To Jupiter in a Week?

 kw: analytical projects, space travel, solar system

I began reading a bit of space opera from the early 1950's, in which a space pilot goes from Earth to Jupiter in seven days. I wondered how that would be possible. My first thought was, "Jupiter is around half a billion miles away. What does it take to speed up to around 70 million miles/day (~3 million mph)?"

For perspective: About fifteen years ago, the New Horizons spacecraft was launched from Earth and boosted to a speed of about 45 km/s (about 100,000 mph). Thirteen months later it approached Jupiter, still going just under 20 km/s, and was directed into a slingshot flyby that boosted its speed to about 22.5 km/s.

From this point we go metric. Jupiter's distance from Earth varies between 588 and 968 million kilometers (Mkm). For what it is worth, when Earth is at quadrature with Jupiter, one can gain an extra 30 km/s of takeoff speed, and at that point the distance is near the average of about 780 Mkm. For this analysis I'll be dividing by 7 and by two numbers divisible by 3, and to have an even-numbered result, I'll pick a distance that is divisible by 126 million (126=9x7x2). I chose 126x6 = 756 Mkm.

First simple analysis: 756/7 = 108. Thus, average speed needs to be 108 Mkm/day or 1,250 km/s. That's almost 28 times as fast as the starting speed of New Horizons (Clearly, science fiction writers in the pre-Sputnik days expected great advances in rocket fuel technology). Suppose the rocket can accelerate at 1G for as long as needed. How long does it take to get up to 1,250,000 m/s?

Basic velocity formulas:

• 1G acceleration (a) = 9.8 m/s²
• Velocity (v) = 9.8*t m/s
• Distance in time t = 4.9*t² m

Turn the second formula around: t = v/9.8, which comes to 1,250,000/9.8 = 127,550 sec = 35.43 hours. It takes about a day and a half to get up to speed. Without going into detail, this means that actual travel time would be more like 8.5 days. But this puts us in the right ballpark.

Let us figure what acceleration is needed to go half the distance at constant acceleration, then turn around and slow down in the same amount of time and distance. Half the distance is 378 Mkm. Solve the third equation for a, the acceleration needed to go 378 Mkm in 3.5 days, or 84 hours, or 302,400 seconds: a = 2*dist/t², which comes to 8.27 m/s². That is about 0.84 G. With that level of acceleration, our pilot can have the comfort of a near-1G environment for the whole trip, except for turnaround at the midpoint, and other maneuvers at both ends. Peak speed would be 8.27*302,400 = 2.7 million m/s or 2,700 km/s, or more than twice the average speed.

So there we have it. We just need a fuel-&-engine system that can accelerate at near-one-G for a total of a week, and repeat the performance for the return trip.

A secondary consideration is, what would be the consequences of hitting a dust particle, or worse, a sand-size particle, at a speed of 2,700 km/s? An average grain of beach or dune sand is half a mm across and weighs about 180 micrograms (µg). 180 µg is 180 billionths of a kg, the unit we need to calculate energy. Silt particles are 1/100 the diameter and weigh one millionth as much, or 180 trillionths of a gram.

Let's start with a sand grain. Kinetic energy E = m*v²/2, or 0.000 000 18*(2,700,000)²/2 = 656 thousand joules; a joule is a watt-second, so this comes to 182 watt-hours. This is the energy of a 1 kg mass at a speed of 1,150 m/s, a little faster than the bullet from an AR-15 rifle, but that bullet weighs only about 4 grams. This sand grain deposits the energy of 250 rifle rounds in an area half a millimeter across. That would melt a chunk of armor plate and make a hole you can stick your finger in. The ship's pilot would feel a bit of a jerk from the impact.

A grain of silt or dust, with one-millionth the weight, has one-millionth the kinetic energy, which comes to 2/3 of a joule. It doesn't sound like much, but that's the energy of a BB dropped about a foot. You'd hear it. It would strike off a bit of material, which a BB wouldn't do. Intermediate-sized grains would do correspondingly more damage. Sand size grains are very scarce in the asteroid belt, but silt-size grains are probably abundant enough that the wear on forward armor would be significant.

So, as enjoyable as such tales are, with people bombing around the solar system as though one were driving from Idaho to Florida, there's a lot of reality in between where we are now and the technology needed to accomplish it.