How can I calculate the orbital elements from two position vectors and a time difference?

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I have two position vectors for my satellite, and I know that the satellite reaches these two positions 15 minutes apart.

I know I can find the inclination using linear algebra and my position vectors, but is there a way to figure out the rest of the orbital elements from this information?

The positions are X,Y,Z coordinates in the Earth-centered inertial (ECI) frame.

orbital-mechanics orbit trajectory calculation

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up vote
4
down vote

favorite

I have two position vectors for my satellite, and I know that the satellite reaches these two positions 15 minutes apart.

I know I can find the inclination using linear algebra and my position vectors, but is there a way to figure out the rest of the orbital elements from this information?

The positions are X,Y,Z coordinates in the Earth-centered inertial (ECI) frame.

orbital-mechanics orbit trajectory calculation

share|improve this question

edited 1 hour ago

uhoh

29.6k1599360

asked 5 hours ago

user27556

212

New contributor

user27556 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

add a comment | 

up vote
4
down vote

favorite

up vote
4
down vote

favorite

I have two position vectors for my satellite, and I know that the satellite reaches these two positions 15 minutes apart.

I know I can find the inclination using linear algebra and my position vectors, but is there a way to figure out the rest of the orbital elements from this information?

The positions are X,Y,Z coordinates in the Earth-centered inertial (ECI) frame.

orbital-mechanics orbit trajectory calculation

share|improve this question

edited 1 hour ago

uhoh

29.6k1599360

asked 5 hours ago

user27556

212

New contributor

user27556 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

I have two position vectors for my satellite, and I know that the satellite reaches these two positions 15 minutes apart.

I know I can find the inclination using linear algebra and my position vectors, but is there a way to figure out the rest of the orbital elements from this information?

The positions are X,Y,Z coordinates in the Earth-centered inertial (ECI) frame.

orbital-mechanics orbit trajectory calculation

orbital-mechanics orbit trajectory calculation

share|improve this question

edited 1 hour ago

uhoh

29.6k1599360

asked 5 hours ago

user27556

212

New contributor

user27556 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|improve this question

edited 1 hour ago

uhoh

29.6k1599360

asked 5 hours ago

user27556

212

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user27556 is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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share|improve this question

share|improve this question

edited 1 hour ago

uhoh

29.6k1599360

edited 1 hour ago

uhoh

29.6k1599360

edited 1 hour ago

uhoh

29.6k1599360

29.6k1599360

asked 5 hours ago

user27556

212

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asked 5 hours ago

user27556

212

asked 5 hours ago

user27556

212

212

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1 Answer
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If you have no other infomration about the orbit of your satellite (e.g. the orbit is circular), I believe you have to solve this problem with the Lambert's theorem assuming an elliptic transfer orbit (see Wikipedia). However, as far as I know there is no analytical solution and either numerical methods or series expansions need to be used.

In this answer I will try to introduce some aspects about this problem and give you some hints about how to approach it.

As stated by the theorem, given a gravitational parameter $mu=GM$, the time $Delta t$ required to perform a given transfer is a function of

• the semimajor axis $a$ of the orbit,


• the sum of $|vecr_1| + |vecr_2|$, and


• the length $c$ of the chord that connects the two positions (see figure below).

This can be expressed as:
$$sqrtmu Delta t = f(a,r_1+r_2,c)$$
In your case, you know $Delta t$ but you need to find $a$. You will see that there are actually two different values of semimajor axis that bring you from one position to the other in a certain $Delta t$. However, since you are describing an orbit around Earth you might encounter that only one makes sense (e.g. the other colides with the Earth surface).

As I introduced before, as far as I know no analytical solution exists to solve this problem. Some proposed numerical methods/series expansions include:

• Lagrange-Battin (1977)


• Gauss-Battin (1971)


• Battin (Elegant Algorithm) (1984) (here)

amongst others. A review of the Lambert's problem is made by D. de la Torre Sangrà and E. Fantino here (and here).

A generic Lambert solution procedure could be:

1. Compute the geometrical parameters of the transfer




2. Obtain an initial guess for the free parameter




3. Iterate on the transfer time equation until convergence




4. Compute the orbital elements

I hope it helps!

(Special credit for this answer is given to D. de la Torre Sangrà.)

share|improve this answer

edited 1 hour ago

uhoh

29.6k1599360

answered 1 hour ago

Xavi

1413

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Xavi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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• 
  
  
  

  
  

  
  
  
  
  
  

  
  Great answer! I added a 2nd link to one of the references because links can break over time and it's a great paper. Do you know if it was ever published somewhere from where it could be cited?
  – uhoh
  1 hour ago
  
  

  
  
  
  

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
4
down vote

If you have no other infomration about the orbit of your satellite (e.g. the orbit is circular), I believe you have to solve this problem with the Lambert's theorem assuming an elliptic transfer orbit (see Wikipedia). However, as far as I know there is no analytical solution and either numerical methods or series expansions need to be used.

