Making reasonable assumptions, estimate the surface...

Making reasonable assumptions, estimate the surface temperature of Neptune. Neglect any possible internal sources of heat. What assumptions have you made about the planet's surface and/or atmosphere? Astronomical data which may be helpful: radius of sun $=7 \times 10^5 \mathrm{~km}$; radius of Neptune $=2.2 \times 10^4 \mathrm{~km}$; mean sun-earth distance $=1.5 \times 10^8 \mathrm{~km}$; mean sun-Neptune distance $=4.5 \times 10^9 \mathrm{~km} ; T_{\mathrm{S}}=6000 \mathrm{~K}$; rate at which sun's radiation reaches earth $=1.4 \mathrm{~kW} / \mathrm{m}^2$; Stefan-Boltzman constant $=$ $5.7 \times 10^{-8} \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}^4$.

Making reasonable assumptions, estimate the surface temperature of Neptune. Neglect any possible internal sources of heat. What assumptions have you made about the planet's surface and/or atmosphere?

Astronomical data which may be helpful: radius of sun $=7 \times 10^5 \mathrm{~km}$; radius of Neptune $=2.2 \times 10^4 \mathrm{~km}$; mean sun-earth distance $=1.5 \times 10^8 \mathrm{~km}$; mean sun-Neptune distance $=4.5 \times 10^9 \mathrm{~km} ; T_{\mathrm{S}}=6000 \mathrm{~K}$; rate at which sun's radiation reaches earth $=1.4 \mathrm{~kW} / \mathrm{m}^2$; Stefan-Boltzman constant $=$ $5.7 \times 10^{-8} \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}^4$.

Key Concepts

Radiative Equilibrium

This concept refers to the balance between the energy a body absorbs and the energy it radiates away. In the context of estimating a planet's effective temperature, it is assumed that the incoming solar energy (after accounting for factors like geometric dilution and possible albedo) is exactly balanced by the energy emitted by the planet. This balance is critical for determining the planet's surface temperature under the assumption of steady state conditions.

Stefan-Boltzmann Law

The Stefan-Boltzmann law is fundamental in calculating the energy radiated by a blackbody as a function of its temperature. It states that the power radiated per unit area is proportional to the fourth power of the temperature, which is central to relating the absorbed solar flux to the effective temperature of a planet. This law is used to equate the energy input from the sun with the energy output of the planet, allowing one to solve for temperature.

Inverse-Square Law

The inverse-square law describes how the intensity of radiation from a point source, such as the sun, decreases with the square of the distance. This law is crucial in determining the solar flux that reaches a planet at a significant distance from the sun, as the flux diminishes rapidly with increasing distance. Understanding this concept allows for adjusting the energy input in energy balance equations based on the planet’s orbital distance.

Idealized Surface and Atmospheric Assumptions

This concept covers the simplifying assumptions made about the planet’s surface and atmosphere when estimating its effective temperature. It typically includes treating the planet as a perfect blackbody that uniformly absorbs and radiates energy, neglecting factors such as atmospheric composition, greenhouse effects, and internal heat sources. These assumptions simplify the energy balance calculation and yield a first-order estimate of the surface temperature.

Transcript

00:01 So on neptune, it radiates out 2 .6 times as much as it absorbs.

00:08 So that's what this is saying here.

00:10 So neptune radiates out 2 .6 times the amount that it takes in.

00:18 So a little bit comes into neptune, and 2 .6 times that amount comes out.

00:28 So, and we want to find what the actual temperature is.

00:32 So we are looking for t -actual.

00:38 So we can set this up using this relationship right here.

00:44 So the t -actual over the temperature expected to the fourth power should equal that to 0 .6.

01:02 And so we want to find the t -a -up top here.

01:07 So in order to do that, we will have to take a fourth root of something.

01:14 So i'm going to use an x because it's going to become a little bit easier.

01:20 So now our equation is going to look like this.

01:21 So it's going to be x.

01:22 This is the one that we're looking for over what we were given is the 47.

01:32 All of this is to the fourth power.

01:35 And that should equal to 0 .6.

01:38 So what's interesting is that we can do here is when you have a full thing to a power like that, you can just apply that power to each top and bottom individually, and it still means the same thing...

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