Is the Analysis of the Space Shuttle's Orbit Correct?

[Imagin_e]

Homework Statement

Hi,
I have a textbook problem from a course that I need help with

Question:
You’ve just completed an analysis of where the Space Shuttle must be when it performs a critical maneuver. You know the shuttle is in a circular prograde orbit and has a position vector of ro=6275.396î+2007.268j +1089.857k

In 55 minutes, you predict the orbital parameters are (ER=1 Earth radius):
a= 1.0470357 ER (apogeum) , e=0.000096 (eccentricity)
i=28.5 degrees (inclination) , M=278.94688 degrees (mean anomaly)

Comments: The initial orbit is circular, but the final orbit has eccentricity different from 0, but it is small, perhaps caused by disturbances.

Is your analysis correct?
Answer with either of the two beginnings

a) The analysis can’t be correct, because ...
b) The analysis is correct, provided that periapsis has been created at ...

Homework Equations

See below

The Attempt at a Solution

Here is my attempt to solve this problem:

I first began to find the position by calculating the absolute value of the position vector ro given earlier:
r=|ro|= 6678.13657 km

Then, we know that 55*60= 3300 seconds, which can be used to calculate the period: p= 3300/1.002737 3290.989 seconds (1.0002737 is how many seconds it is in 1 solar day)

The semi major axis can be calculated by using the formula for a period: p= 2*pi*sqrt(a³/my)
(my=gravitational parameter= 398600.4418 km³/(solar sec)²

I change the equation to get the semi-major axis: a=((p/2*pi)²)³
I inserted the numbers and got a=4782.0080 km. Divide this with the Earth radius and one gets: a=0.7497 ER
This is less than a=1.0470357 ER from above, which means that the analysis isn't correct, i.e. my answer is a). But my solution is wrong and I don't know what to do. I would appreciate if someone could explain (and maybe show a solution of) this problem.

Thanks!

 

[gneill]

In 55 minutes, you predict the orbital parameters are (ER=1 Earth radius):
a= 1.0470357 ER (apogeum) , e=0.000096 (eccentricity)
i=28.5 degrees (inclination) , M=278.94688 degrees (mean anomaly)

You've labelled the quantity "a" as apogeum. Does that mean it's not the semimajor axis but rather the apogee radius of the new orbit?

Then, we know that 55*60= 3300 seconds, which can be used to calculate the period: p= 3300/1.002737 3290.989 seconds (1.0002737 is how many seconds it is in 1 solar day)

Note that 1.0002737 is not the length of a solar day in seconds. It's the ratio of the length of a solar day to a sidereal day.

I think you need to use the fact that the initial orbit was circular and that the location on the "new" orbit 55 minutes later has a given mean anomaly. What's special about the location that the maneuver took place if the orbit was initially circular? What do you know about the motion of a mean anomaly?

 

[Imagin_e]

You've labelled the quantity "a" as apogeum. Does that mean it's not the semimajor axis but rather the apogee radius of the new orbit?

Note that 1.0002737 is not the length of a solar day in seconds. It's the ratio of the length of a solar day to a sidereal day.

I think you need to use the fact that the initial orbit was circular and that the location on the "new" orbit 55 minutes later has a given mean anomaly. What's special about the location that the maneuver took place if the orbit was initially circular? What do you know about the motion of a mean anomaly?

Hi!
Yes, you're correct. It should be the semi-major axis. Hmm, I know that the mean anomaly is the angular distance the S/C has from a body that it rotates. If the eccentricity has increased, it means that the orbit is a little "flatter" than the circular one from the beginning. And what's about the location that the maneuver took place? I can't answer this actually. I guess that it is related to time since the last passage through the semi-minor axis. But I don't know how to relate this to the problem : /

 

[gneill]

When a maneuver is executed that takes an object from a circular orbit to an elliptical one the location where the change occurred is either the perigee or apogee of the new ellipse. For reference see Hohmann Transfer Orbit.

