Travel from Mars to Earth question......

[vicvalis]

TL;DR Summary

Doing some very ball-park amateur calculations, am i even doing it right?

First, i think this is the correct place to ask this question.
So the story is that I'm working on a piece of fiction that I want to be as accurate as possible. One part of it involves an object that travels from Mars to earth. This would have been between February 22, 1901 (beginning of opposition that year, I think) and June 7 of 1901, 104 days if i counted right. So the object leaves Mars and has to race to catch up to earth, because Earth is already moving further away. I've come up with a collection of figures, and made come calculations that I'm trying to keep as estimates... they don't need to be exact to the nth decimal point. But as i don't have the slightest grasp of celestial mechanics, I'm sure I've gotten plenty of stuff wrong.

What I'm hoping to end up with is the total distance the object would have had to travel, and how fast, to make the trip in 104 days. so here's my collection of figures (and they may be wrong, PLEASE feel free to correct me if they are!):

Mars distance from Earth 23:59:59, Feb 22, 1901 0.6774849 AU 62976155.79478 (62,976,156) miles.
Mars distance from Earth 23:59:59, Jun 07, 1901 1.3464453 AU 125159909.81 (125,159,910) miles.

Earth travels about 1.6 million miles per Earth day (1,600,000); in one hundred and four days the Earth traveled one hundred sixty-six million four hundred thousand miles. (166,400,000).
Mars travels about 1.3 million miles per Earth day (1,300,000); in one hundred four days Mars traveled one hundred and thirty-five million two hundred thousand miles (135,200,000).

‭Distance between Earth and Mars after 104 day is thirty-one million two hundred thousand miles (31,200,000)‬.
Speed traveled is 12500 miles per hour.

I KNOW I have something wrong. I'm pretty sure about the Mars distances during the dates at top... I got them off the internet, so they HAVE to be true, right? After that, the rest are averages or estimates... hopefully in the ballpark. Even so, the rest I have no confidence in. I truly suck at math, took me eight years to get through bonehead algebra in college, I'm THAT bad. So correct me! If you can look up more accurate speeds, I'd appreciate them and a citation. If nothing else, I'd like to be able to say the object left Mars on February 22nd 1901, and traveled approximately "x" miles at "x" mph, arriving at Earth 104 days later.

I hope this exercise is not an annoyance. Thanks in advance.

 

[mjc123]

Distance between Earth and Mars after 104 day is thirty-one million two hundred thousand miles (31,200,000)‬.
Speed traveled is 12500 miles per hour.

That is simply the difference between the distances traveled by Earth and Mars in 104 days. It is not the distance traveled by the object. That distance is the length of the orange line in the diagram. (As a ballpark estimate, I would suggest calculating the length of an arc along an orbit halfway between Earth and Mars, between the angles of M1 and E2.)

 

[vicvalis]

Okay, thanks, I could see the orange line in my head as I made the post, and understood that was the distance I was looking for, and needed the speed I would need to travel it in 104 days. Now, after plugging away on the internet, I have finally found that I'm looking to calculate the Hohmann transfer orbit. Yes, knowing that name would have helped in my request, but I'm a little slow sometimes. I have no idea how to calculate that curving path. When I get to work in a little bit, I will see if there are calculators for that... NASA must have something, but in the mean time folks, keep it coming! Teach me stuff!

 

[jbriggs444]

So the object leaves Mars and has to race to catch up to earth, because Earth is already moving further away.

As I understand the expected Hohmann transfer orbit, the object leaves Mars by decellerating away from the near-circular orbit of Mars to an elliptical orbit that will intersect with the near-circular orbit of the Earth.

While the object begins following this path, it is losing ground relative to both Mars and Earth. As it finishes following this path, it has gained velocity and will be out-pacing both. It will decellerate a second time to circularize its orbit.

I do not know enough orbital mechanics to compute the transit time from start point where the transfer orbit begins at Mars and end point, 180 degrees away where the transfer orbit ends at Earth. It is clear that the interval will be more than half an Earth year and less than half a Mars year. So somewhere between 183 and 343 days. Rough order of magnitude is 263 days -- almost enough time to make a baby.

The "launch window" would be chosen such that the Earth will be 180 degrees away (at arrival) from where Mars is (at launch). You do not launch when Mars and Earth are in opposition. You do not arrive when Mars and Earth are in opposition.

 

[vicvalis]

I'm revisiting some much older fiction, and the original author thought such a trip during opposition was the best time, as at that moment the two planets were so close together. He didn't take into account the motion of the two planets. So in this case, I'd have the object accelerating to catch up to Earth in about 100 days, rather than falling back iand intercepting Earth at a certain point in it's own orbit. I'm sure it would take a lot of energy, but if that's what it takes to explain this timing, so it goes. I'm looking to put a plausible description with approximate speeds and distances on what, on it's face, is an implausible trip. Perhaps calling it a Hohmann transfer orbit is the wrong thing. Don't think of tis as me making it hard, think of it as me presenting a challenge! Thanks!

