Math needed for 5-week flight from Earth to Mars

[Staiduk]

'Lo all!
I've been involved in a fun discussion on another forum here.

As some might know; I'm close to finishing my first novel. It's a murder mystery in a S-F environment. I'd started that thread to ask about the feasability of the design of the ship the novel's set on and it's turned into quite an entertaining discussion.
I'm bogged down however because there is one question I can't answer. The Boom-ship Princess Astoria flies a regular run between Earth and Mars; the trip takes between 4 and 8 weeks depending on the planets' position. (Beyond that; she doesn't fly the route - it's simply not economical.) On this particular run, the journey is five weeks. Now, I imagine that in order to make it in that time the Astoria has to be going pretty mutherin' fast. If you haven't clicked the link; the idea is she burns at .5g until she reaches cruise speed, then spins herself to provide .5 gravity for the 5-week transit.
My question is how long would she need to burn at .5g in order to reach a speed that would allow her to arrive at Mars in 5 weeks? Some assumptions must be made for distance; the two planets are just a little past closest approach. Is it possible to calculate the length of that burn?
I don't know if this is necessary; but the mass of the ship is approx. 7,000 metric tonnes (I based this around 2001's Discovery; which is listed at 5440 tonnes. Princess Astoria is 30m shorter; but quite a bit more robust.) In addition; she carries about 60,000 tonnes of fuel (that's just a guess) and will arrive in Martian airspace with about 5/8ths fuel remaining.

So just to restate the question, how long would a .5g burn from Earth to Mars at near-closest approach have to be in order to arrive at Mars in 5 weeks?
Thanx!

 

[dgatsoulis]

Hi Staiduk. That seems to be a fun experiment. To answer your question, we must first find out the ΔV needed for the TMI burn and to do that, we need to know the starting point of the "Princess Astoria".

LEO, GEO, Lunar orbit, other? (She sounds like a space creature, so I don't think you are starting her from a landed position, are you?)

 

[Staiduk]

Hi Staiduk. That seems to be a fun experiment. To answer your question, we must first find out the ΔV needed for the TMI burn and to do that, we need to know the starting point of the "Princess Astoria".

LEO, GEO, Lunar orbit, other? (She sounds like a space creature, so I don't think you are starting her from a landed position, are you?)

Definitely not - She starts from High Heaven; a 15km. long Habitat in a 40,000km. equatorial Earth orbit. Her destination is New Paris; a similar Habitat (though smaller at 10km) 15,000km. over Mars. Thanks for the help - I don't know calculus from the stuff you scrape off teeth; so I appreciate the assistance.

---------- Post added at 07:07 PM ---------- Previous post was at 07:01 PM ----------

Heh - someone just answered this question on that thread but the answer was totally incomprehensible to me. Just to be clear I have no mathematical ability outside of basic arithmetic and certainly no understanding of orbital mechanics so baby talk is appreciated. :lol:

 

[dgatsoulis]

Ok, I will setup an Orbiter scenario and get back to you.
Can you give any infomation about the engine's thrust? If not, I think we can calculate it from the fuel consumption you posted above (3/8 of total fuel for the Trans Mars Injection burn).

 

[Staiduk]

Ok, I will setup an Orbiter scenario and get back to you.
Can you give any infomation about the engine's thrust? If not, I think we can calculate it from the fuel consumption you posted above (3/8 of total fuel for the Trans Mars Injection burn).

Thanks so much. Nope, no idea at all; just that she acellerates at .5g. She could probably burn harder if she needed to; the half-gee figure is there for passenger comfort.

Edit: I never thought of trying to use Orbiter for it. I'm terrible at using TransX; I only ever use it to close on the ISS on a shuttle flight.

 

[dgatsoulis]

From an ideal starting point, (the ship needs no plane change) I find that you need ~15.14 km/s ΔV to get to Mars in 35 days, from a 40000 km orbit around Earth as your starting point.

Even from an equitorial orbit (which needs a plane-change of ~30°) , you don't need more than 16 km/s

If we assume 5 m/s² constant accelaration (~0.5g), the burn will take 16000/5= 3200 seconds to complete.

The numbers above are for the next Mars 35 day trip window, leaving Earth at July 1st 2018 and arrival at Mars on August 5th.

The arrival at Mars is a bit fast (~20km/s).
If you could give me a rough era for the journey, (within a 2 decade span), I can find the launch windows for optimal fuel use.

Will post the scenario and a couple of explanatory pics when I get back, in a couple of hours.

 

[Staiduk]

From an ideal starting point, (the ship needs no plane change) I find that you need ~15.14 km/s ΔV to get to Mars in 35 days, from a 40000 km orbit around Earth as your starting point.