The thing about the mean anomaly is that it represents the motion of an imaginary circular orbit that has the same period as the given orbit. For an elliptical orbit the mean anomaly will alternately lag and lead the true anomaly (which varies in angular speed over the orbit), but both will complete a full 360° in the same period of time. The difference is that the motion of the mean anomaly is uniform (constant angular speed).

 

[Imagin_e]

When a maneuver is executed that takes an object from a circular orbit to an elliptical one the location where the change occurred is either the perigee or apogee of the new ellipse. For reference see Hohmann Transfer Orbit.

The thing about the mean anomaly is that it represents the motion of an imaginary circular orbit that has the same period as the given orbit. For an elliptical orbit the mean anomaly will alternately lag and lead the true anomaly (which varies in angular speed over the orbit), but both will complete a full 360° in the same period of time. The difference is that the motion of the mean anomaly is uniform (constant angular speed).

Thanks for the explanation. My initial though was that it had to do with a Hohmann transfer but I didn't know if it was relevant, I've calculated orbits after Hohmann transfers, so I know how to do it. The equations are easy and straightforward. Does this problem involve calculating the Hohmann transfer? I could calculate it with one or two orbital parameters. Is the inclination or mean anomaly may somehow useful for a conclusion, after calculating the transfer (if it involves this). It somehow feels like this is a trick question

 

[gneill]

I used the Hohmann transfer as an example to provide a mental picture of the of how the locale of the orbit change maneuver becomes the perigee or apogee of the new ellipse (with the assumption that the maneuver is effectively an "impulse" of a short time duration).

The constant speed of the mean anomaly gives you a path to finding the period of the new orbit, since you are given the mean anomaly at a certain time after the maneuver.

I'm not sure what the question needs in terms of proving that the "analysis" is correct or not. Perhaps you need to confirm that the perigee assumption is correct (correct distance for example) and that the calculated period matches that given via another method (you say you're given the semimajor axis...).

 

[Imagin_e]

I used the Hohmann transfer as an example to provide a mental picture of the of how the locale of the orbit change maneuver becomes the perigee or apogee of the new ellipse (with the assumption that the maneuver is effectively an "impulse" of a short time duration).

The constant speed of the mean anomaly gives you a path to finding the period of the new orbit, since you are given the mean anomaly at a certain time after the maneuver.

I'm not sure what the question needs in terms of proving that the "analysis" is correct or not. Perhaps you need to confirm that the perigee assumption is correct (correct distance for example) and that the calculated period matches that given via another method (you say you're given the semimajor axis...).

Okay, thanks! I will give it a try

 

[Imagin_e]

see attempt below

 

[Imagin_e]

New attempt:

Okay, we have the orbital parameters given. My guess is that we need to validate one of them (or all) . The orbit was circular but put into a little elliptical orbit (since e e is not equal to 0) , which means that we have a Hohmann transfer.
The phasing time, τ is 55 minutes (how much it takes for the transfer).
For elliptical orbit, τ= π*sqrt((a_trans)²/μ) . μ= gravitational parameter and a_{_transtrans} is radius of periapsis.
We can calculate a_{_transtrans}from this equation, and I did that and got 1.3.. Earth Radii, which does not match the given one, i.e. answer is b) . Correct or completely wrong? I haven't used the other orbital parameters, which bothers me since I must (?) use them as well. Is it enough to pick whichever orbital parameter to calculate and then give a conclusion without regarding the rest?

 

[Imagin_e]

Maybe I can calculate the rest of the parameters to strengthen my answer?

 

[gneill]

Okay, we have the orbital parameters given. My guess is that we need to validate one of them (or all) . The orbit was circular but put into a little elliptical orbit (since e e is not equal to 0) , which means that we have a Hohmann transfer.
The phasing time, τ is 55 minutes (how much it takes for the transfer).

