Try this some weekend. Go to the highest peak on a nonrotating, nearly spherical moon or small body. Hit a golf ball and watch it fly. If you don’t hit it very hard, it will follow a nearly parabolic path, until it hits the surface.

The path isn’t really parabolic; it’s actually part of an ellipse, which intersects the surface at two points: let’s call them launch and impact. A parabola is what you would get if the gravitational field of the moon was the same strength and in the same direction everywhere. That’s approximately true in the small scale, but on a larger scale, the strength of gravity decreases with increasing altitude, and the direction changes because it always points toward the center.

The trajectory that the golf ball will follow depends on the speed and direction it has when it leaves the launch point. On Earth, the world record fastest golf ball, hit by Jason Zuback, had a launch speed of 91.2 m/sec. If you’re not a pro golfer, or if you’re wearing a cumbersome space suit, you might be lucky to manage 50 m/sec.

During the Apollo 14 mission, astronaut Alan Shepard actually hit a couple of golf balls, on the Moon. He used a six-iron, and had to swing with one hand because of the space suit. It’s not known how far the golf balls traveled, but Shepard said the second one went “mile and miles and miles.” That’s an exaggeration; encumbered as he was, Shepard probably had a shot that would have gone no more than 200 m on Earth, so 1200 m on the Moon. Still, it must qualify as the longest golf shot in history, on level ground.

If you hit your golf ball horizontally, at just the right speed, it will go into a circular orbit, just above the surface of the moon. This circular orbit is just a special case of an elliptical orbit. The speed required for this is called the circular orbit speed, or v_circ. It depends on the mass of the moon, and the distance from the center. Since we are launching from the highest point on an almost spherical body, the distance from the center is just slightly more than the radius.

Phoebe, one of Saturn’s moons, has dimensions 230 x 220 x 210 km, and it is one of the smaller nearly spherical objects in the Solar System. Maybe it formed out of molten rock from some catastrophic event, and has escaped being smashed up too badly since then. The circular orbit speed at the surface of Phoebe is about 70 m/sec. Standing on a mountain top on Phoebe, a very strong golfer could put a golf ball into orbit.

Mimas, another of Saturn’s moons, has dimensions 414.8 x 394.4 x 381.4 km, so it qualifies as nearly spherical. The circular orbit speed at the surface of Mimas is 112 m/sec. A golf ball would not have enough speed to go into a circular orbit, but it could easily travel 100 km or more. An 18-hole golf course on Mimas would have to cover pretty much the entire surface: about half a million square kilometers. That’s a big golf course!

If you hit your golf ball horizontally, with a little more than circular orbit speed, it will go into an elliptical orbit, which has the launch point as its perigee, or point with lowest altitude. So its launch and impact point are the same.

Now, if you hit it with a lot more than circular orbit speed, the ball will follow a hyperbolic trajectory, and eventually escape from the moon altogether. Specifically, the launch speed must be greater than the escape speed, v_esc, which is equal to v_circ, multiplied by the square root of 2. So it’s about 1.4 times as fast as v_circ.

If the launch speed is v_launch, then after a long time, the golf ball will be moving in essentially a straight line, with speed v_launch – v_esc leftover, and essentially free of the moon’s gravitational influence. It has given up v_esc of its speed to get away from the moon.

The borderline case between elliptical orbits, with v_launch < v_esc, and hyperbolic orbits, with v_launch > v_esc, is the parabolic case, with v_launch = v_esc. The speed has to be just right. So there are parabolic orbits around spherical bodies, just as there are in uniform gravitational fields. However, in this case, as the golf ball gets farther away, it slows down instead of speeding up, as it gives up all its speed to get away.

Of course, no body in our solar system is perfectly spherical, and the oblateness and lumpiness of bodies make orbits around them more complicated than simple ellipses and hyperbolas. Still, the above description is a useful approximation, sufficiently far above a sufficiently spherical body.

Also, none of the bodies in our solar system are perfectly nonrotating. A golfer has to take the Coriolis Effect into account. An easier way is to compute the velocity of the launch point relative to a nonrotating coordinate system, and use vector addition to add this to the launch velocity, relative to the ground. This becomes very complicated, if the body is highly irregular, because it may not simply spin on an axis. It may have a more complex spin state, like a precessing football, or worse!

