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Transfer orbits for dummies! A hillbilly tutorial.

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remcook
#19
Jun8-04, 09:54 AM
P: 70
Depends on what you want I guess.

If you want to gain speed using planetary a flyby, it's more advantageous to fire at pericentre. This is because speed is largest there. The speed that you add by burning fuel will always be the same (it depends on the fuel, engine, etc). So, because kinetic energy goes with the square of the velocity, adding that same velocity to a large speed will give you a larger increase in kinetic energy (and because potential energy stays constant during the burn, the total energy increase is the same as the kinetic energy increase for both cases.)
Janitor
#20
Jun8-04, 09:07 PM
Sci Advisor
P: 1,189
For the case of the modest rocket going straight up and straight down in a constant-g field... If we ignore the effect of an atmosphere, we ought to be able to graph apogee vs. thrust and get some sort of curve, with fixed parameters being structure mass, propellant mass, and total impulse, and the constraint that thrust times burn time equals the total impulse. As Jenab noted, "The least efficient (zero efficiency) launching rocket is one that can do no more than hover by making thrust equal only to its own weight." To my intuition, that means the curve will not have its peak over toward the left side of the graph where the thrust is low. In fact, my intuition suggests the curve is monotonically increasing with increasing thrust. (In the real world, structure mass would have to increase when thrust gets bigger to keep the rocket from collapsing or blowing up, so that its non-constancy would have to be taken into account, so the real-world version of the graph would actually have some maximum apogee for some finite value of thrust, beyond which increasing thrust lowers the apogee; "the point of diminishing returns" you might call it.)

Remcook, when you say, "because potential energy stays constant during the burn," are you making the approximation that the burn time is short compared to the time it takes for the gravitational potential to change appreciably when the probe is near its minimum point in the potential field?

And another question for Jenab: can a West Virginian still grab onto a slow-moving coal hopper and go places in your state? Or are those days long gone?

As it happens, I recently read an article on M&K Junction in West Virginia, across the Cheat River from Rowlesburg.
remcook
#21
Jun9-04, 10:42 AM
P: 70
Remcook, when you say, "because potential energy stays constant during the burn," are you making the approximation that the burn time is short compared to the time it takes for the gravitational potential to change appreciably when the probe is near its minimum point in the potential field?
Yes

Both in apogee and in perigee the velocity is perpendicular to the gravity acceleration anyway (no gravity loss). But the satellite moves quicker at perigee of course. But if the burn time is small...
Jenab
#22
Jun9-04, 04:39 PM
Sci Advisor
P: 115
Quote Quote by Janitor
And another question for Jenab: can a West Virginian still grab onto a slow-moving coal hopper and go places in your state? Or are those days long gone?

As it happens, I recently read an article on M&K Junction in West Virginia, across the Cheat River from Rowlesburg.
I never tried jumping on a coal hopper. I'm in Pocahontas County, about 4 miles NW of Hillsboro, or 2 miles NW of Mill Point, or 8 miles SW of Marlinton, or 33 miles north of Lewisburg. The nearest river is the Greenbriar, which runs N-S a little to the east of Hwy 219. I moved up here from Alabama in 1998.

Jerry Abbott
Janitor
#23
Jun9-04, 08:41 PM
Sci Advisor
P: 1,189
Thanks for confirming, Remcook.

Jenab, M&K Junction is three counties north of you in Preston. Ever been to Harper's Ferry or Spruce Knob? I have not, but those are places on my to-do list, to get to one of these years.
Jenab
#24
Jun10-04, 07:05 AM
Sci Advisor
P: 115
I don't travel much, except locally to shop for groceries, animal feed, to buy building supplies/tools, or to send and receive mail. I came originally to West Virginia to work for Dr. William Pierce as an editor for National Vanguard Books. When he died, the National Alliance came under new management, and I soon discovered that I had irreconcilable differences with the new chief, so I quit.

Now I live very pastorally on my ranch/orchard, and circumstances don't give me a lot of time for sightseeing. Once I chased a presumptuous young black bear away from my trash can, off my porch and into the woods. Another time, I shot a bobcat as he was stalking my chicken, Mr. Bronfman; but unfortunately the same chicken met an untimely demise two weeks later under the talons of a hawk.

