Navigation in space

[steve_bank]

That is great, now we just nee a starship and warp drive.

 

[Tigers!]

For pulsar-pulse navigation, I will see if the aliasing problem can be resolved.

One can find one's position with a least-squares fit:
\( E = \sum_i w_i (n_i \cdot x - t_i)^2 \)
for error E, individual pulsars i with arrival time t(i) at the coordinate origin and weighting w(i), and position x. A good choice of weight is
w = 1/(pulse_size)^2

E can be found from a quadratic equation in x:
\( E = T - 2(X.x) + (x.N.x) \)
where
\( T = \sum_i w_i (t_i)^2 ,\ X = \sum_i w_i t_i n_i ,\ N = \sum_i w_i (n_i \otimes n_i) \)

The least-squares solution is
\( x = N^{-1} \cdot X \)
with error
\( E = T - X \cdot N^{-1} \cdot X \)

The aliasing problem is essentially which pulse is one observing. For pulse periods P and pulse count m we have
\( t_i \to t_i + m_i P_i \)

This gives us
\( E = E(0) + 2 \sum_i m_i E'_i + \sum_{ij} m_i m_j E''_{ij} \)
where
E(0) is E for all the m's = 0,
\( E'_i = w_i P_i (t_i - n_i \cdot N^{-1} \cdot X) \)
\( E''_{ij} = w_i (P_i)^2 \delta_{ij} - (w_i P_i) (w_j P_j) (n_i N^{-1} n_j) \)

One can do a least-squares solution here also,
\( m = - (E'')^{-1} E' \)
but it has the problem that the m's are constrained to be integers. But one can take such a solution and round it to create an initial one, and then search around it. This problem is sometimes called a problem in integer quadratic programming.

Yes but by the time I work out my position doing all that maths I will be in a different position.

 

[bilby]

For pulsar-pulse navigation, I will see if the aliasing problem can be resolved.

One can find one's position with a least-squares fit:
\( E = \sum_i w_i (n_i \cdot x - t_i)^2 \)
for error E, individual pulsars i with arrival time t(i) at the coordinate origin and weighting w(i), and position x. A good choice of weight is
w = 1/(pulse_size)^2

E can be found from a quadratic equation in x:
\( E = T - 2(X.x) + (x.N.x) \)
where
\( T = \sum_i w_i (t_i)^2 ,\ X = \sum_i w_i t_i n_i ,\ N = \sum_i w_i (n_i \otimes n_i) \)

The least-squares solution is
\( x = N^{-1} \cdot X \)
with error
\( E = T - X \cdot N^{-1} \cdot X \)

The aliasing problem is essentially which pulse is one observing. For pulse periods P and pulse count m we have
\( t_i \to t_i + m_i P_i \)

This gives us
\( E = E(0) + 2 \sum_i m_i E'_i + \sum_{ij} m_i m_j E''_{ij} \)
where
E(0) is E for all the m's = 0,
\( E'_i = w_i P_i (t_i - n_i \cdot N^{-1} \cdot X) \)
\( E''_{ij} = w_i (P_i)^2 \delta_{ij} - (w_i P_i) (w_j P_j) (n_i N^{-1} n_j) \)

One can do a least-squares solution here also,
\( m = - (E'')^{-1} E' \)
but it has the problem that the m's are constrained to be integers. But one can take such a solution and round it to create an initial one, and then search around it. This problem is sometimes called a problem in integer quadratic programming.

Yes but by the time I work out my position doing all that maths I will be in a different position.

That's what He(isenberg) said.

 

[lpetrich]

For pulsar-pulse navigation, I will see if the aliasing problem can be resolved. (snipped for brevity)...

Yes but by the time I work out my position doing all that maths I will be in a different position.

If you do it by hand, that is.

 

[steve_bank]

A practcal system would require detailed error analyses. How accurate?

How accurate is our estimate of distance between distant moving objects?

You are in deep space and want to travel 10ly to an object, or where you think it will be. What is the initial pointing accuracy and accuracy of the thrust vector of the ship?.