The asteroid Eros, visited by the NEAR mission in 2000, is shaped like a bent potato. Mission planners had to continually revise the orbit around Eros, especially when the spacecraft got close.

Eros is only 34 x 11 x 11 km in size, with escape speed at the surface of about 10 m/sec. This could be expected to vary considerably over the surface, but it is safe to say that any golfer could hit or throw a ball fast enough to escape Eros. It might not be the best place for a golf course – you would lose a lot of golf balls! Besides, most of the surface is covered with a fine powder, pulverized by millions of years of bombardment by particles large and small. Your ball may be embedded in a powder sand trap. That could be a problem if you have to play it where it lies! In fact, in very low gravity, this powder may actually behave somewhat like a fluid. So the sand traps are like water hazards! Here is a very interesting article on this and related topics.

Hyperion, another moon of Saturn, is one of the largest highly irregular bodies in the Solar System, at 360 x 280 x 225 km. Generally speaking, objects with diameter larger than 400 km tend to be pulled into a roughly spherical shape by their own gravity, while objects less than 200 km tend to be very irregular.

The escape speed on the surface of Hyperion varies from 45 to 100 m/sec. A golfer, standing on one of the high points (far away from the center of mass), could hit a ball into orbit around Hyperion, and it just might escape Hyperion and go into Saturn orbit. At one of the low points, where the escape speed is 100 m/sec, even the best golfer would have trouble sending a tee shot into Saturn orbit.

Even though Hyperion is larger than Phoebe, its irregularity and lower density causes lower escape speeds at some points on the surface. This makes sporting events more complicated.

Playing golf from moon to moon would be very challenging. If you tee off from Atlas, one of the inner moons of Saturn, with a pretty impressive 60 m/sec tee shot, then since Atlas has an escape speed of about 6 m/sec, you end up in Saturn orbit, with a speed of 54 m/sec relative to Atlas, having spent only 6 m/sec getting away from Atlas. However, Atlas is moving around Saturn at 16.6 km/sec, over 300 times your speed relative to Atlas. So you have escaped Atlas, but you are travelling in almost exactly the same orbit. Your puny 54 m/sec may not be enough to get you to another moon. Luckily, there are a few other moons pretty close by. But getting out as far as Phoebe doesn’t seem feasible.

If that doesn’t seem cool enough, you might want to venture into Saturn’s rings. Some of the ring “particles” are as big as buildings. You could jump from rock to rock. Or, more precisely, ice chunk to ice chunk. This would be a very dangerous sport! Better stick to a nice game of catch, with your feet anchored so you don’t accidentally step out into space.

If we venture out to 23 million km, near the outer edge of Saturn’s system of moons, we find 3 or 4 small moons. They don’t get close together very often, because their orbits have different inclinations. For example, Ymir is only about 18 km in diameter, with an escape speed of 8.7 m/sec. A strong golfer could hit a ball off Ymir and into Saturn orbit, with 50 m/sec or more to spare. However, even though Ymir is very far from Saturn, it is still moving at over 1 km/sec relative to Saturn (the speed varies because the orbit is not circular), and that is 20 times the golf ball’s speed. So a golfer might be able to reach one of the other outer moons, if the timing is just right, but targets that never get close to Ymir’s orbit would be out of range. It would be a very tricky shot!

A high power rifle bullet, at around 1 km/sec, would be able to hit pretty much anything in the Saturn system, if fired from one of the outermost moons. Anyone in the inner Saturn system would need a lot more than a 1 km/sec gun if they want to shoot back. If you’re a military planner, you want to occupy the high ground.

A few of the outer moons of Uranus and Neptune slow down to 400 m/sec or less. So escaping from Uranus or Neptune, even from the orbit of an outer moon, would require around 600 m/sec. That’s well beyond the capability of any golfer. So I’m afraid interplanetary golf is not happening. But there is always the Kuiper Belt.

Fore!

Articles, September 3, 2017

2 weeks ago

Maybe not have the golf ball shot from moon to moon, but have 2-3 holes per moon, with their varying gravity? That would be a pretty difficult round!

ReplyDelete