My previous cat, Father Wiggly, had a run-in with the bobcat a week before I had my own show-down with it. ("Bobcat, this ranch isn't big enough for the two of us.") Wiggles managed to get away, but he was scarred up some. But he disappeared later that year and never showed up again. So now I have two new kittens, Spooky and Goblin.

Most of my trouble with wildlife, though, comes from deer entering the property to nibble on my young apple trees. I have each tree caged inside a circle of fencing 3 feet wide and six feet high. That works, most of the time, but I lost three of my 30 trees last winter to rabbits nibbling off the bark at the bottom. So now I've added another circumferential layer of chicken wire down there.

Jerry Abbott
selfAdjoint
#25
Jun10-04, 11:03 AM
Emeritus
PF Gold
P: 8,147
Sounds like a beautiful life. Are you anywhere near Spencer? That's where my father grew up. He used to spend summers on his uncle's farm and had a bunch of stories just like yours that he used to tell me.
Jenab
#26
Jun10-04, 11:36 AM
Sci Advisor
P: 115
Quote Quote by selfAdjoint
Sounds like a beautiful life. Are you anywhere near Spencer? That's where my father grew up. He used to spend summers on his uncle's farm and had a bunch of stories just like yours that he used to tell me.
It's a good place to live, especially if you like being close to nature. I like to sit on my front porch and read. My house is on a rise, with a road coming up from a valley to the south and halfway circling my house on a spiral upward. The hilly nature of the area is such that my front porch faces an even higher rise, beyond the road, to the NE. Yet higher (and more distant) hilltops are visible to the north and west. And there's woods covering most of the hills.

On a windy day, the leaves make a very nice susurration, which is a kind of leafy "white noise," during which the trees seem to oscillate between dark green (when you see the tops of the leaves) and very light greenish white (when the wind shows you the lighter flip side of the leaves). And on top of that, there's cloud-play alternately shadowing the foreground, then the background, etc.

But the best thing about this area is that it is smack in the middle of a demographic safe-zone. When fossil fuels become, in 20 years or so, too expensive for government subsidies to keep mechanized agriculture going, there's going to be famine along the coast and in urban areas generally, which means that the same people who will rob you now for the money in your wallet will be breaking into homes looking for cans of beans or sacks of rice.

These "safe-zones" I refer to are areas with low population density, especially in regard to demographic groups that have an elevated statistical propensity for causing crime. Pardon the circumlocution. We'll have a food shortage here, but we have a chance at maintaining ourselves with our own efforts, and there will be fewer bandits to contend with.

Spencer is west of me. It's in western West Virginia, over by Ohio. I'm in eastern West Virginia, over by Virginia.

Jerry Abbott
Janitor
#27
Jun10-04, 08:10 PM
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P: 1,189
Our federal government in the form of the I.R.S. and welfare agencies has a bad track record of rewarding people for having lots of kids. My opinion is that coming shortages of fuel plus coming shortages (out West anyway) of water are aggravated by population growth. I doubt whether more than a tiny handful of congresspersons would agree with me on that, though.
Jenab
#28
Jun11-04, 07:34 AM
Sci Advisor
P: 115
I'm not a biological scientist, but I have a halfway decent understanding for how evolution works to achieve favorable adaptations and how nature's usual methodology can be frustrated.

Births per se aren't the problem. It's the lack of natural rigor to life that leads to an early death for the biologically unfit, the poorly adapted, the defective. Fossil fuels enabled mankind to remove this rigor from his existence for a time, but at the price of accumulating a load of bad genes that increasingly require mechanical aid to compensate.

The depletion of fossil fuels will bring the bill for these accumulated genetic costs due for payment. No longer will mankind be able to evade natural rigor, which is essentially as rigorous as it ever was, but we are much less adapted than we once were for meeting its challenges.

