Star Trek Space Jump

While I am still fresh on the Space Jump topic, let me take it to the extreme. Star Trek extreme.

SPOILER ALERT

But really, is this a spoiler alert if it is from the trailer of a movie that has been out forever? Of course, I talking about the latest Star Trek movie where three guys jump out of a shuttle and into the atmosphere.

So, in light of the Red Bull Stratos jump, how would this jump compare? First, my assumptions:

• This Star Trek jump is on the planet Vulcan. I am going to assume this is just like Earth in terms of gravity and density of air.
• The jumpers in Star Trek have on stuff that is different than what Felix will wear in the Stratos jump - but I will assume these guys will have similar falling characteristics.
• The jumpers start from a low orbit similar to the orbit of the space station. I will use a starting height of 300 km above the surface.
• The jumpers are not in orbit. I will assume their initial starting speed is 0 m/s.
• The model I am using for the density of air is only valid to about 36 km above the surface of the Earth. Higher than that, I am just going to have to estimate the density of air (see below)
• The drag coefficient is constant. This is really not true, but it is the best I can do. Sorry, I will try harder next time.

Ok, now what do I want to look at? I will compare this Star Trek jump to the Red Bull Stratos Jump in several ways:

• Maximum acceleration
• Maximum speed
• Speed compared to the speed of sound

Density of air

Since my model for the density of air seems to only be valid up to 36 km, I need to do some thing else for the other 250 km. My first thought was jut to put the density at zero. But then I thought that might not be the best thing. Even a very low density can make a big difference dropping that first 250 km. Here is a graph from Wikipedia showing the density as a function of height.

Actually, I have a new plan. This was not trivial to find (lots of broken links) but here is NASA's MSIS-E-90 Atmosphere Model. What a find. Using this I can generate air density as a function of altitude to 300 km. Here is a plot of that data:

And here is a plot of the old density model I used in the last Red Bull post along with the new NASA approved one.

Those are close enough for me. I will just use the NASA-Navy model (well, I will use select points from that model).

Maximum acceleration

I already did this for Felix and the stratos jump. Here is what I got:

So, not too bad. The maximum acceleration is less than 1 g. He could easily handle that (even I could). Now, for the Star Trek guys, I just need to change the initial height to 300 (and change the density model).

This looks crazy. Part of the problem is that in order to get density data over 300 km, I had it broken into big chunks (10 km sized chunks). Obviously, that is too big. Also, another problem. The acceleration never goes to zero. This means that the jumper wouldn't reach terminal velocity. I just don't think that would happen. Even meteors usually hit terminal velocity (I think). Here is what I am going to do. I am going to use these big chunks for stuff greater than 39 km and then use the old Red Bull way of calculating the density for stuff below that. Doing that, I get:

I like this one better. There might still be a problem with the density around 39 km. I am a little worried about the sharp increase in acceleration. I changed my density model so it was much more "detailed" at the higher altitudes. I am still using the old density model for heights less than 30 km.

So, what does this mean? This means that for most of the jump (above 39 km) there is so little air resistance, the jumpers just super speed up. Like ZOOM. After 39 km altitude, the air resistance really starts to increase. It is almost like hitting a wall since they are falling so much faster than terminal velocity. This make the air resistance force ginormous and the resulting acceleration deadly. Well, maybe not deadly. The Wikipedia g-force tolerance page says that an acceleration of 25 g's is possible for about 1 second. However, in this fall, the jumpers will have over 20 g's for over 4 seconds. Maybe they have special Star Fleet issue suits that allow them to experience higher accelerations. I mean, if they can make inertial dampeners for a ship, they surely can do this.

Maximum speed

Now that my air density model seems to be working well enough, it is relatively simple to look at the speed of the star trek jumpers.

Top speed just over 2,200 m/s (4900 mph). In physics, we call that zoom-fast. Remember that from 120,000 feet, a jumper would get around 250 m/s.

Comparing the speed to the speed of sound

If I use the most basic model of the speed of sound, it only depends on the temperature of the gas. This is a problem when you get up to 300 km above the Earth. So, instead of plotting the speed of sound, I am just going to calculate the speed of sound at the height where the jumper will be going the fastest. From the previous plot, I get a max speed of about 2,200 m/s at about 36,000 km. The speed of sound at this height is about 200 m/s. The answer to the question: the star trek jumpers are going way faster than the speed of sound, about mach 11.

Ok - I think what I need to do is to implement NASA's atmospheric density model in python rather than discreetly take data points from their online thing.