 

[Shadowy Man]

A practcal system would require detailed error analyses. How accurate?

How accurate is our estimate of distance between distant moving objects?

You are in deep space and want to travel 10ly to an object, or where you think it will be. What is the initial pointing accuracy and accuracy of the thrust vector of the ship?.

Presumably this will depend on how much fuel you have; that is, how much capacity for course correction you have.

 

[steve_bank]

Ther would have to be condtnt correction. I don't know what the current GPS error bubble is.

As I remember it para lax is used to measure distance of an object out to the technique limit set by the diameter of the Earth orbit and instrumentation. That calibrates distance versus radiated energy. Then radiated energyis used to estimate distance outside of parallax based on predicted radiated energy based on star models.

Relative distance estmates do not seem precise, many error sources.

 

[skepticalbip]

A practcal system would require detailed error analyses. How accurate?

How accurate is our estimate of distance between distant moving objects?

You are in deep space and want to travel 10ly to an object, or where you think it will be. What is the initial pointing accuracy and accuracy of the thrust vector of the ship?.

A sphere with a radius of 10LY around Earth is well within the local area so we have a pretty good values for distance and relative motion. Parallax is useful for measuring distance to stars out to 100 Parsecs (326LY) but accuracy decreases with distance.

Since a 10LY trip would take several centuries, the original crew wouldn't have to worry about making corrections. It would be many generations later before any corrections would be needed and the kids would be asking, "Are we there yet?".

 

[lpetrich]

A practcal system would require detailed error analyses. How accurate?

How accurate is our estimate of distance between distant moving objects?

You are in deep space and want to travel 10ly to an object, or where you think it will be. What is the initial pointing accuracy and accuracy of the thrust vector of the ship?.

steve_bank, I suggest doing some research into astrometry and astrometric satellites. It's possible to get the directions of stars to very high accuracy and parallaxes and proper motions of the nearer ones to a fair degree of accuracy.

 

[Tigers!]

For pulsar-pulse navigation, I will see if the aliasing problem can be resolved. (snipped for brevity)...

Yes but by the time I work out my position doing all that maths I will be in a different position.

If you do it by hand, that is.

Yes my slide rule speeds skills have deteriorated over the decades. Lack of practice and reliance on devices.
"Age shall weary me and the years condemn." (With apologies to Laurence Binyon)

 

[steve_bank]

A practcal system would require detailed error analyses. How accurate?

How accurate is our estimate of distance between distant moving objects?

You are in deep space and want to travel 10ly to an object, or where you think it will be. What is the initial pointing accuracy and accuracy of the thrust vector of the ship?.

steve_bank, I suggest doing some research into astrometry and astrometric satellites. It's possible to get the directions of stars to very high accuracy and parallaxes and proper motions of the nearer ones to a fair degree of accuracy.

Ipeyttrich, I spent my career doing analysis on real systems.

Locally Newtonian mechanics gets satellites in orbit , gets probes to Mars, and gets us to the moon.

I would want to see actual detailed error analysis on how one would derive a relative position that is always changing. Analysis incorporating the nav system would not be trivial. What is the pointing accuracy of the telescopes on Earth. How accurate is the parallax orbital baseline diameter which changes.

It would have to incorporate thrust vectoring accuracy of a space ship, electronics and drift, and the initial position of objects in meters. You initialize position in orbit and vector away, the practical problems begin immediately. How do you get on course? The errors begin to add up.

As I have said before IMO space travel without some new science is impractical.


A claim of accuracy of star positions without citing an error bound has no value.

The Apollo moon missions carried an optical telescope to make sights on stars for position checks in flight.

Inertial navigations systems used in submarines drift over time from an initialization point in port. The periscopes are precisly aligned to the sub's axis so they can take a star or noon sun sight fix. Always back ups. On Navy ships someone always takes a daily manual position fix to cross check GPS. Sextants.

I worked for Kolmorgen who made periscopes.