 

[jbriggs444]

I'm revisiting some much older fiction, and the original author thought such a trip during opposition was the best time, as at that moment the two planets were so close together. He didn't take into account the motion of the two planets. So in this case, I'd have the object accelerating to catch up to Earth in about 100 days, rather than falling back iand intercepting Earth at a certain point in it's own orbit. I'm sure it would take a lot of energy, but if that's what it takes to explain this timing, so it goes. I'm looking to put a plausible description with approximate speeds and distances on what, on it's face, is an implausible trip. Perhaps calling it a Hohmann transfer orbit is the wrong thing. Don't think of tis as me making it hard, think of it as me presenting a challenge! Thanks!

A "Hohmann" transfer specifically refers to an elliptical transfer trajectory that is tangent to both the original orbit and to the intended new orbit. It is an unpowered (ballistic) trajectory involving only two brief burns, one to begin the transfer and one to end it. Both burns are parallel (or anti-parallel) to the craft's existing velocity.

Yes, you can transfer more quickly if you have lots delta-V to squander.

 

[vicvalis]

The way I'm framing it, I will be writing something to the effect of "if the departure from Mars happened on February 22, and arrival at Earth was on March 7, then the object would have had to travel X miles at x mph to complete the trip in 104 days, a difficult but not impossible trip." And I like your phrase "lots delta-V to squander." There are lots of other aspects of what I'm revisiting that straight-up defy known physics and biology; at some point I'd be writing "our current understanding still does not explain this." I'd go into more details, but next week I could end up putting this project back into a desk drawer for another few years, and in the mean time someone might take the idea and run with it.

 

[Janus]

I'm revisiting some much older fiction, and the original author thought such a trip during opposition was the best time, as at that moment the two planets were so close together. He didn't take into account the motion of the two planets. So in this case, I'd have the object accelerating to catch up to Earth in about 100 days, rather than falling back iand intercepting Earth at a certain point in it's own orbit. I'm sure it would take a lot of energy, but if that's what it takes to explain this timing, so it goes. I'm looking to put a plausible description with approximate speeds and distances on what, on it's face, is an implausible trip. Perhaps calling it a Hohmann transfer orbit is the wrong thing. Don't think of tis as me making it hard, think of it as me presenting a challenge! Thanks!

Below is a diagram of orbital trajectories. Blue circle is Earth orbit, Red circle Mars orbit.

The green ellipse is a Hohmann minimum energy transfer orbit. Your craft will intersect Earth's orbit at the Green arrow after ~258 days. What you want is a trajectory more like the violet ellipse. It full orbit would pass inside the Earth's orbit, which it crosses along the way. You time things so that the Earth and craft meet at the violet Arrow.
If you were to just completely stop your craft relative to the Sun and just let it fall in towards the sun, then it would take ~ 84 days to reach Earth orbit. You specified 104 days, so you wouldn't have to go to that extreme.

Working out the exact trajectory needed for a 104 day trip would be a bit tricky. Unfortunately there is no direct equation for determining how much of an elliptical orbit an object will travel in a give amount of time. ( there are methods, but many of them involve iteration)So even if you knew the exact shape of the orbit to start with, you need have to use one of these indirect methods to find how far you traveled along the orbit in 104 days. In your case however, the added requirement that the change of radial difference over the trip also has to be a fixed distance complicates things quite a bit more. It can be done, but it would take a fair amount of number crunching.

 

[vicvalis]

You know, I think you've about hit the nail on the head for me. I'd be writing from the point of view of a historian looking back on events, writing for the average reader (thus, not needing exact details). If I describe it generally as you did, with a more extreme elliptical orbit than a a Hohmann minimum energy transfer orbit, meeting the Earth in about 100 days... almost stopping the object relative to the sun and letting it fall towards the sun until it reached Earth orbit, that would be easy for a reader to visualize.
That the trip was accomplished in the world of this book would not be in question... how it accomplished, the sort of path, and how it compares to current understanding of orbital mechanics, it would just make it easier for the reader to believe such a thing happened.

 

[Ibix]

I suspect something on that purple orbit would make a fairly respectable crater if it hit Earth. Depending on what Earth knows about the object and its intentions, this might be a concern for your characters.

 

[vicvalis]

Yes, in the original, it was somehow a soft enough landing to survive in one piece, but still big enough to make a crater. Other people have written about this in the past, mentioning that the acceleration (for other dates, the exact dates are up for interpretation), could have killed anything inside. But, as the saying goes, it's not the fall that kills you, it's hitting the ground.