Even from an equitorial orbit (which needs a plane-change of ~30°) , you don't need more than 16 km/s

If we assume 5 m/s² constant accelaration (~0.5g), the burn will take 16000/5= 3200 seconds to complete.

The numbers above are for the next Mars 35 day trip window, leaving Earth at July 1st 2018 and arrival at Mars on August 5th.

The arrival at Mars is a bit fast (~20km/s).
If you could give me a rough era for the journey, (within a 2 decade span), I can find the launch windows for optimal fuel use.

Will post the scenario and a couple of explanatory pics when I get back, in a couple of hours.

OK so just shy of an hour if I read you correctly; I was imagining closer to two. The year this takes place is 2283. Knowing my luck that'll be the one year Earth and Mars are on opposite sides of the sun. :lol:

One thing: High Heaven is basically a port city; taking in trade and tourism and servicing inbound and outbound deep-space vessels. If an equatorial orbit isn't ideal; would an orbit of 30deg. inclination (as you mentioned) be a better orbit for The Big Barbell (Earth slang - drives locals insane) to be in?

 

[dgatsoulis]

OK so just shy of an hour if I read you correctly; I was imagining closer to two. The year this takes place is 2283. Knowing my luck that'll be the one year Earth and Mars are on opposite sides of the sun. :lol:

If you MUST leave on 2283 the best date to do it is April 2nd with an arrival on May 7th. The ΔV cost for the Trans Mars Injection burn will be 24.17 km/s. At a constant accelaration of 5 m/s² it will take 24170/5 = 4834 seconds to complete the burn. An orbit 40000 km above the Earth's surface has a period of ~99800 seconds, so you will complete the burn in about (4834/99800 = 0.048) 1/20th of a full orbit.

The best date to make the journey within a reasonable time from the date you suggested is on 2287 with a departure on August 3d and arrival on September 7th. The ΔV cost is 14.88 km/s with a burn-time of 2976 seconds. (From equatorial orbit).

One thing: High Heaven is basically a port city; taking in trade and tourism and servicing inbound and outbound deep-space vessels. If an equatorial orbit isn't ideal; would an orbit of 30deg. inclination (as you mentioned) be a better orbit for The Big Barbell (Earth slang - drives locals insane) to be in?

No need to change the inclination. Some dates will be good and some will be bad, to use it as a starting point. What I am curious about is the altitude. Just ~4214 km lower and you would be at a [ame="http://en.wikipedia.org/wiki/Geostationary_orbit"]geostationary[/ame] orbit, which (IMHO) would make a little bit more sense for an orbiting station.

-------------------------------------EDIT----------------------------------------------

Here are some pics. First the 5 week journey on 2283. On the left (heliocentric); Earth's orbit in green and Mars' orbit in orange, transfer in blue.
On the right (geocentric); the initial orbit (40000 km equitorial) in green and the trajectory after the TMI burn in blue.

Now the 5 week journey on 2287

 

[Staiduk]

The best date to make the journey within a reasonable time from the date you suggested is on 2287 with a departure on August 3d and arrival on September 7th. The ΔV cost is 14.88 km/s with a burn-time of 2976 seconds. (From equatorial orbit).

OK, great - No problem changing the date by 5 years and the specific dates will be pretty handy too; since Wilson's case file provides an important part of the storyline. I'd thought to fudge the dates; thanks so much for figuring out the real ones.

What I am curious about is the altitude. Just ~4214 km lower and you would be at a geostationary orbit, which (IMHO) would make a little bit more sense for an orbiting station.

Yeah; I'd started thinking about that a few days ago; something along the lines of 'hey, isn't geostationary somewhere around 36,000 or something?' Another excellent point; thanks! :thumbup:

Hehe - now I have to go back to the other site and let 'em know I intend to keep my ship just the way I designed her in the first place. Not that the current argument about offset engines and structural stress isn't fascinating, but they're kinda ignoring my original question - is the basic design feasible? The one providing the math telling me to 'fix my mistakes' raised my hackles a bit as well, since he ignored my original parameters. (Edit: Whoops, my mistake there. I just read the post again; he was asking others to correct any errors he made. Good thing I looked before posting; might have made a nasty mistake there.)
Just speaking seriously for a moment; they all know far more about space than I do; their design and math might be more accurate but I have no intention of spending a few pages trying to explain all the gobbledygook to readers who just want to find out who killed the dead guy. They've got a great argument going, but you've given me what I need. If this ever gets as far as an E-reader upload (and it will; publishers be damned), would you mind if I included you on the 'Acknowlegements' page? Thanks.

