How Do You Calculate Travel Time Between Orbiting Planets?

[ckirmser]

(Not sure if I selected the proper prefix for this thread or not. If not, could an admin alter it, if necessary?)

Hello, all -

I'm trying to come up with a general formula to determine the travel time for a spaceship to travel from one planet in orbit around a star to another planet in a different orbit.

I can't just use a straight-line measurement and then apply a standard T=2*sqr(D/A) (for a midpoint turnaround and then applying thrust in the reverse direction to slow down, reaching V=0 at the destination), because the destination planet will have moved in its orbit which may increase or decrease the length of that direct line as the ship travels.

I figure it will involve the initial angular separation of the two planets, but I don't know for sure.

I looked at a Hohman transfer, but that seemed to be designed for a ship at a constant velocity, not one under acceleration.

Any ideas?

Thanx in advance for any help.

 

[phinds]

I doubt you'll find a useful general equation. Even if you get a general equation for one planet to another, it's likely to give you a bad answer in some cases because it won't include the possibility of a slingshot maneuver.

 

[Vanadium 50]

I don't think you will find a simple and accurate formula.

First, you need additional information, like the relative phases (as you suggested), and the amount of acceleration available.

Second, if you look at NASA probes, the results are not simple. On average, the closest planet to the Earth is Mercury. It took MESSENGER 7 years to get there. Mariner 9 reached Martian orbit in about half a year.

 

[ckirmser]

I doubt you'll find a useful general equation. Even if you get a general equation for one planet to another, it's likely to give you a bad answer in some cases because it won't include the possibility of a slingshot maneuver.

For what I need, a slingshot maneuver can be ignored. I'm just looking for something where I can plug in an acceleration of, say, 1g, and determine how long it will take to get from Earth to Jupiter, for example. This is just for an RPG, so details can be glossed over.

 

[ckirmser]

I don't think you will find a simple and accurate formula.

First, you need additional information, like the relative phases (as you suggested), and the amount of acceleration available.

Second, if you look at NASA probes, the results are not simple. On average, the closest planet to the Earth is Mercury. It took MESSENGER 7 years to get there. Mariner 9 reached Martian orbit in about half a year.

I had thought the acceleration would be just another variable to plug into the formula, along with the angular separation. Then, I could just plug in the masses and distance from the Sun along with the angular separation of the two planets to get a basic idea of the length of time for the trip.

This is just for an RPG, nothing spectacular. If worse comes to worst, I'll just measure the straight-line distance and maybe add 10% or something to account for the destination's movement along its orbit, but I was hoping for a bit more accuracy.

 

[Vanadium 50]

1g? For the whole trip? That will give you a VERY short travel time. About 100x faster than anything done before less than 2 days. For that kind of unrealistic acceleration, you can assume that the planets are stationary and the time can be calculated as accelerating full blast to the half way point and deaccelerating full blast to the target,

 

[DaveC426913]

...I can plug in an acceleration of, say, 1g, and determine how long it will take to get from Earth to Jupiter, for example.

OK, this is a powered trip. That's a completely different kettle of fish than anything we'd be dealing with here in the "real" Astro forum.It should be possible to compose a formula that calculates the straight line distance and time between any two points on any two orbits. But: that might be onerous to use in real time while your players wait.

I have a suggestion that might be more useful of your time.

I am a big fan of analogue computation.

1. Plot all relevant (circular) orbits on a map.
2. Make a ruler whose units are days/weeks/months.

To use:

1. Decide where two planets are in relation to each other on the map.
2. Measure it with your ruler.
3. Profit!

^{* ruler is roughed-in but accurate enough.}
So: if you the GM decide that Earth and Mars are, say, (125-20=)105 degrees apart in their orbits, and your players navigated in a straight line, accelerating at 1g for half the trip then decel'ing at 1g, that should be a trip time of just over 4 days.

Note: the solar system is averse to being rendered in just one scale; you'll need a whole another calculator if they want to visit the outer planets.If you want to pursue this line of thought, I'd suggest this thread would be served better in the Sci-Fi Fantasy Forum, where our math isn't constrained by reality.

