Orbital mechanics and flying to the Sun

When I watch a science fiction television show or movie, I tend to divide the physics errors into two categories.  There are the ones that I ascribe to narrative or stylistic necessity, and there are the unnecessary mistakes.

In the first group I put things like sound effects in space scenes.  Of course, we all know that sound is not transmitted in a vacuum, but it is often an artistic choice to put sound into these scenes.  Certainly, there are cases where it is not done, the first example that springs to mind being “2001: A Space Odyssey“.  There, the sudden dead silence in some scenes was both realistic and dramatic.

In the second group, though, are things that generally don’t matter to the plot, that could be depicted realistically, but are not.  That brings us to flying into the Sun.

If one were to posit a spacecraft, starting at the Earth, with a trajectory intended to bring it close to the Sun in a flight time of weeks to months, how do you direct the thrust of your engines?  The common depiction treats a flight to the Sun as essentially driving a truck on an invisible highway.  You point your direction of thrust toward the centre of the Sun, turn on the engine, and fly straight there.  This is incorrect, such a trajectory is extremely inefficient, if achievable at all.

While there are many ways we can examine this problem, the simplest one I can think of uses very little math and fairly basic physics.  We will consider only angular momentum.

Angular momentum is a conserved quantity.  If I choose a point in space, the magnitude of my angular momentum relative to that point is equal to the product of my mass, speed and the distance of closest approach between my straight-line extended trajectory and the point of interest.  In mathematical terms, it is the cross product of my momentum and the vector connecting me with the point in space.

Just as momentum is conserved in the absence of a force, angular momentum is conserved in the absence of a torque.  Torque is, similarly, the cross product of the force applied and the vector connecting me with the point in space.

So, let’s consider our angular momentum relative to the centre of the Sun.  Now, when we begin our journey, we are traveling at the same speed as the Earth (approximately 30 km/s), around the Sun.  Our speed is roughly perpendicular to the line connecting the Earth to the Sun, as the Earth’s orbit is quite close to circular.  The magnitude of our angular momentum is, from basic trigonometry, about equal to mass times speed times distance.  That amounts to 4.5E+15 kg m^2/s per kilogram of spacecraft.

Now, we’ve pointed out engine straight away from the Sun, so that our applied force is exactly on the line connecting the spacecraft to the centre of the Sun.  That means that the torque is zero, because the cross product of parallel vectors is zero.  So, in this configuration, the engine cannot change the angular momentum of the spacecraft.  Yes, it can push us toward the Sun, but our initial sideways deflection keeps trying to push us out of the way.  Imagine that you just kept adjusting your engine so the thrust was always directed straight into the Sun, and you managed to just barely touch the edge of the Sun while swinging by.  The Sun’s radius is about 700000 km.  If, for simplicity, we ignore any change in the mass of the spacecraft, that means that it must have a speed along the surface of the Sun of 6430 km/s.  Now, it’s deeper in the Sun’s gravity.  An object starting at the Earth’s orbit and ending at the surface of the Sun would be expected to gain no more than 620 km/s in speed due to gravitational forces.  That leaves a deficit of about 5800 km/s that must come from the spacecraft’s engines.  That is a stupendous delta-V.  Holding onto this number, we now look at a better option.

If, instead, the spacecraft were to fire its engines so as to oppose its motion around the Sun, a much different result is seen.  We already mentioned that the Earth is moving at 30 km/s around the Sun.  So, we now set our engines to drive us backwards along the Earth’s orbit around the Sun.  We run the engines long enough to apply a delta-V of 30 km/s.  Our spacecraft is now sitting still in space, with zero angular momentum relative to the Sun.  But it can’t stay there.  The Sun’s gravity is pulling, and the spacecraft will fall into the Sun.  First slowly, then faster and faster, it will hit the Sun in just under 6 months.  If the trip is intended to take less time, then after the first burn has completed, the engines can now be directed to thrust toward the Sun, and thrust applied as needed.  Earlier thrust is more efficient than later thrust.

So, pushing toward the Sun, about 5800 km/s.  Pushing at 90 degrees to the Sun, about 30 km/s.  And yet you rarely see this handled correctly.

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