June 1, 2015

WolframAlpha And The Trip To Mars

When I was a kid I was really into science fiction.
Robert Heinlein was my hero, Arthur C. Clarke close behind.
When I have some down time I still daydream about stuff, like this afternoon when it was raining and dreary.

I was thinking about a trip to Mars and what if an engine was developed that could propel a rocket at the acceleration of gravity, a constant of one g or 32 feet per second squared. This would alleviate the need to induce centripetal force for gravity. This constant acceleration would also work in reverse, just maneuver the rocket for a 180 degree shift and decelerate at one g. Half the flight in acceleration and the other half decelerating.

How long would it take to get to Mars?
First, some spreadsheet calculations. An acceleration of 32 feet per second squared means you will gain 32 fps, every second. Thus at the end of 60 seconds you will be traveling at 1,920 fps or 1,309.1 miles per hour. Since gravity is a constant, you will gain an additional 1,309 mph for every minute you maintain this acceleration.

After an hour your speed will be 78,546 mph.
After 24 hours accelerating at one g, your speed will be 1,885,104 mph.
Sweet, sweet gravity.

Now we bring WolframAlpha into play.
Q: How far is it to planet Mars?
A: Mars' average distance from Earth: 14.1 light minutes.

Back to the spreadsheet.
Speed of Light: 186,282 mps
One light minute: 11,176,920 miles
Average distance to Mars: 157,594,572 miles

Back to WolframAlpha to ask a series of questions on how far you would travel when accelerating at one g (phrase it as "distance traveled at acceleration x for y minutes/hours/days").
After one minute: 10.9 miles
After one hour: 39,273 miles
After 24 hours: 22,600,000 miles
After 48 hours: 90,480,000 miles (with a terminal speed of 5,655,312 mph)
Now let's turn the rocket around and decelerate at the same constant. At the end of four days (2 days accelerating and 2 days decelerating) you would have traveled a total of almost 181 million miles. More than enough to cover the average distance from Earth to Mars but average distances can be very deceiving and we are aiming at a moving target - but let's keep it simple.

The timing of the launch is critical. The closest Mars has been to Earth in the last twelve years is 34.6 million miles. And that only happens when Mars is in opposition to Earth. For that to happen, three conditions must exist:
  1. the Earth and Mars are on the same side of the Sun,
  2. Earth is at it's farthest from the Sun (aphelion) and
  3. Mars is at its closest to the Sun (perihelion).
Right now Mars is almost 240 million miles away as shown below.

But in July of 2018 Mars will be only 35.8 million miles away. At that distance at a constant 1g acceleration you could reach Mars in just 30 hours. But you'd be traveling at 2.3 million miles per hour when you crashed. So if we cut our acceleration time down to 21 hours, we'd travel 17.3 million miles reaching a top speed of 1.65 million mph. Decelerating at 1g for another 21 hours should get us to Mars in 42 hours. So, if we leave Earth on July 27, 2018 at noon, we'd reach Mars on July 29, 2018 at 5 AM. Pack your bags, we're going.

Oh well, it's fun to dream on a rainy Sunday.


michigan doug said...

Red Thunder by John Varley.

underground pewster said...

I can hear the kids now, "Are we there yet?"