 

[Drakkith]

For what I need, a slingshot maneuver can be ignored. I'm just looking for something where I can plug in an acceleration of, say, 1g, and determine how long it will take to get from Earth to Jupiter, for example. This is just for an RPG, so details can be glossed over.

For constant 1g acceleration, you're looking at a trip time of just a few days to a few weeks at most to get anywhere. By my calculation it would take about 6.5 days to travel from the Sun to Jupiter, with constant acceleration to the midpoint and then having the spaceship flip and do a constant deceleration until it reaches Jupiter.

I'm not quite sure my formula is exactly correct, but it should give you a rough idea of travel times. Travel to Saturn from Earth should take roughly a week (7-9 days) depending on where in their orbits Earth and Saturn are when you leave. Jupiter 4-7 days, Mars 2-4 days, Venus and Mercury 1.5-2.5 days. Neptune takes barely two weeks, sitting at roughly 15 days while Uranus takes about 12 days.

If someone wants to check this formula, here it is: $t=\frac{1}{43,200}\sqrt{\frac{2d}{9.98*2}}$
With $t$ being time in days and $d$ is the distance you're traveling. Other numbers are constants to get the time in days and other required factors.

Basically, you start with the formula for getting the time to reach a distance $d$ under constant acceleration of 9.98 m/s: $t=\sqrt{\frac{2d}{9.98}}$ But we aren't just rushing past that point, we are going to reach halfway and then slow down. So you actually want to take the time to get to the halfway point and double it, adding a factor of 2 outside the square root. But I don't want to plug in half of some distance, I just want to plug in the distance, so I added a factor inside the square root, which is the factor of 2 under the $d$. Then to get the time in days instead of seconds I added a factor of $\frac{1}{60*60*24}$ out front of the square root.

 

[Hornbein]

I don't think you will find a simple and accurate formula.

First, you need additional information, like the relative phases (as you suggested), and the amount of acceleration available.

Second, if you look at NASA probes, the results are not simple. On average, the closest planet to the Earth is Mercury. It took MESSENGER 7 years to get there. Mariner 9 reached Martian orbit in about half a year.

Mercury is difficult because the craft has to kill its orbital momentum first so it can drop toward the Sun. For the other planets that momentum can be useful.

 

[Vanadium 50]

That's kind of the point - before the problem was clarified and realistic constraints removed, things were not simple or even monotonic. Mercury is 4x closer than Mars on average and it takes 15x longer to get there.

 

[DaveC426913]

Mercury is difficult because the craft has to kill its orbital momentum first so it can drop toward the Sun. For the other planets that momentum can be useful.

The OP specified he's blasting at 1g the whole time, so he doesn't really have to "drop" toward the sun.

It takes about two days to reach Mercury's closet orbital point at full 1g burn (with turnover). The difference between Earth's and Mercury's orbital velocities is about (105k-67k) 38k mph. Only 29 minutes of your two day crossing is necessary to match velocities. At 1g, your velocities are so huge it doesn't much matter whether you choose a pro- or anti-orbital path; the boost or "knock" you get translates to a total of 58 minutes - or 1/48th - of your trip.

And that's the worst case. Any longer trips (including Mercury in a less convenient location) make the boost/knock even more negligible.

The upshot is that the Solar system can be treated as a static array of destinations, navigated to and from by effectively straight trajectories. There's a little bend due to various gravity wells, but it too is only a fraction of your journey.

 

[DaveC426913]

One question I don't have an answer to @ckirmser : what year(s) is your campaign set in? It makes a big difference in Pluto's travel time.

In AD 2100, it's about 50 AUs out, which is about 3 weeks trip time,
but if it's set nearer AD 2200, Pluto is only about 30AUs out, which is more like 2 weeks.

 

[DaveC426913]

Here is the final calculator, barring tweaks.

 

[ckirmser]

1g? For the whole trip? That will give you a VERY short travel time. About 100x faster than anything done before less than 2 days. For that kind of unrealistic acceleration, you can assume that the planets are stationary and the time can be calculated as accelerating full blast to the half way point and deaccelerating full blast to the target,

I figured on 1g to make any calculations easier. Also, this is for an RPG, so 1g is not unusual.

