View Full Version : Minimum planetary mass to be trapped on

Matthias

2006-Nov-05, 08:49 PM

How much mass would a dwarf planet or small solar system body have to have in order for it to be physically impossible for a human being to achieve escape velocity by jumping? (I don't know what the world record for speed achieved by jumping but I figure that would be the value to use.)

I know the surface radius and density would also be relevant ... we could assume an average density for the planetoid as a main belt asteroid, yes? Also we would assume the planetoid was a perfect sphere for the sake of simplicity.

With a value for density and the highest velocity a human can achieve without machinery, it would just be a matter of solving for whatever radius would result in a mass great enough to have an escape velocity higher than what a human being could achieve.

I ask the question since I imagine "real" planets as being objects big enough that you couldn't escape without mechanical assistance, whereas some asteroid would be something you could escape by jumping off or pushing yourself off its surface. It's the idea of the 'gravity well' (something you can't climb out of by yourself).

I'm curious where that line might be, roughly speaking.

antoniseb

2006-Nov-05, 10:20 PM

We've nailed down that the density could be three and the shape could be a sphere. Now all we need is to know the speed you get to when jumping. I'm not sure how you could measure that, since all the jumping we do on Earth is with weighty bodies against stronger gravity than your situation.

Just as a wild guess, I'm going to say that the fastest a human could get going on one jump would be 20 miles per hour, but I could be off by a factor of two.

For the rest of your calculation, a minimal elevation orbit around a spherical object with a density of 3 gm/cc will take about 2 hours, no matter what the diameter is. Escape velocity is 1.4 times the closest orbit velocity.

So, if you can jump up to 20 MPH, you can escape something you can orbit at 14 MPH, which means the circumference is 28 miles.

PhantomWolf

2006-Nov-06, 12:26 AM

I'm going to say that the fastest a human could get going on one jump would be 20 miles per hour

IIRC Mythbusters measured it at about 8 mph when they did their elevator jump experiment.

antoniseb

2006-Nov-06, 12:44 AM

Mythbusters measured it at about 8 mph when they did their elevator jump experiment.

Even if that's what they came up with, they didn't take into account how fast a human could jump against almost no weight.

That's a relatively easy calculation. Against basically no weight, the human would accelerate at 1G faster than they would otherwise. So, if you know the time they are in contact with the ground, you can figure it out. I would guess that the excess speed would be about 4 or 5 mph. So, a human jump speed would be around 12-13mph against only their own inertia.

PhantomWolf

2006-Nov-06, 12:56 AM

Need to think about the fact our astronaut is going to be in a restrictive suit and life support as well. ;)

Matthias

2006-Nov-06, 07:29 AM

For simplicity's sake let's say the EVA suit masses at 10 kg, the human themselves is about 60 kg, and the suit is advanced enough that it wears as well as a spandex jogging suit with the weight distributed fairly evenly.

Since we are dealing with a human being who had a spaceship that could drop them off on the planet, I don't think it's unreasonable to assume they've gotten better at making vacuum suits too. :)

antoniseb

2006-Nov-06, 01:09 PM

I would guess that the excess speed would be about 4 or 5 mph.

Just to muddy the water a little bit here, some athletes can hurl baseballs at over 100 miles per hour. It is plausible that some clever people could find styles of 'jumping' that did much better than 14 miles per hour.

Jeff Root

2006-Nov-08, 12:51 AM

You could find your jumping speed quite easily by lying on a

wheeled, low-friction dolly and pushing away from a wall, like

a swimmer pushing off from the side of the pool.

-- Jeff, in Minneapolis

PhantomWolf

2006-Nov-08, 12:57 AM

Or you could just set up a highspeed camera and a scale. ;)

pghnative

2006-Nov-08, 01:50 AM

Even if that's what they came up with, they didn't take into account how fast a human could jump against almost no weight.

Would it be easier to use energy considerations? I think the best athletes can achieve a vertical leap of over 3 feet, but probably less than 4 ft. (Obviously the world record high jump is more, but the increase in center of mass probably is much less.