In this answer I will try to introduce some aspects about this problem and give you some hints about how to approach it.

As stated by the theorem, given a gravitational parameter $mu=GM$, the time $Delta t$ required to perform a given transfer is a function of

• the semimajor axis $a$ of the orbit,


• the sum of $|vecr_1| + |vecr_2|$, and


• the length $c$ of the chord that connects the two positions (see figure below).

This can be expressed as:
$$sqrtmu Delta t = f(a,r_1+r_2,c)$$
In your case, you know $Delta t$ but you need to find $a$. You will see that there are actually two different values of semimajor axis that bring you from one position to the other in a certain $Delta t$. However, since you are describing an orbit around Earth you might encounter that only one makes sense (e.g. the other colides with the Earth surface).

As I introduced before, as far as I know no analytical solution exists to solve this problem. Some proposed numerical methods/series expansions include:

• Lagrange-Battin (1977)


• Gauss-Battin (1971)


• Battin (Elegant Algorithm) (1984) (here)

amongst others. A review of the Lambert's problem is made by D. de la Torre Sangrà and E. Fantino here (and here).

A generic Lambert solution procedure could be:

1. Compute the geometrical parameters of the transfer




2. Obtain an initial guess for the free parameter




3. Iterate on the transfer time equation until convergence




4. Compute the orbital elements

I hope it helps!

(Special credit for this answer is given to D. de la Torre Sangrà.)

share|improve this answer

edited 1 hour ago

uhoh

29.6k1599360

answered 1 hour ago

Xavi

1413

New contributor

Xavi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Great answer! I added a 2nd link to one of the references because links can break over time and it's a great paper. Do you know if it was ever published somewhere from where it could be cited?
  – uhoh
  1 hour ago
  
  

  
  
  
  

add a comment | 

up vote
4
down vote

If you have no other infomration about the orbit of your satellite (e.g. the orbit is circular), I believe you have to solve this problem with the Lambert's theorem assuming an elliptic transfer orbit (see Wikipedia). However, as far as I know there is no analytical solution and either numerical methods or series expansions need to be used.

In this answer I will try to introduce some aspects about this problem and give you some hints about how to approach it.

As stated by the theorem, given a gravitational parameter $mu=GM$, the time $Delta t$ required to perform a given transfer is a function of

• the semimajor axis $a$ of the orbit,


• the sum of $|vecr_1| + |vecr_2|$, and


• the length $c$ of the chord that connects the two positions (see figure below).

This can be expressed as:
$$sqrtmu Delta t = f(a,r_1+r_2,c)$$
In your case, you know $Delta t$ but you need to find $a$. You will see that there are actually two different values of semimajor axis that bring you from one position to the other in a certain $Delta t$. However, since you are describing an orbit around Earth you might encounter that only one makes sense (e.g. the other colides with the Earth surface).

As I introduced before, as far as I know no analytical solution exists to solve this problem. Some proposed numerical methods/series expansions include:

• Lagrange-Battin (1977)


• Gauss-Battin (1971)


• Battin (Elegant Algorithm) (1984) (here)

amongst others. A review of the Lambert's problem is made by D. de la Torre Sangrà and E. Fantino here (and here).

A generic Lambert solution procedure could be:

1. Compute the geometrical parameters of the transfer




2. Obtain an initial guess for the free parameter




3. Iterate on the transfer time equation until convergence




4. Compute the orbital elements

I hope it helps!

(Special credit for this answer is given to D. de la Torre Sangrà.)

share|improve this answer

edited 1 hour ago

uhoh

29.6k1599360

answered 1 hour ago

Xavi

1413

New contributor

Xavi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Great answer! I added a 2nd link to one of the references because links can break over time and it's a great paper. Do you know if it was ever published somewhere from where it could be cited?
  – uhoh
  1 hour ago
  
  

  
  
  
  

add a comment | 

up vote
4
down vote

up vote
4
down vote

If you have no other infomration about the orbit of your satellite (e.g. the orbit is circular), I believe you have to solve this problem with the Lambert's theorem assuming an elliptic transfer orbit (see Wikipedia). However, as far as I know there is no analytical solution and either numerical methods or series expansions need to be used.

In this answer I will try to introduce some aspects about this problem and give you some hints about how to approach it.