 

[ckirmser]

OK, this is a powered trip. That's a completely different kettle of fish than anything we'd be dealing with here in the "real" Astro forum.It should be possible to compose a formula that calculates the straight line distance and time between any two points on any two orbits. But: that might be onerous to use in real time while your players wait.

I have a suggestion that might be more useful of your time.

I am a big fan of analogue computation.

1. Plot all relevant (circular) orbits on a map.
2. Make a ruler whose units are days/weeks/months.

To use:

1. Decide where two planets are in relation to each other on the map.
2. Measure it with your ruler.
3. Profit!

View attachment 328693


^{* ruler is roughed-in but accurate enough.}
So: if you the GM decide that Earth and Mars are, say, (125-20=)105 degrees apart in their orbits, and your players navigated in a straight line, accelerating at 1g for half the trip then decel'ing at 1g, that should be a trip time of just over 4 days.

Note: the solar system is averse to being rendered in just one scale; you'll need a whole another calculator if they want to visit the outer planets.If you want to pursue this line of thought, I'd suggest this thread would be served better in the Sci-Fi Fantasy Forum, where our math isn't constrained by reality.

I had firgured that the destination planet would move enough to throw off a direct measurement, but, if not, I'll just go with that.

There's a SciFi forum? I had no idea - 'course, didn't look for one, though...

Thanx!

 

[ckirmser]

One question I don't have an answer to @ckirmser : what year(s) is your campaign set in? It makes a big difference in Pluto's travel time.

In AD 2100, it's about 50 AUs out, which is about 3 weeks trip time,
but if it's set nearer AD 2200, Pluto is only about 30AUs out, which is more like 2 weeks.

This is for an RPG - and, I just discovered there is a specific SciFi forum that I was not aware of - so the specific date is 2192 - but, I may adjust that based upon how I postulate certain technologies advancing.

 

[ckirmser]

For constant 1g acceleration, you're looking at a trip time of just a few days to a few weeks at most to get anywhere. By my calculation it would take about 6.5 days to travel from the Sun to Jupiter, with constant acceleration to the midpoint and then having the spaceship flip and do a constant deceleration until it reaches Jupiter.

I'm not quite sure my formula is exactly correct, but it should give you a rough idea of travel times. Travel to Saturn from Earth should take roughly a week (7-9 days) depending on where in their orbits Earth and Saturn are when you leave. Jupiter 4-7 days, Mars 2-4 days, Venus and Mercury 1.5-2.5 days. Neptune takes barely two weeks, sitting at roughly 15 days while Uranus takes about 12 days.

If someone wants to check this formula, here it is: $t=\frac{1}{43,200}\sqrt{\frac{2d}{9.98*2}}$
With $t$ being time in days and $d$ is the distance you're traveling. Other numbers are constants to get the time in days and other required factors.

Basically, you start with the formula for getting the time to reach a distance $d$ under constant acceleration of 9.98 m/s: $t=\sqrt{\frac{2d}{9.98}}$ But we aren't just rushing past that point, we are going to reach halfway and then slow down. So you actually want to take the time to get to the halfway point and double it, adding a factor of 2 outside the square root. But I don't want to plug in half of some distance, I just want to plug in the distance, so I added a factor inside the square root, which is the factor of 2 under the $d$. Then to get the time in days instead of seconds I added a factor of $\frac{1}{60*60*24}$ out front of the square root.

I was thinking that I'd just circularize the orbits - for ease of calculation - and determine the distance between the planets based on angular separation and using the formula for a circle - in some fashion. I just have been frying my 66-yr old brain and when my eyes started crossing, I figured I'd post here, thinking it might be something simple that I'm just not seeing.

 

[DaveC426913]

...determine the distance between the planets based on angular separation and using the formula for a circle - in some fashion....

Don't like my calculator in post 13?

 

[ckirmser]

Don't like my calculator in post 13?

Oh, no, I liked it, but I already have that sort of tool built in with Astrosynthesis. I was hoping there'd be a formula where I could plug in the angular separation and the difference in orbits that would give me a distance.