GPS works because it can be refenced to specific points on the surface. Accurate to a radius of several meter or so, I'd have to look up current specs.

Going to a star in a multigenerational ship would be best done by dead retconning. Derive an estimate of projected position and head to it making course corrections zeroing in on the star.

 

[lpetrich]

A practcal system would require detailed error analyses. How accurate?

How accurate is our estimate of distance between distant moving objects?

You are in deep space and want to travel 10ly to an object, or where you think it will be. What is the initial pointing accuracy and accuracy of the thrust vector of the ship?.

steve_bank, I suggest doing some research into astrometry and astrometric satellites. It's possible to get the directions of stars to very high accuracy and parallaxes and proper motions of the nearer ones to a fair degree of accuracy.

Ipeyttrich, I spent my career doing analysis on real systems.

...
It would have to incorporate thrust vectoring accuracy of a space ship, electronics and drift, and the initial position of objects in meters. You initialize position in orbit and vector away, the practical problems begin immediately. How do you get on course? The errors begin to add up.

...
A claim of accuracy of star positions without citing an error bound has no value.

Interstellar space is not low Earth orbit. Work out the numbers.

 

[lpetrich]

 Astrometry  Star catalogue

• [1104.4554v2] Astrometry during the past 2000 years - 2011
• [1706.08097] Selected astrometric catalogues - 2017
• [1707.01020] Astrometric accuracy during the past 2000 years -2020
• (Several other papers by Erik Høg at arxiv)
• Erik Høg – University of Copenhagen - the author's home page

The 2017 paper has a list of accuracy values for various star catalogs. The oldest one is Claudius Ptolemy's, published around 150 CE, with 1028 entries, and with a standard error of 25 arcminutes. In 1437, Ulugh Beg did a little better, at 20 arcmin. The last notable astronomer to do only unaided-eye observations was Tycho Brahe, and he got 2 arcminutes. That's about the angular resolution of his eyes.

Not surprisingly, one could do much better with a telescope. By the 19th cy., one could get to an arcsecond or better, and with photography even better: around 0.05 arcsec.

In 1989, the  Hipparcos was launched. It observed the positions of a large number of stars with very high precision for 4 years, and it got the positions of 120,000 stars to within 1 milliarcsecond (mas, 1/1000 arcsec).

In 2013,  Gaia (spacecraft) was launched, and it has observed some 23 million stars to within 10 microarcseconds (mcas, uas, 1/1000,000 arcsec), and some 1.2 billion stars to within 300 mcas.

Note: 1 arcsec is about 1/206,265 radians or 5 microradians. The best ground-based measurements are 300 nanoradians, Hipparcos's are 5 nanoradians, and Gaia's 50 picoradians.

 

[lpetrich]

Parallaxes have the complication that many measurements have been relative to nearby stars. This had made possible much greater accuracy than for overall-coordinate position measurements. But that is only true of visual or photographic measurements - astrometric satellites don't have that advantage. But even without it, they can outperform most photographic nearby-star measurements.

Proper motions are another complication. Stars move relative to each other, and their motions have been observed, both as radial velocity and as angular velocity.


A big problem is identifying the stars that one observes, especially if one's navigation software does not have strong AI. I think that such an interstellar navigation system would observe the stars every now and then and use previous observations to identify the stars. Earth-neighborhood-based observations should be good enough to get good starting values of the positions and velocities of the stars, and one can improve those measurements as one travels.

To show how big this problem is, consider traveling to Alpha Centauri, the closest star system to the Sun. It is 1.4 parsecs (4.3 light-years) away, about 300 thousand times farther from the Sun than the Earth is. Going to A-Cen will give a parallax baseline that much longer, though with the Earth, one can use opposite sides of its orbit to get a factor of 2.

Let's say we use Gaia parallaxes with errors of 10 mcas. At A-Cen, those errors will have become magnified to 3 arcsec. Spacecraft celestial navigation ought to do at least as good as 1 arcsec, and likely much better. For 0.1 arcsec, that means that the Gaia errors will be 30 times larger than the observational errors.

