Surely you have heard all of the news about Planet Nine? If not, here are a few of the major points:

There are many objects in the Kuiper belt (the stuff past Pluto) that have orbits around the Sun but are sort of bunched to one side of the solar system.

The motion of these Kuiper belt objects could be explained by the gravitational influence of an unknown planet—Planet Nine.

Planet Nine might not be the best name. I would suggest something cooler like Planet Awesome or maybe Planet Pokemon.

Based on the motion of the Kuiper belt objects, Planet Nine would have an orbital period of 10,000 to 20,000 years with a semi-major axis of 700 AU (1 AU equals the distance from the Sun to the Earth).

The estimated mass of Planet Nine is 6 x 1025 kg (more massive than Earth) with a radius of 13,000 to 26,000 km (compared to 6,371 km for the Earth).

Now that we know at least something about this (theoretical) planet, let's consider some questions.

How Long Would it Take New Horizons to Get to Planet Nine?

Really, there are two versions of this question. First you could ask how long it would take New Horizons (the spacecraft that just flew past Pluto) to get to Planet Nine from its current location. I suspect the answer to this question is an infinite amount of time. New Horizons doesn't really have much fuel to mess around on any extra trips. In order to get to Planet Nine it would likely have to make a huge course adjustment—and it probably doesn't have enough fuel.

The second version of this question is "how long would a spacecraft like New Horizons take to get to Planet Nine?" New Horizons doesn't travel at a constant speed—however, I can still use an estimate of about 16 km/s (16,000 m/s). If this spacecraft targeted Planet Nine during its closest approach to the Sun, that would be a distance of about 200 AU (2.99 x 1013 m). Now we can use the standard definition for average velocity (in one dimension).

I can convert 1.87 x 109 seconds into years (you can check the math for homework) and I get 59.3 years. Doing the same thing for the case of Planet Nine being the farthest from the Sun (1200 AU), I get a travel time of 355.7 years. Oh, the distance from the Sun is approximately the same as the distance from the Earth to Planet Nine. But you can see that if we want to launch a spacecraft to fly past Planet Nine, we should either start with young scientists or find a way to make a faster (much faster) spacecraft.

Could You Walk on Planet Nine?

Let's assume that Planet Nine has a solid surface. What would it be like to walk on it (other than being cold)? Really, this has to do with the surface gravitational field. On the surface of the Earth, the gravitational field is 9.8 Newtons per kilogram. We all know what it feels like on the surface of the Earth—so this will be a good number to use in a comparison.

The surface gravitational field for a planet depends on two things (well, at least two things): the mass of the planet and the radius of the planet. The more massive the planet, the greater the gravitational field. The larger the radius of the planet, the lower the gravitational field (because you are farther from the center). The magnitude of the gravitational field on the surface can be found as:

Using the high and low estimates for the radius of Planet Nine, I get a gravitational field between 5.9 and 23.7 N/kg. So, it's possible that the field could be just like on Earth (but much colder). Of course if the gravitational field is 23.6 N/kg, it's going to suck trying to walk.

How Cold Is It on the Surface of Planet Nine?

OK, let's start off with the assumption that Planet Nine has no atmosphere. This is important because as we all know, the atmosphere of a planet can have a large impact on the surface temperature (see for example Venus, Earth, and Mars). If you also assume that the Sun is a blackbody (all radiation is from surface temperature) then you can estimate the power radiated from the Sun. If all energy is radiated equally in every direction then the intensity of light will decrease as one over distance squared.

As a planet receives radiation from the Sun, the surface warms up. When this happens, the planet also becomes a blackbody radiator. The equilibrium temperature of the planet is the point at which the emitted and received radiation is equal. There's a little bit of math, but that's the general idea.

Using this temperature model and a distance of 200 AU, I get a surface temperature of 20.5 Kelvin (-422.8 Fahrenheit). That's colder than Hoth. If you assume that Planet Nine doesn't absorb all the radiation (which it wouldn't), then the temperature is a little bit lower—but you would have to guess at some properties of the planet to get an actual value.

Oh, just for fun if you use that same model for the Earth you get a surface temperature of 288 K—but it's actually lower than that because the Earth isn't a perfect absorber. Without an atmosphere, the Earth's temperature would be about 255 K (right around 0 Fahrenheit).

Homework

Clearly there are other questions to answer. Here they are as homework questions for you. Yes, these will all be on the test.

Some planets were discovered after finding perturbations in other planet's orbits. The changes in orbit of Uranus was due to Neptune—that's how it was discovered. Suppose we just watched Pluto's orbit. How much of a deviation would be created by Planet Nine? You should try this as a numerical calculation. You can put Planet Nine wherever you like.

The Earth's atmosphere warms it up from about 255 K to about 288 K. Could you add enough carbon dioxide to Planet Nine to give it a nicer temperature? This is a pretty tough problem, so you will need to make some estimates. Here is a hint: Venus without an atmosphere is about 300 K and then with an atmosphere it is over 700 K. What if you put an atmosphere as thick as Venus on Planet Nine? Suppose it increased the temperature by the same multiplicative factor—how hot would it be?

Technically, modeling the temperature effect of Venus's atmosphere isn't so easy because it's so thick. The Earth's atmosphere effect can be estimated with a single layer model—via ACS. Still, it's fun to make estimations.