I don't think we have a full Hohmann transfer here; there's no evidence of a second maneuver to establish a third orbit (original orbit, transfer orbit, final orbit). So if anything we have a simple orbit maneuver that results in a new orbit with a bit of eccentricity. We don't know if there was a plane change involved.

Also, something that's been bothering me about the phrasing of the original problem statement is that it's unclear whether any maneuver has been performed yet! The text reads:

You’ve just completed an analysis of where the Space Shuttle must be when it performs a critical maneuver.

Which due to sloppy grammar can be interpreted to mean that the critical maneuver is to be completed at some time in the future and the analysis was meant to determine where and when that location would be achieved on the shuttle's current orbit.

If it's true that the eccentricity can be discounted as being due to anomalous perturbations (the influence of so-called mascons, for example), then perhaps you need to only look for consistency in the given information. For example, is the semimajor axis value consistent with the given position vector? Could a touch of eccentricity account for the discrepancy? Can you find a value for the True Anomaly from the given information so that you can place the periapsis?

 

[Imagin_e]

I don't think we have a full Hohmann transfer here; there's no evidence of a second maneuver to establish a third orbit (original orbit, transfer orbit, final orbit). So if anything we have a simple orbit maneuver that results in a new orbit with a bit of eccentricity. We don't know if there was a plane change involved.

Also, something that's been bothering me about the phrasing of the original problem statement is that it's unclear whether any maneuver has been performed yet! The text reads:

Which due to sloppy grammar can be interpreted to mean that the critical maneuver is to be completed at some time in the future and the analysis was meant to determine where and when that location would be achieved on the shuttle's current orbit.

If it's true that the eccentricity can be discounted as being due to anomalous perturbations (the influence of so-called mascons, for example), then perhaps you need to only look for consistency in the given information. For example, is the semimajor axis value consistent with the given position vector? Could a touch of eccentricity account for the discrepancy? Can you find a value for the True Anomaly from the given information so that you can place the periapsis?

Thanks for the help! Yes, I was thinking the same thing about the phrasing. It is a little difficult to really understand what they want.
Okay, I can try that. Should it be for a circular orbit or for an elliptical (different equations when it comes to the elliptical one) ? If the eccentricity is almost zero, then we are dealing with orbital elements for an elliptical orbit I guess.

 

[gneill]

Thanks for the help! Yes, I was thinking the same thing about the phrasing. It is a little difficult to really understand what they want.
Okay, I can try that. Should it be for a circular orbit or for an elliptical (different equations when it comes to the elliptical one) ? If the eccentricity is almost zero, then we are dealing with orbital elements for an elliptical orbit I guess.

As a first approximation, and what should be a very good one given the tiny eccentricity, you might assume a perfectly circular orbit. How are Mean Anomaly, Eccentric Anomaly, and True Anomaly related for a circular orbit? Can you place the designated periapsis direction?

Then, do the same thing assuming an elliptical orbit with the given eccentricity. The relationships between the anomalies are more complex for this case (you will likely need to use numerical methods: iterative solutions).

 

[Imagin_e]

As a first approximation, and what should be a very good one given the tiny eccentricity, you might assume a perfectly circular orbit. How are Mean Anomaly, Eccentric Anomaly, and True Anomaly related for a circular orbit? Can you place the designated periapsis direction?

Then, do the same thing assuming an elliptical orbit with the given eccentricity. The relationships between the anomalies are more complex for this case (you will likely need to use numerical methods: iterative solutions).

Thanks for making it clear! I now have a better understanding of what to do.