 

[Loren Pechtel]

A practcal system would require detailed error analyses. How accurate?

How accurate is our estimate of distance between distant moving objects?

You are in deep space and want to travel 10ly to an object, or where you think it will be. What is the initial pointing accuracy and accuracy of the thrust vector of the ship?.

steve_bank, I suggest doing some research into astrometry and astrometric satellites. It's possible to get the directions of stars to very high accuracy and parallaxes and proper motions of the nearer ones to a fair degree of accuracy.

Ipeyttrich, I spent my career doing analysis on real systems.

Locally Newtonian mechanics gets satellites in orbit , gets probes to Mars, and gets us to the moon.

I would want to see actual detailed error analysis on how one would derive a relative position that is always changing. Analysis incorporating the nav system would not be trivial. What is the pointing accuracy of the telescopes on Earth. How accurate is the parallax orbital baseline diameter which changes.

It would have to incorporate thrust vectoring accuracy of a space ship, electronics and drift, and the initial position of objects in meters. You initialize position in orbit and vector away, the practical problems begin immediately. How do you get on course? The errors begin to add up.

As I have said before IMO space travel without some new science is impractical.


A claim of accuracy of star positions without citing an error bound has no value.

The Apollo moon missions carried an optical telescope to make sights on stars for position checks in flight.

Inertial navigations systems used in submarines drift over time from an initialization point in port. The periscopes are precisly aligned to the sub's axis so they can take a star or noon sun sight fix. Always back ups. On Navy ships someone always takes a daily manual position fix to cross check GPS. Sextants.

I worked for Kolmorgen who made periscopes.

GPS works because it can be refenced to specific points on the surface. Accurate to a radius of several meter or so, I'd have to look up current specs.

Going to a star in a multigenerational ship would be best done by dead retconning. Derive an estimate of projected position and head to it making course corrections zeroing in on the star.

I think you're making the problem a lot more complex than it really is.

You're trying to solve the problem of getting a very accurate position in 3D space in interstellar realms. You're probably right that we can't do this. However, I don't see a need to solve the problem.

Since I recently reread Rocheworld lets consider that voyage.

Voyage #1: Laser-pumped lightsail flyby probe, humanity's first interstellar mission.

To aim: Look at the proper motion of the star, project the bearing it will be on at the time of the flyby given the best estimate of the range (and since it's nearby parallax will give a pretty good range.) Nothing is known about what's there so there's nothing beyond the star to aim for. During the boost the probe can tack a bit in cooperation with the forts that are driving it. (The scenario was that they used anti-missile laser defenses to boost it.)

On arrival: The probe has no meaningful maneuvering capacity. It can see the star to aim it's instruments and look for planets.

Voyage #2: Laser-pumped lightsail starship, humanity's first manned interstellar mission.

To aim: The flyby gave exact information on distance, they know exactly when to cut the ring off their sail for the braking maneuver. They don't need to know anything more than that. They have been in unpowered flight since the laser turned off decades before, Earth knows the bearing to fire upon. The ship has no ability to maneuver nor can a maneuver even be considered due to the communications lag with Earth. The laser that will stop them must be fired years before the encounter.

On arrival: All they need is to get capture. While the book didn't bother with all the details it's obvious the beam will continue for some extra time, when their velocity is what they want they tack away from the beam and become a simple lightsail ship. Accuracy only matters to the extent that they are close enough to the star to use their lightsail--and note that even if they're a bit far out they can slow enough to put their periapsis as low as they want. Miss distance only matters if they are so far out the inward fall takes too long.

 

[Swammerdami]

<sub><sup>Let me present a mini-project I once wanted to pursue,. I didn't. For me the task would have been tedious, error-prone, and a doubtful use of scarce time. But it might be fun and easy for someone at ease with 3D navigation.

The task is to draw pictures of the Orion constellation, as it would be seen from this solar system, and from any of (your choice of) several grid locations, say 12-20 light-years distant from Sol.