So let's say ~ 4 feet. Assume a good athlete has a mass of 180 lbm. So energy input of ~ 2000 ft2 lbm / s2. Now, how to translate this into escape from a planet. Needs some integration I think.

pghnative

2006-Nov-08, 02:16 AM

OK, after a visit to wikipedia (http://en.wikipedia.org/wiki/Potential_energy) to remind myself how the math works, it seems that the energy needed to escape a body is given by

Energy = G * m1 * m2 / r,

Where G, m1, m2 are obvious, and r is (in this case) the radius of the planet(oid).

So, if we change my units from the previous post to SI, I get an energy input by jumping for a 180 lb athlete with a 4 ft vertical leap of ~ 1000 m2 kg /s2, (or N*m, whichever you prefer)

I was lazy and used a spreadsheet instead of working out the math directly, but though quick iteration, I get a limiting planet(oid) mass of ~4E14 kg, and a radius (at SG = 3) of a little over 3000 meters. Circumference is about 20000 meters, or about half of antoniseb's guess of 28 miles. So we're in the same ball park.

edited to add: feel free to check my math, as I didn't double check and I'm tired. I assumed a 120 kg astronaut + spacesuit.

BigDon

2006-Nov-08, 04:18 AM

Don't any of you people have jobs?

(Welcome Matthias)

TriangleMan

2006-Nov-08, 04:44 AM

Don't any of you people have jobs?

Says the man who averages almost 8 posts a day. :D

BigDon

2006-Nov-08, 04:53 AM

:D Not all moving jobs last all day, and I've been laid off, (Though I did spend five hours doing some demolition work yesterday. Breaking up reinforced concrete for five hours with a sledgehammer. Hands are so sore that I can only type with the ring finger of one hand and the middle finger of the other.)

On the topic, I recall reading that you could jump off one or both of Mars' moons.

PhantomWolf

2006-Nov-08, 11:52 PM

Though I did spend five hours doing some demolition work yesterday. Breaking up reinforced concrete for five hours with a sledgehammer.

I did that about 15 years back. It's a terrible job, I spend the nights dreaming about cracks in concrete.

BigDon

2006-Nov-09, 12:21 AM

Yeah, but I'm going to be 50 in less than four years. Still I did a good days work. The job was scheduled for 16 hours.

But anywho, does anybody know if what I read was true? That you can jump off either Phobos or Deimos?

Jeff Root

2006-Nov-09, 02:30 AM

I'm certain that both Phobos and Deimos have gravity too strong

for a human to jump off and stay off, but I haven't done the

calculations.

-- Jeff, in Minneapolis

antoniseb

2006-Nov-09, 12:50 PM

I'm certain that both Phobos and Deimos have gravity too strong

for a human to jump off and stay off, but I haven't done the

calculations.

I wouldn't be so certain. In the case of Phobos, you might have to work just to stay on it. IIRC, it is close to the Roche limit, which may explain why it is undusty enough that we can see cracks on it.

Matthias

2006-Nov-09, 12:59 PM

What does everyone think about this math problem? Would it be useful as a criterion for a defining whether something is "just a rock" in orbit around the sun or one of its satellites, or something "special" (a dwarf planet or a moon)?

After all this time people are still finding new things in orbit around the gas giants. The count is about 63 for Jupiter, 56 for Saturn, 27 for Uranus, and 13 for Neptune. A significant number of these 'moons' are irregular captured asteroids and it's always possible they will capture more as time goes on. Since there is no minimum size for a moon, it remains ambiguous just how many 'moons' every planet has - the better our instruments get, the better we get at finding these orbiting bits of debris, and so we have to add one more moon to the list and come up with a name for it.

Hopefully there will be two types of planetary satellites eventually defined: one for the 'true' moons (Luna, Io, Titan, Charon, etc.) and one for everything else. The list of 'moons' can then become static, and the list of "dwarf satellites" can expand without end, and not bother having to name them all unless we need to. (A right to be reserved, perhaps, for the first human beings to set foot on them.)

pghnative

2006-Nov-09, 02:22 PM

I'm certain that both Phobos and Deimos have gravity too strong

for a human to jump off and stay off, but I haven't done the

calculations.