(P.S. - ahh; I'll hold off telling 'em - let 'em argue for a while. :thumbup: )

 

[dgatsoulis]

I can't say that I read the whole thread on that forum, but I did read your initial post.

Is the "thrown pencil" concept similar to this? (jump to 26 minutes)

And a clarification. I only calculated the ΔV for the journey and the burn-time for a constant 0.5g accelaration. If you want to calculate what kind of engine thrust will be needed, we can do that too. But as you said, there's no need for several pages of technical data, when your goal is to write a murder mystery in a sci-fi setting.

Even Ray Bradbury got his science wrong sometimes. Who cares?

 

[Staiduk]

Yes that's exactly the idea; though in my universe things have progressed quite a bit (evidenced by the fact Princess Astoria is flying to an orbiting city of 350,000 people). The ships have progressed beyond the 'girders and balloons' concept of early transplanetary flight and have begun to take on a more attractive appearance. Especially this one - designed and owned by Princess Cruise Lines to cater to the wealthy; it's designed to look good as well as fly good. Therefore it has a certain elegant grace, even with those two great big whompin' fuel tanks in the middle.
This era of spaceflight roughly equates to the Age of Sail - trade is exploding, colonising is heavy as is shipping. The ships have a long way to evolve yet; solar system travel still takes a long time but now that Humanity has settled other worlds they are drastically improving.
Cheers!

Edit: Here's a couple of pictures I placed on that thread. I'm probably the world's worst artist and I don't have a scanner but it was the best I could do.

Princess Astoria in cruise configuration:
http://farm9.staticflickr.com/8230/8410347954_696f4a5a0d_m.jpg
...and in manoeuver configuration:
http://farm9.staticflickr.com/8506/8410347970_c8dc424d9e_m.jpg

 

[dgatsoulis]

Princess Astoria in cruise configuration:
http://farm9.staticflickr.com/8230/8410347954_696f4a5a0d_m.jpg
...and in manoeuver configuration:
http://farm9.staticflickr.com/8506/8410347970_c8dc424d9e_m.jpg

Shouldn't there be a 180° difference between the cruise and maneuver configs at the passenger module? The "floor" in the cruise config is the very front of the pic you drew, right?

But in the maneuver config the "floor" is actually the "back of the chairs".
Or am I missing something?

Also, I didn't catch this the first time:

would you mind if I included you on the 'Acknowlegements' page?

I'd be honored, but only if you also mention the whole community. :thumbup:

 

[Staiduk]

Shouldn't there be a 180° difference between the cruise and maneuver configs at the passenger module? The "floor" in the cruise config is the very front of the pic you drew, right?

But in the maneuver config the "floor" is actually the "back of the chairs".
Or am I missing something?

No, you're not - given that the vessel would be under thrust for a relatively short time and all personnel would be belted in for accelleration; I felt it appropriate to have the vessel nose-forward for thrust. After all; thrusting back into the seat is safer on the body than up into the spine and the ship will take some serious gees aerobraking at Mars and Earth (2 or 3gs; or thereabouts) Still; I don't suppose rotating the life module 180deg. would be much more technically difficult than rotating it 90degrees; I'll think on that a while.

 

[dgatsoulis]

No, you're not - given that the vessel would be under thrust for a relatively short time and all personnel would be belted in for accelleration; I felt it appropriate to have the vessel nose-forward for thrust. After all; thrusting back into the seat is safer on the body than up into the spine and the ship will take some serious gees aerobraking at Mars and Earth (2 or 3gs; or thereabouts) Still; I don't suppose rotating the life module 180deg. would be much more technically difficult than rotating it 90degrees; I'll think on that a while.

The explanation you gave for the "back of seat/floor" configuration is fine, but I don't think that aerobraking that kind of vessel is an option. Not at those speeds. Since you have 5/8 of fuel still left, it would make much more sense to brake the ship with the engines, close to -but not inside- the atmosphere. This way you can take advantage of the [ame="http://en.wikipedia.org/wiki/Oberth_effect"]Oberth effect[/ame] and continue with your rendezvous to the station orbiting Mars*.

*An Areostationary orbit at ~17000 km altitude, would make a nice symmetry to a Geostationary station on Earth.

 

[Staiduk]

The explanation you gave for the "back of seat/floor" configuration is fine, but I don't think that aerobraking that kind of vessel is an option. Not at those speeds. Since you have 5/8 of fuel still left, it would make much more sense to brake the ship with the engines, close to -but not inside- the atmosphere. This way you can take advantage of the Oberth effect and continue with your rendezvous to the station orbiting Mars*.