I'll be happy to provide (x, y, z, abs_mag) quadruplets for each star in Orion, along with Sirius and any other desired stars.</sup></sub>

Give it a snazzy interface. Set some parameters with sliders; Click; and -- with today's high speeds -- the desired picture (a 3x3 array of Orion views arranged into a Gif) appears at once.

Come to think of it, there ought to be an App like that already. Anyone have a link?

 

[lpetrich]

If one has some programming ability, and some experience with computer graphics, it ought to be easy to do simulations of what one would see during an interstellar voyage.

 

[Swammerdami]

If one has some programming ability, and some experience with computer graphics, it ought to be easy to do simulations of what one would see during an interstellar voyage.

Yes, it would be easy to do ... for those who do these things easily! I've programmed successfully in the past, but am increasingly old, stupid, lazy and distracted.

Browsing a folder just now, I see I did some simple star-mapping 4 years ago. I'm attaching one of the resulting lists; I never did anything with it beyond this. The required algebraic manipulations for that were much simpler than drawing constellations.

The Orion-drawing task is simple to state. Given (x_I, y_I, z_I) and (x_B, y_B, z_B), respectively the coordinates of an observer and Betelgeuse (or any other target star), output the two coordinates of Betelgeuse in some viewing plane. (I'll also want Betelgeuse's relative magnitude, but that's trivial. Approximating a sphere as a plane may seem absurd, but Orion is small enough and 2-D images are the target result anyway.

There seem to be TWO different ways to proceed. (1) Calculate the angle of each target star relative to two reference points; or (2) Determine some reference plane perpendicular to the Observer-Orion line, transform all coordinates to give that plane a simple equation (z = 1) and project each target star to that plane.

Perhaps indecision between which method would be simpler to program is part of my hesitation. But either way would be a tedious effort for me. I'd probably pursue this IF someone provided detailed pseudo-code to perform the task.

~ ~ ~ ~ ~

Here's output from a mini-project I did 4 years ago. It shows the brightest stars encountered on a straight-line journey from Antares to the Pleiades. Three coordinates are provided for each star (z, r, theta), respectively the point on the journey of nearest approach, distance at nearest approach, angle w/r projection of Sol. For example, Sirius was 46.2 ly away at closest approach, and was 55th brightest at brightest approach -- or perhaps less bright since there'd be small but very close stars not in the giant star catalog I started with.

 

[Loren Pechtel]

If one has some programming ability, and some experience with computer graphics, it ought to be easy to do simulations of what one would see during an interstellar voyage.

Yes, it would be easy to do ... for those who do these things easily! I've programmed successfully in the past, but am increasingly old, stupid, lazy and distracted.

Browsing a folder just now, I see I did some simple star-mapping 4 years ago. I'm attaching one of the resulting lists; I never did anything with it beyond this. The required algebraic manipulations for that were much simpler than drawing constellations.

The Orion-drawing task is simple to state. Given (x_I, y_I, z_I) and (x_B, y_B, z_B), respectively the coordinates of an observer and Betelgeuse (or any other target star), output the two coordinates of Betelgeuse in some viewing plane. (I'll also want Betelgeuse's relative magnitude, but that's trivial. Approximating a sphere as a plane may seem absurd, but Orion is small enough and 2-D images are the target result anyway.

There seem to be TWO different ways to proceed. (1) Calculate the angle of each target star relative to two reference points; or (2) Determine some reference plane perpendicular to the Observer-Orion line, transform all coordinates to give that plane a simple equation (z = 1) and project each target star to that plane.

It feels like the easiest approach would be to convert the star location to x, y, z, coordinates, offset for the ship's location relative to Earth and then convert back.

 

[steve_bank]

I expect that would be a life project evolving over generations.

Start with a model of the solar system that for any [x,y.z] you see the motion of the solar around you. When that is working start expanding incrementally.


It sounds simple, just coordinate and reference frame transformations. I expect complexity wou ldgrow quickly.

Compensation for the finite speed of light and time dilation.