-- Jeff, in MinneapolisFor what it's worth, using 2E15 kg as mass of Deimos, and 13 km as diameter (nine planets gives it as 11 x 12 x 15 km, so let's model as a 13 km sphere), then the escape energy is 2000 N*m for a 100 kg object. This is about double my estimate of what a top athlete could generate. Of course, I somewhat rounded up in those estimates, so maybe it is more like a factor of 3 or 4, and this doesn't account for how much easier it is to jump in track shoes compared to a spacesuit. But it would seem that you could jump quite a ways.

Phobos is a little more, with a 6000 N*m escape energy.

Peter Wilson

2006-Nov-10, 12:40 AM

Does anybody know if what I read was true? That you can jump off either Phobos or Deimos?

I do recall reading that Phobos or Deimos would make a good "jumping-off" point.

Van Rijn

2006-Nov-10, 01:23 AM

I do recall reading that Phobos or Deimos would make a good "jumping-off" point.

In the sense that they would be really handy for construction and probably fuel. They appear to be carbonaceous and likely have some hydrogen as well as carbon, oxygen, and so on. If earth had low moons like these, it would be much easier to develop a space infrastructure.

Even if you can't completey jump off a small asteroid, you could accidentally jump hard enough to be off the surface for a half-hour or more. Astronauts would likely either need safety lines or something like the MMU. (http://en.wikipedia.org/wiki/Manned_Maneuvering_Unit) You couldn't really walk and even hopping would be difficult. You might push off very lightly and glide, and quite likely you would want to be parallel to the surface, not "standing up" when you moved around.

BigDon

2006-Nov-10, 09:07 AM

Whoa, I followed your link about the MMU Van Rijn. Do you think McCandless might have had a small adrenlin rush going when he seperated himself from the shuttle with no tether? Thats a loooong way down.

closetgeek

2006-Nov-13, 07:11 PM

Let's just say "my friend" found questions and equations like this to be incredibly interesting, exciting to think about, and challenging to figure out. However "my friend" does not do well with absorbing a lot of information at once. So let's say "friend" can put aside time to take one class per semester, not for a degree, but just for the sake of knowing, with no particular rush on completion. What class would this person have to start with and what ones should follow, if this person admits that the furthest they got was basic geometry (in high school)?

OK, after a visit to wikipedia (http://en.wikipedia.org/wiki/Potential_energy) to remind myself how the math works, it seems that the energy needed to escape a body is given by

Energy = G * m1 * m2 / r,

Where G, m1, m2 are obvious, and r is (in this case) the radius of the planet(oid).

So, if we change my units from the previous post to SI, I get an energy input by jumping for a 180 lb athlete with a 4 ft vertical leap of ~ 1000 m2 kg /s2, (or N*m, whichever you prefer)

I was lazy and used a spreadsheet instead of working out the math directly, but though quick iteration, I get a limiting planet(oid) mass of ~4E14 kg, and a radius (at SG = 3) of a little over 3000 meters. Circumference is about 20000 meters, or about half of antoniseb's guess of 28 miles. So we're in the same ball park.

edited to add: feel free to check my math, as I didn't double check and I'm tired. I assumed a 120 kg astronaut + spacesuit.

Tobin Dax

2006-Nov-13, 07:34 PM

Let's just say "my friend" found questions and equations like this to be incredibly interesting, exciting to think about, and challenging to figure out. However "my friend" does not do well with absorbing a lot of information at once. So let's say "friend" can put aside time to take one class per semester, not for a degree, but just for the sake of knowing, with no particular rush on completion. What class would this person have to start with and what ones should follow, if this person admits that the furthest they got was basic geometry (in high school)?

First, closetgeek, there's no need for any hypothetical friends around here (if that is the case). We only judge the cranks, not anyone interested in gaining knowledge. ;)

I assume that this person has no algebra background, if they only got to geometry? That would be a good place to start. If you understand algebra, you can understand the equations and what they mean. After that, any introductory (but somewhat math-intensive) astro or physics course would work (one without calculus, of course, unless one really wants to put the time in to learn it and get in to the nitty-gritty).

Jeff Root

2006-Nov-14, 05:22 AM

I think that the algebra required is entirely covered in 9th-grade

math. That would probably be called "Introductory algebra".

Intermediate algebra, which I got in 11th grade, covers much of

that same material, and develops a lot of the ideas in more detail.