As stated by the theorem, given a gravitational parameter $mu=GM$, the time $Delta t$ required to perform a given transfer is a function of

• the semimajor axis $a$ of the orbit,


• the sum of $|vecr_1| + |vecr_2|$, and


• the length $c$ of the chord that connects the two positions (see figure below).

This can be expressed as:
$$sqrtmu Delta t = f(a,r_1+r_2,c)$$
In your case, you know $Delta t$ but you need to find $a$. You will see that there are actually two different values of semimajor axis that bring you from one position to the other in a certain $Delta t$. However, since you are describing an orbit around Earth you might encounter that only one makes sense (e.g. the other colides with the Earth surface).

As I introduced before, as far as I know no analytical solution exists to solve this problem. Some proposed numerical methods/series expansions include:

• Lagrange-Battin (1977)


• Gauss-Battin (1971)


• Battin (Elegant Algorithm) (1984) (here)

amongst others. A review of the Lambert's problem is made by D. de la Torre Sangrà and E. Fantino here (and here).

A generic Lambert solution procedure could be:

1. Compute the geometrical parameters of the transfer




2. Obtain an initial guess for the free parameter




3. Iterate on the transfer time equation until convergence




4. Compute the orbital elements

I hope it helps!

(Special credit for this answer is given to D. de la Torre Sangrà.)

share|improve this answer

edited 1 hour ago

uhoh

29.6k1599360

answered 1 hour ago

Xavi

1413

New contributor

Xavi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

If you have no other infomration about the orbit of your satellite (e.g. the orbit is circular), I believe you have to solve this problem with the Lambert's theorem assuming an elliptic transfer orbit (see Wikipedia). However, as far as I know there is no analytical solution and either numerical methods or series expansions need to be used.

In this answer I will try to introduce some aspects about this problem and give you some hints about how to approach it.

As stated by the theorem, given a gravitational parameter $mu=GM$, the time $Delta t$ required to perform a given transfer is a function of

• the semimajor axis $a$ of the orbit,


• the sum of $|vecr_1| + |vecr_2|$, and


• the length $c$ of the chord that connects the two positions (see figure below).

This can be expressed as:
$$sqrtmu Delta t = f(a,r_1+r_2,c)$$
In your case, you know $Delta t$ but you need to find $a$. You will see that there are actually two different values of semimajor axis that bring you from one position to the other in a certain $Delta t$. However, since you are describing an orbit around Earth you might encounter that only one makes sense (e.g. the other colides with the Earth surface).

As I introduced before, as far as I know no analytical solution exists to solve this problem. Some proposed numerical methods/series expansions include:

• Lagrange-Battin (1977)


• Gauss-Battin (1971)


• Battin (Elegant Algorithm) (1984) (here)

amongst others. A review of the Lambert's problem is made by D. de la Torre Sangrà and E. Fantino here (and here).

A generic Lambert solution procedure could be:

1. Compute the geometrical parameters of the transfer




2. Obtain an initial guess for the free parameter




3. Iterate on the transfer time equation until convergence




4. Compute the orbital elements

I hope it helps!

(Special credit for this answer is given to D. de la Torre Sangrà.)

share|improve this answer

edited 1 hour ago

uhoh

29.6k1599360

answered 1 hour ago

Xavi

1413

New contributor

Xavi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|improve this answer

share|improve this answer

edited 1 hour ago

uhoh

29.6k1599360

edited 1 hour ago

uhoh

29.6k1599360

edited 1 hour ago

uhoh

29.6k1599360

29.6k1599360

answered 1 hour ago

Xavi

1413

New contributor

Xavi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

answered 1 hour ago

Xavi

1413

answered 1 hour ago

Xavi

1413

1413

New contributor

Xavi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

New contributor

Xavi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

Xavi is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Great answer! I added a 2nd link to one of the references because links can break over time and it's a great paper. Do you know if it was ever published somewhere from where it could be cited?
  – uhoh
  1 hour ago
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Great answer! I added a 2nd link to one of the references because links can break over time and it's a great paper. Do you know if it was ever published somewhere from where it could be cited?
  – uhoh
  1 hour ago
  
  

  
  
  
  

Great answer! I added a 2nd link to one of the references because links can break over time and it's a great paper. Do you know if it was ever published somewhere from where it could be cited?
– uhoh
1 hour ago

Great answer! I added a 2nd link to one of the references because links can break over time and it's a great paper. Do you know if it was ever published somewhere from where it could be cited?
– uhoh
1 hour ago

add a comment | 

user27556 is a new contributor. Be nice, and check out our Code of Conduct.

 

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