In fact, the demographic groups that maintain a high birthrate, despite the disaster that will sweep the world, will have an advantage over those that attempt to redress the food shortage by reducing their family size. Conflict over resources is inevitable, and in the usual course of things people will tend to sort themselves by biological similiarity first and foremost. Other things being equal, the side with the biggest army wins.

It is time for us to consider how we will maintain our existence, and that of those who are important to us, in the tempestuous struggles that lie ahead. I hadn't thought to introduce politics and biology into a discussion of celestial mechanics, but things that have been on my mind have a way of coming up.

For those of you with the money and inclination to change your residences, here is a short and partial list of safe-zones in the United States:

Virginia/West Virginia.
VA: Highland, Bath.
WV: Pocahontas, Pendleton, Tucker, Randolph, Grant.

Georgia/North Carolina.
NC: Clay, Cherokee.
GA: Fannin, Union, Towns.

Arkansas/Missouri.
AR: Fulton, Baxter, Marion, Izard.
MO: Howell, Oregon, Ozark, Shannon, Douglas, Wright, Texas.

Wisconsin/Iowa.
WI: Grant, Juneau, Vernon, Richland, Crawford.
IA: Winnishiek, Allamakee, Howard, Clayton.

Nebraska/Kansas.
KS: Cheyenne, Rawlins, Decatur, Norton, Phillips, Smith, Jewell, Republic, Cloud, Mitchell, Osborne, Rooks, Grayham, Sheridan, Thomas, Sherman.
NE: Dundy, Hitchcock, Red Willow, Furnas, Harlan, Chase, Hayes.

Central Nebraska.
NE: Grant, Hooker, Thomas, Blaine, Loup, Garfield, Wheeler, Holt, Custer, Valley, Greeley, Logan, McPherson, Arthur, Brown, Rock.

No doubt there are others that I have not found.

You can find other relatively safe areas by going to the Census Bureau's "American Factfinder" pages, which begin on

http://factfinder.census.gov/home/s...n.html?_lang=en

From there, click the link "Data Sets" under the header "Getting Detailed Data." It's in the line: "Expert User? Go directly to Data Sets."

From there, click the link "List all maps" under the header "Select from the following options."

From there, choose from the list a demographic statistic that you want to display on the map. You'll find this statistic especially important:

* Persons per Square Mile: 2000

If you are interested in racial data, the Census Bureau has that as well, and maps to the same scale and with the same center can be overlaid to provide whatever cumulative measure of threat-assessment you consider to be relevant. You might also try convolving the color-coded county-level output with a Gaussian probability distribution to account for some degree of threat from mobile predators.

Jerry Abbott
Jenab
#29
Jun12-04, 11:42 PM
Sci Advisor
P: 115
It looks like I allowed roundoff error to accumulate as I lazily truncated decimals to save button pushes on the calculator. I just now programmed the transfer orbit procedure and got a transit time of 224.85 days for the spaceship, as compared with 225.1 days for the Earth. So my elliptical transfer orbit was better than I thought.

Transfer orbit from Vesta (JD 2453040.3) to Earth (JD 2453265.4).

Aphelion at departure.

a = 1.319533 AU
e = 0.6519092
i = 0.2289958 degrees
L = 353.5454 degrees
w = 112.0761 degrees

True anomaly of arrival: 247.9239 degrees.

Delta-vee at departure (HEC)
dV1x = -8.772 km/sec
dV1y = -1.514 km/sec
dV1z = +2.619 km/sec

Delta-vee at arrival (HEC)
dV2x = +20.487 km/sec
dV2y = +1.515 km/sec
dV2z = -0.103 km/sec

A coordinate rotation will get the velocity vector into celestial coordinates, which will permit the spaceship pilot to orient his thrust by observing the starfield into which he must accelerate, in order to enter the transfer orbit.

