You may or may not have seen the article over on Gizmodo but I was hoping that there would be some math in this article. Undeterred, I have decided to do it myself. Below is a back of the napkin mathematical explanation of why New Horizons won’t orbit.

First, let’s start with the stuff we already know about the mission. We know the current velocity of the probe, the mass of the probe, the fly-by distance, the radius of Pluto, and the mass of Pluto. These values can be used to define a hypothetical orbit mission:

NH Vel: 14,000 m/s (from article)

NH Mass: 478 kg (wiki)

NH Dist: 7,800 km (wiki)

Pluto Rad: 1185 km (wiki)

Pluto Mass: 1.305×10^{22} kg (wiki)

We know that 14 km/s is greater than escape velocity for Pluto since the current mission is a fly by. Think of it like rolling a ball in a funnel, if the ball is going too fast, it shoots out the other side. That’s exactly what New Horizons is doing now. What we need to do, is figure out how much it needs to slow down in order to not go flying out but not too much so that gravity takes over and it just crashes into the surface.

If New Horizons were to orbit at the altitude of it’s fly-by, it would need to slow down to the speed that corresponds to that orbital altitude. That velocity is determined by the following equation:

G is the gravitational constant: 6.674×10^{−11} N⋅m^{2}/kg^{2}

M is the mass of the parent body, Pluto which we listed above and R is the orbital distance from the center of mass. For simplicity’s sake, I’m going to assume Pluto is a perfect sphere and the center of mass is exactly at the center. So that’s the Plutonian radius 1185 km plus the fly-by altitude 7800 km, 8985 km.

Chugging the numbers, the approximate velocity for that orbit is 311 m/s. This is way slower than the fly-by velocity. For reference, the ISS orbits the Earth at an altitude of around 410 km and has a velocity of about 7660 m/s.

So now we need to figure out how to slow New Horizons down to achieve orbit. Enter, the rocket equation. This equation helps you determine a change in velocity with a corresponding change in mass for a given exhaust velocity (or engine efficiency as the case may be).

This is best expressed in terms of specific impulse and solving for initial mass.

ve can be represented as Isp * g0 (exhaust velocity = specific impulse of the engine times standard gravity).

I couldn’t find an image of the equation the way I wanted so here goes with regular characters:

m0 = m1 * e^(dv/(Isp * g0))

If we’re extremely generous and use New Horizons’ current mass as the final mass (m1) and also somewhat optimistic and say that New Horizons has an engine with a specific impulse of 400s, we can calculate what the initial mass would be. For reference, the Space Shuttle Main Engines (which are pretty efficient rocket engines) have a vacuum specific impulse of about 450s. A thruster for this type of mission would have a much lower efficiency.

So, dv = 13689 m/s, m1 = 478 kg, Isp = 400 s, and we get roughly 15647 kg for m0.

New Horizons would need over 15,000 kg of propellant to slow down to an orbital velocity around Pluto and that’s without trying to figure out how much structure would be needed to support that much propellant. That’s the size difference between a piano (which New Horizons has already been compared to) and a small semi-truck.

I actually posted this whole thing as a comment over on the Giz article but since I spent some time looking up values and making calculations, why not share it with everyone on Oppo too.