0.15)^(1/4) * (3.828 x 10^26 W / (16 * 5.67 x 10^-8 W/(m^2*K^4) * π * ((1.52 AU * 149.6 million km/AU * 1000 m/km)^2))^0.25
Simplifying the equation and solving it will give us the mean surface temperature on Mars.
Performing the calculations, we find that the mean surface temperature on Mars is approximately 210 Kelvin.
Therefore, the estimated mean surface temperature on Mars is 210 Kelvin.