Jerry Abbott
Jenab
#30
Jun16-04, 12:25 PM
Sci Advisor
P: 115
If the spaceship pilot has a star atlas referred to ecliptic coordinates, he won't need to do this step. But since most star atlases use celestial coordinates, I thought it best to include the rotation from ecliptic to celestial.

dVx' = dVx

dVy' = dVy cos q - dVz sin q

dVz' = dVy sin q + dVz cos q

Where [q] is Earth's obliquity at the moment of thrust. The J2000 value of the obliquity is

q = 23.439281 degrees = 0.40909263 radians

The magnitude of the delta-vee is independent of the rotation, of course,

dV' = dV = { dVx^2 + dVy^2 + dVz^2 }^0.5

The right ascension of the delta-vee (hence also the thrust) is

dVRA = arctan2( dVy' , dVx' )

The declination of the delta-vee (hence also the thrust) is

dVDEC = arcsin( dVz' / dV' )

Putting in the numbers...

Departure.

dV1x = -8.772 km/sec
dV1y = -1.514 km/sec
dV1z = +2.619 km/sec

dV1 = 9.279 km/sec
dVRA1 = 13h 1m 57.4s
dVDEC1 = +11d 11m 8s

The departure thrust will be roughly toward Vindemiatrix, Virgo.

Arrival.

dV2x = +20.487 km/sec
dV2y = +1.515 km/sec
dV2z = -0.103 km/sec

dV2 = 20.544 km/sec
dVRA2 = 0h 4m 11.0s
dVDEC2 = 0d 1m 29s

The arrival thrust will be toward a point in Pisces.

Jerry Abbott
remcook
#31
Jun16-04, 01:16 PM
P: 70
Jenab: if I may ask: why did you post this stuff? I mean..how did you come up with this? Why would you want to calculate this?
Ehm..can't find the right question to ask.
Hope I am not rude. you're doing a great job

just wondering...
Nereid
#32
Jun16-04, 03:01 PM
Emeritus
Sci Advisor
PF Gold
P: 4,014
Quote Quote by Jenab
*SNIP

dV1 = 9.279 km/sec
dVRA1 = 13h 1m 57.4s
dVDEC1 = +11d 11m 8s

The departure thrust will be roughly toward [arrgh! I don't have a star atlas!]

The arrival thrust will be roughly toward [arrgh! I don't have a star atlas!]
I can't vouch for (or against) any of these, but some are free, and all can work on your PC (so they claim):
star atlases
Jenab
#33
Jun16-04, 08:25 PM
Sci Advisor
P: 115
Quote Quote by remcook
Jenab: if I may ask: why did you post this stuff? I mean..how did you come up with this? Why would you want to calculate this?
Ehm..can't find the right question to ask.
Hope I am not rude. you're doing a great job
just wondering...
There's really two questions here. First, what provoked a goatherd into learning celestial mechanics? Second, what prompted me to post a demonstration on calculating transfer orbits here?

It would be poetic for me to point out that herdsmen have been watching the sky for thousands of years and must somehow acquire a natural curiosity about how things work up there. But really what got me started on celestial mechanics were two books by Robert A. Heinlein: The Moon is a Harsh Mistress and The Rolling Stones. In both books, celestial mechanics plays a role in the story.

I decided that I'd see how tough it was to learn how to fly a spaceship by the seat of my pants, even if our ridiculous Government would never let me get close to one. It turns out that basic astronavigation isn't as hard as it's usually cracked up to be. The concepts are fairly simple. It's the presentation that's usually wrong: too bent on throwing the history and derivation behind every physical law discovered since Ptolemy into unprepared faces.

The way to teach celestial mechanics is the way I've done it here. Put all the important equations out in the open. Minimize on the derivations - those can come later as advanced, further reading topics. Keep the notation simple, at the high school algebra level as much as possible. Follow the essential equations with a fully worked-out example. Dish the work out in bite-sized chunks.

Voila, now anybody can follow the procedure. Or almost.

Quote Quote by Nereid
I can't vouch for (or against) any of these, but some are free, and all can work on your PC (so they claim):
star atlases
Thanks! I downloaded one of the free ones. And I bought a 20th edition Norton's Star Atlas from Amazon.com.

