Help My Extraterrestrial Friend Arrive to Madrid's New Year's Eve Party

[DrSirius]

I have a friend of mine which is an extraterrestrial. She travels in a ship around the Milky Way. She was likely in the Solar System, and she wants to attend tomorrow New Year's Eve at Plaza del Sol in Madrid, Spain.
Her last communication was that she had faced a violent cosmic ray storm, and she is inside the panic room aboard the ship. She cannot see outside to see where she is, and can only control the drive force, direction and use a very old fashioned radio receiver/transmitter.
She doesn't want to arrive too late to tomorrow party and not too soon.
The automatic guidance will control the ship correcting the course as if there would be no gravity (i.e., following as rectilinear path as possible) and she can only adjust the speed and direction of the travel. Currently, she had also lost the ability to calculate speed relative to any other point of reference.
How can I help her?

 

[mfb]

The automatic guidance will control the ship correcting the course as if there would be no gravity (i.e., following as rectilinear path as possible)

Following the path given by gravity gives a completely straight line in four-dimensional spacetime. Using thrusters gives deviations from this.

Locate her radio signal, measure her position and velocity, ideally measure the spacecraft orientation (otherwise: tell her to accelerate along different axes to establish a common coordinate system), then tell her which velocity correction is necessary? If the spacecraft is not limited in acceleration, the whole procedure can be done in 3 times the light travel time. New year in Spain starts in 26 hours, so let's hope she is within 8.6 light hours of Earth (~10¹⁰ km, 3 times the distance to Uranus - the Voyager probes are too far away). This limited be improved to 13 light hours if we can measure the spacecraft orientation here right now.

 

[DrSirius]

Following the path given by gravity gives a completely straight line in four-dimensional spacetime. Using thrusters gives deviations from this.

Locate her radio signal, measure her position and velocity, ideally measure the spacecraft orientation (otherwise: tell her to accelerate along different axes to establish a common coordinate system), then tell her which velocity correction is necessary? If the spacecraft is not limited in acceleration, the whole procedure can be done in 3 times the light travel time. New year in Spain starts in 26 hours, so let's hope she is within 8.6 light hours of Earth (~10¹⁰ km, 3 times the distance to Uranus - the Voyager probes are too far away). This limited be improved to 13 light hours if we can measure the spacecraft orientation here right now.

Well, if she is at 8.6 light hours from Earth at most, and she would be unlucky to recede from the Earth in the direction of the line connecting Earth-ship, depending on her speed it could take days and even weeks to contact her if we sent a signal right now.
If we are lucky that she is coming to us at a significant fraction of c, we could contact here in about a little more than 4 hours.
I would like you elaborate more on

1. Method for determining the distance.
2. Method for determining the relative speed.
3. Method for determining the orientation (just provided). Can be obtained from 1) or 2) or do we need, necesarily, another method?
4. Correction, if needed, for the distance if we get that we need relativistic speed,
5. We asume that we would appear in the neighbourhood of the Earth if we don't take into account orbit around Sun, and we need just to correct a little the journey path. Is that correct?

PS: Path is referred to 3D path, not 4D. Also speed is referred to spatial speed, not 4-speed.

 

[mfb]

Well, if she is at 8.6 light hours from Earth at most, and she would be unlucky to recede from the Earth in the direction of the line connecting Earth-ship, depending on her speed it could take days and even weeks to contact her if we sent a signal right now.
If we are lucky that she is coming to us at a significant fraction of c, we could contact here in about a little more than 4 hours.

Sure. The 8.6 hour limit (now ~7) applies only if she is not moving too fast.

To determine distance and relative speed, you can use the usual techniques:
- the direction of the signal gives two parameters for the position
- red/blueshift gives distance changes
- a known signal strength of the spacecraft (radio, visible, ...) can give distance - parallax is an alternative if the spacecraft is close, or bright enough to be visible to optical telescopes.
- proper motion gives motion orthogonal to the line of sight. If relativistic, this will also go into the Doppler shift calculation, but the system has a unique solution so this is not an issue.

If the spacecraft emissions are directed, this angular characteristic can help to determine its orientation.
Actually, we don't have to communicate that in advance: she is asking for help, so she should perform some acceleration pattern that reveals the spacecraft orientation, we can directly observe it.

Correction, if needed, for the distance if we get that we need relativistic speed,

Well, plug everything into the formulas of special relativity.

We asume that we would appear in the neighbourhood of the Earth if we don't take into account orbit around Sun, and we need just to correct a little the journey path. Is that correct?

For an actual application, you probably want to take into account that Earth is accelerated by Sun and Moon (leads to a deviation of ~2 Earth diameters over a day). As this is a purely hypothetical situation, I guess we can neglect it.

 

[pervect]

Well, if she is at 8.6 light hours from Earth at most, and she would be unlucky to recede from the Earth in the direction of the line connecting Earth-ship, depending on her speed it could take days and even weeks to contact her if we sent a signal right now.
If we are lucky that she is coming to us at a significant fraction of c, we could contact here in about a little more than 4 hours.
I would like you elaborate more on

1. Method for determining the distance.
2. Method for determining the relative speed.
3. Method for determining the orientation (just provided). Can be obtained from 1) or 2) or do we need, necesarily, another method?
4. Correction, if needed, for the distance if we get that we need relativistic speed,
5. We asume that we would appear in the neighbourhood of the Earth if we don't take into account orbit around Sun, and we need just to correct a little the journey path. Is that correct?

PS: Path is referred to 3D path, not 4D. Also speed is referred to spatial speed, not 4-speed.

This is a little more definite, I can start to provide some general answers

For distance, we have a couple of possibliites that I can think of. There may be more. The round-trip time for an exchange of radio messages could be measured. Or we could use triangulation methods on her radio signal from multiple receivers, and tell her.

For relative speed, again a few possibilities. A couple of distanceand direction methods at different times, or using the doppler effect and accurate measurements of the received frequency of her signal, using the doppler effect. The later would require her to know, accurately in our units, her transmission frequency. It would also only give the component of her velocity towards or away from us, not the tangential component.

For orientation, it's going to be up to antennas. Basically the antenna is pointed in the right direction when you get the strongest signal. Details and accuracy depend on the radio frequency and the size of the antennas. Presumably our ground-mount antennas are bigger than the ones on her ship?

There are two general high-level approaches. She can figure out where the Earth is based on her messages, or we can figure out where she is and tell her. Some methods (like triangulation) would seem to require that we make the measurements, unless her ship has widely-spaced antennas.If we figure out where she is, she has to have enough knowledge of how we set up coordinate systems to understand our answer when we radio it back. She also has to have some idea of the travel time of the signal to account for the propagation delay, she can either calculate it based on the observations we give her, or attempt a signal-travel time measurement. Probably she would use multiple techniques and try to make sure they gave the same answer.