Hitching a ride on a black hole, part II: The basics of star-surfing

Last time we started by trying to reduce the effects of g-forces on the crew and ended up considering a star corpse as our star ship. Here is a schematic drawing of how it looks:

star surfing

Yes, this is a 5-min drawing in Google docs. If you are inspired to do better, let me know and I will gratefully replace this “drawing” with yours, and give you credit, of course.

Here is what is happening on this picture: A star is used as a propulsion source controlled by as-yet-unspecified technology and emits its content at or near light speed in one direction. As a result, it is pushed in the opposite direction, like a rocket. A crew module is positioned at a respectable distance (which we will shortly calculate) and is in a free fall toward the star. While the star in question keeps getting away. If you position the crew ship just right, it will be stationary relative to the star, even if the acceleration of the star itself is many times Earth’s gravity.

Let’s do a few simple calculations to see how far from the star we should place our ship. We assume that the star used as a star-drive is roughly one solar mass. This is not an unreasonable assumption, as all white dwarfs and neutron stars and even newly formed stellar-remnant black holes are in that range. Instead of dealing with the large and inconvenient numbers, like the gravitational constant, or the Sun’s mass, I will use only a few basic small ones I remember:

  • Free-fall acceleration at earth’s surface (1g).
  • Earth’s speed in its orbit around the Sun (v = 30 km/s).
  • Distance from Earth to Sun (r = 1 astronomical unit = 150 million kilometers).
  • Newton’s law of gravitation: the attraction force falls as the square of the distance between two bodies.

For now, we will do only the basic Newtonian physics, no relativistic corrections whatsoever. Because none are needed just yet: we are dealing with relatively slow speed and weak gravity. First, let’s calculate the centripetal acceleration due to the Earth-Sun attraction force: it is \frac{v^2}{r} = \frac{g}{1600}. So, to get a 1g acceleration we need to get 40 times closer to the Sun,down  to only 4 million kilometers from its center, or barely 3 million km from its surface, 10 times closer than Mercury. Clearly this is not a healthy place to be in. That’s one reason that a white dwarf would work better. Hopefully. If we want to have our star drive to work at something like 10g, we’d need to be 3 times closer still, down to just over 1 million km from the center. This close even relatively dim objects like white dwarfs and neutron stars would probably make your life uncomfortable, no matter how good your radiation shielding is. So, we are likely down to our last star drive candidate, the black hole. Unless we use a huge crew module… like an asteroid-sized one? Made from an actual asteroid, maybe? But will a relatively large object like that withstand the tidal forces exerted on it by the star’s gravity? Let’s do another simple calculation!

Given gravitational acceleration a and distance r from the massive object, what is the tidal force per unit length? Well, dimensional analysis to the rescue! There is only one way to get acceleration per unit length from these two numbers, and it is to divide them: \frac{a}{r}! (An honest calculation which involves a small amount of calculus gives us an extra factor of 2.) For an asteroid 100 km in diameter the resulting tidal gravity between its “bow” and “stern” is \frac{2 \cdot 100 km \cdot 10 g}{1,000,000 km} = \frac{g}{5000}. This is a rather inconvenient number. On one hand, it is too small to provide a comfortable level of gravity to the crew, on the other hand, it is large enough to tear our gravitationally-bound asteroid apart, or at least make it shed most of its mass, until it is small enough for the cohesion forces to dominate self-gravity, a few dozen kilometers at best. How did I calculate that last number? I didn’t. I cheated. I looked up shapes of some moons and asteroids, like Phobos, and picked the size where they stop being more-or-less round.

Let’s review what we have figured out so far. If we get a white dwarf-powered star drive provide a 10 g acceleration, we can make a Phobos-sized crew module surf its gravitational wake some 1 million km behind while nearly weightless. This answers our first question posed in the previous post: “How far behind the star-ship is the sweet spot for the crew?”. We shall discuss the next question, how stable the ride is, in the next post. We will also talk about ways to provide Earth-like gravity for the crew without expending a lot of on-board fuel.

 

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