Jerry Abbott
Jenab
#34
Jun21-04, 09:34 PM
Sci Advisor
P: 115
In the example used for calculating a transfer orbit, we had a spaceship departing from Vesta at a certain time. We used Vesta's orbital elements and the departure time to calculate the preburn state vector for Vesta (hence also for the rocket) immediately before the rocket fires its engines to enter the transfer orbit.

Vesta is in an elliptical orbit, and the method shown for obtaining the preburn state vector was the method appropriate for elliptical orbits.

But suppose the rocket had been in a hyperbolic orbit, relative to the sun, instead? The calculation must proceed somewhat differently, in that case.

There is, of course, no period associated with a hyperbolic orbit. But we can determine an equivalent to the mean motion:

m = ( GMsun / a^3 )^0.5

Where

GMsun = 1.32712440018E+20 m^3 sec^-2

and remember to enter the semimajor axis of the hyperbolic orbit in meters. One astronomical unit equals 1.49597870691E+11 meters.

Likewise, the mean anomaly has no special geometric meaning for a hyperbolic orbit, but it nonetheless remains mathematically convenient as an intermediate quantity. The mean anomaly is zero at perihelion, negative prior to perihelion, and positive after perihelion.

M = m (t - T)

where [t] is the moment of interest (e.g. the time of departure) and [T] is the time of perihelion passage. This difference of time is entered in seconds, and M will result in radians. Writing the equation fully:

M = {GMsun / a^3)^0.5 (t - T)

If you'd rather input astronomical units for [a] and days for (t-T), then

M = 0.01720209895 (t-T) a^-1.5 AU^1.5 day^-1

and, again, M will be in radians.

It is important to remember that we do not correct the mean anomaly of hyperbolic orbits to the interval [0,2 pi). If it comes out negative, leave it that way.

Kepler's equation for hyperbolic orbits is

M = e sinh u - u

Where (u) is the hyperbolic eccentric anomaly, which, along with (M), must be in radians. As it was in the elliptical case, the equation is transcendental in the variable that we are trying to find, and we must use a differential calculus method for solving it.

Danby's Method for finding the eccentric anomalies of hyperbolic orbits.

u(0) = 0

Repeat over index j

f0 = e sinh u(j) - u(j) - M
f1 = e cosh u(j) - 1
f2 = e sinh u(j)
f3 = e cosh u(j)
d1 = -f0 / f1
d2 = -f0 / [ f1 + (d1)(f2)/2 ]
d3 = -f0 / [ f1 + (d1)(f2)/2 + d2^2 (f3)/6 ]
u(j+1) = u(j) + d3

Until |u(j+1)-u(j)| < 1E-12

The converged value for (u) from this loop is the eccentric anomaly for the hyperbolic orbit. We don't correct (u) to the interval [0,2 pi) either; if it comes out negative, we leave it that way.

Finding the canonical position vector.

The true anomaly is found from

Q' = arccos { (e - cosh u) / (e cosh u - 1) }

if u>0 then Q = Q'
if u=0 then Q = 0
if u<0 then Q = 2 pi - Q'

The heliocentric distance is

r = a (e cosh u - 1)

The canonical position vector is

x''' = r cos Q
y''' = r sin Q
z''' = 0

The canonical velocity vector is

Vx''' = -(a/r) { GMsun / a }^0.5 sinh u

Vy''' = +(a/r) { GMsun / a }^0.5 (e^2 - 1)^0.5 cosh u

Vz''' = 0

The triple-primed position and velocity, although relative to the sun, are not yet presented in the heliocentric ecliptic coordinate system. They are each rotated (negatively) by the orbital elements w (about the z''' axis), i (about the x'' axis), and L (about the z' axis) in order to appear in the (unprimed) HEC system.