 

[Imagin_e]

It really is hard when you only have to use the eccentricity and nothing else

 

[Imagin_e]

All I got so far is that the three anomalies is undefined for a circular orbit and that the longitude's equation is λ=tan^-1(r_j/r_k) . r is also the absolut value of ro. I noticed in the textbook that the period for an elliptic and circular orbit is the same, which means that a is the same for them . Then I used (2π/P)=sqrt(μ/a³) and got the value for a .
Afterwards, I found this equation that I used to calculate the true anomaly:
sqrt((2μ/r)-(μ/a))=sqrt((μ/r)*(2-(1-e²)/(1+ecos(ν)))). This equation comes from the velocity equation, which is:
v=sqrt(2*(μ/r)+ξ)=sqrt((2μ/r)-(μ/a))=sqrt((μ/r)*(2-(1-e²)/(1+ecos(ν)))). I only took the sqrt((2μ/r)-(μ/a))=sqrt((μ/r)*(2-(1-e²)/(1+ecos(ν)))) part, which should be the right approach. I substituted all the know data so far (a and e) and got the true anomaly v (about 89 degrees more or less).
Then, for the Eccentric anomaly: sin(E)=(e+cos(v))/(1+ecos(v))
And mean anomaly: M=E-e*sin(E)

Is this a valid solution?

 

[gneill]

For a circular orbit the speed is constant over the entire orbit, so the mean motion is the same as the true motion. While technically the eccentric anomaly and mean anomaly aren't defined for the circular orbit, you should be able to see that the geometry of the situation has them all merge as the eccentricity goes to zero.

Rather than a periapsis (since a circular orbit has no variation in radial distance) a reference direction can be specified arbitrarily, but often to correspond to a feature of the orbit such as its ascending node in equatorial coordinates, or an epoch of note where some maneuver or event took place.

 

[Imagin_e]

Okay, so my solution was completely wrong xD

 

[Imagin_e]

I guess that I am expecting this problem to be too difficult. It is just too hard when we only have the eccentricity. Frustrating !

 

[gneill]

I think you have to check whether the given information is consistent. For example, is the position vector's magnitude and direction consistent with an orbit with the given a, e, and i?

 

[Imagin_e]

Can I do that? I was only using the eccentricity. To be honest, that was what I did from day one after guessing that it was wrong.

 

[gneill]

Well for example, given a and e, what is the possible range of values for r? Does the given r fit in that range? For the given i, does the given r's direction lie within the allowable latitude range?

 

[Imagin_e]

It should be for a circular orbit, right?

 

[gneill]

It should be for a circular orbit, right?

Take the given predicted orbit parameters (which corresponds to an ellipse with a small eccentricity) and see if the given position vector is consistent with those parameters.

 

[Imagin_e]

Okay. Thanks for taking your time and helping me, appreciate it!

 

[Imagin_e]

Okay, so I did the following, I used the formulas for the perigee and apogee radius to get rp and ra. They are: rp=a(1-e) and ra=a(1+e) . I also calculated the absolute value of ro.
By inserting the values in the equations above I got: rp<ro<ra , the difference is a couple of meters. This makes it consistent, so this is good news.
I also took the rp and ra values to see if it gave a close value to a : a=(rp+ra)/2 , and it gave something VERY close. So far so good.
I then calculated the velocity it has with the given a and my ro, which is about 7.7 km/s more or less. The final step was to calculate the true anomaly (with an equation that I mentioned in a previous comment): v²=((μ/r)*(2-(1-e²)/(1+ecos(v))) - -> v is around 89 degrees .
I then calculated semi-parameter: p=a(1-e²) . And lastly, I used this equation: r=p/(1+ecos(v)) and got something VERY close to ro . In conclusion, it all seems to work and the analysis is correct. But my guts tells me that I need to use the given M and altitude as well. Is it necessary when I already see that it seems to be correct?

 

[gneill]

Okay, so you haven't found any glaring reason why the analysis would not be correct. You should confirm that the given position vector is consistent with the given inclination (it looks like it should be, given the "k" component of the position vector).

Now, if everything looks okay with the analysis you are faced with finding a location for the periapsis. If you have a true anomaly for the given position vector then you should be able to "locate" the periapsis.

 

[Imagin_e]

Thank you for the help!