*An Areostationary orbit at ~17000 km altitude, would make a nice symmetry to a Geostationary station on Earth.

OK - my vague idea was to armour the backsides of the giant bulbous fuel tanks and use them as a sort of ballute system but your way makes more sense. Incidentally; for someone with no concept of astronautics; I seem to be making some pretty close guesses - missed HH's ideal orbit by 4000kms, missed New Paris's by 2000. Not bad for a total tyro. :thumbup::lol:

Edit: Oberth Effect - kewl! This is why I love projects like this, and fora like this - learning so much in one day. Thanks!

 

[RGClark]

Nice calculation dgatsoulis. I really don't like the 6 to 8 month travel times that have been suggested for manned flights to Mars, either for the first astronauts to be sent or for the Mars colonists to travel there. So I've been investigating delta-v's for shorter travel times.

If instead of leaving from GEO or a near GEO orbit you left from LEO then you would already have a ca. 7.8 km/s orbital speed rather than the ca. 3 m/s at GEO, so you would need less additional delta-v to make that short travel time flight.

Based on the nearly 5 km/s higher orbital speed, I assume you would need that much less delta-v to make the 35 day flight, so only a ca. 10 km/s delta-v, much more easily achievable. You could in fact do this in a single stage.

ULA has estimated by scaling up the Centaurs, switching to aluminum-lithium alloy rather than the steel now used, and going to a wide-body design you can increase the mass ratio to 20 to 1 rather than the 10 to 1 they now have. So let's use for the Isp the highest available now at 465.5 s for the RL-10B2 engine. And set the propellant load at the payload capacity of the later version of the SLS at 130 mT. At a 20 to 1 mass ratio this is a dry mass of about 7 mT. Actually since mass ratio improves as you scale up a rocket stage, it'll probably have an even better mass ratio than this. Then you could get 9 mT payload to 10 m/s:

465.5*9.81ln(1 +130/(7 + 9)) = 10.1 km/s .

This would be sufficient for a [ame="http://en.wikipedia.org/wiki/Sundancer"]Bigelow Sundancer[/ame]-sized inflatable habitation module.

You would be able to do much better than this payload mass if you were able to use the Oberth effect. Launching from GEO toward Earth to get the velocity boost of the Earth's gravity as you skim past, might be sufficient for the 11 km/s escape velocity of the Oberth effect. Launching from L1, L2 or the Moon's orbit certainly would be. Using the Oberth effect then you would only have to supply 7.7 km/s delta-v to get the 15 km/s you need.

Then you could get 22 mT to Mars in 35 days:

465.5*9.81ln(1 + 130/(7 + 22)) = 7.77 km/s.

Bob Clark

 

[dgatsoulis]

If instead of leaving from GEO or a near GEO orbit you left from LEO then you would already have a ca. 7.8 km/s orbital speed rather than the ca. 3 m/s at GEO, so you would need less additional delta-v to make that short travel time flight.

The starting point of the journey was provided by the OP.

Based on the nearly 5 km/s higher orbital speed, I assume you would need that much less delta-v to make the 35 day flight, so only a ca. 10 km/s delta-v, much more easily achievable. You could in fact do this in a single stage.

Yes, you have a higher orbital velocity but you also have a higher escape velocity from that altitude. Still, for that journey there are significant savings leaving from LEO instead of GEO.

Using the numbers you posted, at a 200 km LEO the escape velocity is

At 40000 km alt the escape velocity is

The ΔV for the injection burn is

The ΔVtr for the 35 day journey on 2287 is 17.17 km/s. The Injection ΔV from LEO is

(Ideal orbit with no plane-change in the Injection burn)

The same transfer from 40000 km orbital altitude costs

(Again, ideal orbit with no plane change in the TMI burn).

You would be able to do much better than this payload mass if you were able to use the Oberth effect. Launching from GEO toward Earth to get the velocity boost of the Earth's gravity as you skim past, might be sufficient for the 11 km/s escape velocity of the Oberth effect. Launching from L1, L2 or the Moon's orbit certainly would be. Using the Oberth effect then you would only have to supply 7.7 km/s delta-v to get the 15 km/s you need.

This is one of the cases that even dropping from close to GEO → LEO altitude and make the TMI burn at periapsis, would make sense, since the total ΔV for the 2 burns is smaller than leaving directly from 40000 km. Here are the numbers:

You would need 1.47 km/s to drop from 40000 km x 40000 km to 40000 km x 200 km. At periapsis the velocity would be 10.31 km/s
The ΔV for the second burn would be

which seems to agree with your calculation. Total ΔV for the 2 burns: 1.47 + 10.09 = 11.56 km/s
saving more than 3km/s from leaving directly from the 40000 km circular Earth orbit.