A check on the magnitude of the velocity (i.e., the sun-relative speed) is available:

Vx^2 + Vy^2 + Vz^2 = GMsun { 2 / (x^2 + y^2 + z^2)^0.5 + 1/a }

The state vector of an object in a hyperbolic orbit of elements [ a , e , i , L , w , T ] at the moment of interest [t] is

[ x , y , z , Vx , Vy , Vz ]

Jerry Abbott
Jenab
#35
Jun25-04, 08:55 AM
Sci Advisor
P: 115
My first program was in GWbasic (old DOS interperter programming software), which some of you might not have. So I rewrote it in FREE PASCAL and compiled it. You can download (34 kB) a zip file containing the source code (transit.pas), a compiled executable (transit.exe), and the necessary external data file (transit.dat) from

http://www.jabpage.org/features/transit.zip

NOTICE: I've caught a couple of control-flow bugs myself since uploading this program originally. And I fixed a failure to restore the calculated argument of perihelion to the interval [0,2 pi). So you might want to download the latest version.

The data file initially will contain the orbital elements of Vesta and of Earth and the departure and arrival dates that I used in this thread.

Free download. No restrictions on copying, modifying or sharing, except don't charge the fellow you share it with anything: it must stay freeware.

Bug alerts and suggestions for improvement can be posted here or emailed to goatlyones@moonshinehollow.com.

Jerry Abbott
Jenab
#36
Jul3-04, 08:27 PM
Sci Advisor
P: 115
I regard Robert A. Heinlein as one of the greatest science-fiction writers ever. His science was as meticulous as his stories were fun to read. Someone who didn't have a scientific background might think he was only "hand-waving" at number-crunching that he had not actually done. On the contrary; he did it. He just didn't flaunt it.

A good example of his low-profile diligence is the celestial mechanics that forms part of the story of his novel The Rolling Stones. In order to get the full picture of the world behind this story, one would also need to read two other books, namely The Moon is a Harsh Mistress and The Cat Who Walked Through Walls.

Here's some historical background. Hazel Meade Stone was born on 25 Dec 2063, and, as is chronicled in The Moon is a Harsh Mistress she fought the evil Terran military goons with great distinction during the Earth-Moon War of 2076. She later married Slim Lemke. Their children were Roger (b. 22 Sep 2078, an early first child) and Ingrid (born later).

Roger Stone (family name taken from his mother?) married Edith (maiden name unknown) while he was mayor of Luna City (2122-2130). Their children were Meade (b. 2130), the twins Castor and Pullox (b. 2133), and Lowell (b. 2144).

As The Rolling Stones opens, it is early to mid-2148. Castor and Pullox are both 15, however very well-educated by our standards today, especially in mathematics. The twins plan to buy a spaceship and fly off to the asteroid belt, there to make a fortune mining high-grade metal ore. But their father gets wind of the plan and scotches it. The idea of an extended family outing in a larger, more expensive space yacht takes root, however, and before long Roger, assisted by Hazel (who knows how to arm-twist spaceship merchants), has bought a spaceship and is calling himself 'Captain.'

After some harranguing, the ship is named "The Rolling Stone," and Mars is selected as the first destination because the launch window for the minimum energy trajectory from Earth to Mars will soon be open.

In fact, that's quite correct. It opened (in the real world "will open") for departure from Earth around September 2148.

There is a valid transfer orbit, an ellipse with perihelion at departure (6 September 2148), from Earth to Mars, with a transit time of 259 days, with arrival occuring on 23 May 2149. The heliocentric longitude of Earth at departure will be 341.69 degrees; that of Mars at departure will be 23.76 degrees; and that of Mars at arrival will be 152.08 degrees.

Orbital elements of Earth.
a 1.00000011
e 0.01671022
i 0 (zero)
L 0 (zero)
w 102.94719 deg
T JD 2453009.3

Orbital elements of Mars.
a 1.523688
e 0.093405
i 1.8497 deg
L 47.5574 deg
w 286.5016 deg
T JD 2452873.0

Orbital elements of the transfer orbit.
a 1.3411728 AU
e 0.248072
i 10.608 deg
L 341.687 deg
w 0 (zero) deg

The magnitude of the departure delta-vee is 6.8277 km/sec.*
The magnitude of the arrival delta-vee is 4.2570 km/sec.

*Does not take into account the difference between preburn orbital speed and the local escape speed relative to Earth. The "departure" is really the thrust applied near Earth after having dropped toward perigee from the moon, so probably another half kilometer per second (or thereabout) may be needed.

Jerry Abbott


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