I took physics in 11th grade, so obviously the 11th-grade math

was not required for the physics course. Physics covered all the

ideas involved in the problem here, including velocity, acceleration,

forces, gravity, energy, and potential energy.

Intermediate algebra is widely offered in university extension

courses. Introductory algebra and an equivalent of 11th-grade

physics are probably available, too. They are very likely also

offered by public school adult education, which can be quite

inexpensive.

-- Jeff, in Minneapolis

closetgeek

2006-Nov-14, 02:32 PM

Yeah, I failed 9th grade algebra three years in a row, which is why they put me back to basic geometry. There were underlying reasons why that had little to do with academics. Anyway, not to get too far off topic, thanks for the advice. I now have a path. BTW the hypothetical friend was meant to be humorous.

I think that the algebra required is entirely covered in 9th-grade

math. That would probably be called "Introductory algebra".

Intermediate algebra, which I got in 11th grade, covers much of

that same material, and develops a lot of the ideas in more detail.

I took physics in 11th grade, so obviously the 11th-grade math

was not required for the physics course. Physics covered all the

ideas involved in the problem here, including velocity, acceleration,

forces, gravity, energy, and potential energy.

Intermediate algebra is widely offered in university extension

courses. Introductory algebra and an equivalent of 11th-grade

physics are probably available, too. They are very likely also

offered by public school adult education, which can be quite

inexpensive.

-- Jeff, in Minneapolis

agingjb

2006-Nov-14, 02:49 PM

It was said that Deimos was small enough to jump off (Arthur C. Clarke suggests this in "The Exploration of Space"), but not Phobos.

If (suitably attired) someone could and did jump off Deimos, then I suppose they would orbit Mars in the general region of Deimos' orbit - perhaps returning to Deimos in a millenium or so (back of envelope, order of magnitude, guess...).

Anyway, Wikipedia gives the escape velocities of Deimos and Phobos as 6.9 m/s and 11 m/s; hmm.

antoniseb

2006-Nov-14, 03:19 PM

Wikipedia gives the escape velocities of Deimos and Phobos as 6.9 m/s and 11 m/s; hmm.

Wikipedia's author in this case did not take into account the differential gravity from one end of Phobos being further from Mars than the other.

...the differential gravity from one end of Phobos being further from Mars than the other.

That's a trivial effect for a body as small as Phobos. The range of escape velocities for Phobos is due to from where on its irregular surface you take off. Phobos has "radii" of 13.5x10.8x9.4 km and a mass of 1.08e16 kg. You can then calculate a minimum escape velocity of 10.3 km/sec , a maximum of 12.4, and an average of 11.3.

As for whether you could jump off it, the answer is no. Standing vertical leaps are often measured as an indication of athletic ability, and a jump of 1 meter or so is outstanding. If you use the equation v(0) = sqrt (2gh), you find the athlete can get his body moving initially about 4.5 m/sec. Not enough to escape Phobos, but bring a lunch. It will be a long time before you come down.

Matthias

2006-Nov-22, 05:07 PM

How 'high' would the athlete go if they waited till Mars' center of mass was directly overhead?

antoniseb

2006-Nov-22, 06:50 PM

Phobos is 1.4 times the roche limit away from Mars.

If you are on the tip of Phobos, you experience .00395 Newtons per kilogram force toward Phobos, and .00142 Newtons per kilogram away from phobos. So the tidal forces reduce your apparent weight by a third.

That's a trivial effect for a body as small as Phobos. The range of escape velocities for Phobos is due to from where on its irregular surface you take off. Phobos has "radii" of 13.5x10.8x9.4 km and a mass of 1.08e16 kg. You can then calculate a minimum escape velocity of 10.3 km/sec , a maximum of 12.4, and an average of 11.3.

As for whether you could jump off it, the answer is no. Standing vertical leaps are often measured as an indication of athletic ability, and a jump of 1 meter or so is outstanding. If you use the equation v(0) = sqrt (2gh), you find the athlete can get his body moving initially about 4.5 m/sec. Not enough to escape Phobos, but bring a lunch. It will be a long time before you come down.

I believe you mean m/s, not km/s...

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