 

[Staiduk]

Hi guys, thanks again for the help; it's quite fascinating.

Just in case you're interested, here's a couple of pictures of the Princess Astoria.

I'm an absolutely terrible artist; but I was totally bored and began doodling with MS Paint for a little while.

The Living module came out a little more nautical than I intended but you get the general idea.

Oh - BTW: the booms are paired; two extending from each side of the vessel. In this view, it might look like one.

Enjoy.

 

[Staiduk]

Sorry for necroposting; this is a couple months old but I didn't fully understand what RGClarke said. I've had a few troubles in real-life and my novel has been left on the back-burner. I have solved my personal problems to the point where I feel capable of looking at my novel again. I wanted to come back to this because I think RGClarke made a very important point, and I didn't fully understand it.

Which is a little like saying my cat didn't understand when I tried to explain Bernoulli's Principle to her - and please don't ask why I'm trying to explain flight theory to a miniature mountain lion (otherwise known as an Abyssinian.) I'm hopeless at math and while I am very grateful for the assistance provided, I find myself struggling to understand the simplest concepts. This is great for me, IMO - I'm learning a lot.

RGClarke: Putting it in the simplest possible terms (the only ones I'm capable of understanding) you are saying that the higher initial velocity provided by a low orbit will be a benefit to a ship leaving Earth orbit for Mars. Did I get that right? DgatSoulis countered by pointing out that the required escape velocity would cancel out the advantage?

OK - so what if Princess Astoria departed High Heaven's geostationary orbit and burned retrograde; rather than forward? She would dive down towards Earth and could use the speed provided by the descent to assist her on the burn to Mars, couldn't she? If she swooped down into LEO she could burn at the bottom of her arc and sail off towards Mars with a signifigant fuel-saving, couldn't she?

It seems sensible to me; but there's bound to be a hitch. I'd be grateful to know what the hitch is.

Thanx.

 

[dgatsoulis]

OK - so what if Princess Astoria departed High Heaven's geostationary orbit and burned retrograde; rather than forward? She would dive down towards Earth and could use the speed provided by the descent to assist her on the burn to Mars, couldn't she? If she swooped down into LEO she could burn at the bottom of her arc and sail off towards Mars with a signifigant fuel-saving, couldn't she?

Short answer is yes. Starting from a 40000x40000km orbit around Earth, for a 5 week transfer to Mars, you have significant savings by making two burns; One retrograde to drop to a low perigee and a second one to escape. For comparison here is the ΔV required for all three cases:

1. Single burn from a 40000x40000km orbit: 14.73 km/s

2. Single burn from a 200x200km orbit: 12.61 km/s

3. Two burns. To drop from 40000x40000 to 40000x200km you need 1.47km/s. Then -at periapsis- you need an additional 10.09 km/s. Total ΔV for two burns is 1.47+10.09 = 11.56 km/s

It seems sensible to me; but there's bound to be a hitch. I'd be grateful to know what the hitch is.
Thanx.

You are right. There is a hitch. The hitch is that dropping from a high orbit to a low perigee and then making the burn is NOT always "cheaper".

Here is a graph showing the Injection ΔV needed vs the transfer ΔV, for all three cases; starting from a transfer ΔV of 0 m/s (escape velocity) to 20 km/s.

The red line is showing the Injection Burn ΔV (in m/s) from a 200x200km orbit.

The blue line is showing the Injection Burn ΔV (in m/s) from a 40000x40000km orbit.

The green line is showing the total ΔV (in m/s) for a two burn solution.(Drop from 40000x40000 to 40000x200km and a second burn at periapsis).

Here are a few things to notice:

If we compare the 200x200km burns to the two burn solution we find that the two burn solution is ALWAYS cheaper. (Ofcourse we are not taking into account the ΔV you'd need to get into a 40000x40000km orbit in the first place).

If we compare the 200x200km burns to the 40000x40000km burns we see that it is cheaper to leave directly from the high orbit, for transfer ΔVs up to ~7.5 km/s. Anything above 7.5 km/s and it is cheaper to leave from the low orbit.

If we compare the 40000x40000km burns to the two burn solution we see that it is cheaper to leave directly from the high orbit, for transfer ΔVs up to ~4.5 km/s. Anything above 4.5 km/s and it is cheaper to use two burns.

In the case of the flight you describe in your novel, the transfer ΔV is 17.17 km/s (17170 m/s), so you can see in the graph that the method with the least amount of Injection ΔV is the two burn solution.

(Again, we are not taking into account the initial ΔV needed to get into either of the orbits)