The purpose of this post is to record a number of equations that are useful in designing transfer manoeuvres that are relevant for the reduction of delta-V requirements for both:
a. the 'Begin Game' problem - i.e., using gravity assists to boost the spacecraft's kinetic energy so as to reduce the overall mission delta-V requirements (e.g. leaving Earth and setting up a transfer trajectory to, say, Jupiter or Saturn);
b. the 'End Game' problem - i.e, using gravity de-assist to shed kinetic energy upon arrival at the target planetary system in order to slow the spacecraft sufficiently to permit final orbit insertion around one of the moons).
Such transfer manoeuvres are generally called VILTs ("V-ininity Leveraged Transfers"). They form the basis of most modern real-world mission planning. Without them, many of the missions currently underway (or on the drawing board) simply wouldn't be possible.
This post will have a sequel which will apply the equations presented in this post to perform a 'first principles' calculation of a real VILT sequence that can be implemented in Orbiter 2010 using TransX and IMFD.
And there will be a sequel to that post where a VILT sequence to reduce the delta-V requirements of getting to Jupiter will be designed. Moreover, this first principles design of a VILT will be shown to replicate the essential features of the Begin Game sequence of the current Juno mission to Jupiter.
But, to begin....
A very quick background to VILTs
VILTS are widely used by NASA and ESA in designing missions to planetary systems - e.g., Jupiter, Saturn and Mercury. They employ a specialised a specialised class of manoeuvre whose purpose is to efficiently increase (or decrease) the encounter velocity of a spacecraft during a fly-by of a gravitating body.
A VILT trajectory usually consists of splicing two half ellipses together connecting an encounter with one gravitating body with another gravitating body. By performing a small burn at the apoapsis of those two half-ellipses, the encounter velocity of the craft at the next encounter with than body can be made to be substantially larger or smaller. By a judicious choice of orbits, one can use the gravity assists to add energy to the craft (a gravity assist), or to slow it down (gravity de-assist).
The equations
Although the focus of this work is on the design and implementation of a single VILT, the broader motivation for this post is to develop the mathematical machinery for designing more complicated VILT sequences - e.g., VEEGA-like (Venus-Earth-Earth-Gravity-Assists) sequences; or the gravity de-assist sequences for approaching Jovian or Saturnian moons.
In this respect, a useful tool for designing a complicated sequence of VILTs is something called the Tisserand Graph. Without going into a lot of unnecessary detail at this stage, all that one needs to know about the Tisserand Graph for now is that all elliptical orbits around the Sun can be represented as points on a 2 dimensional graph - spanned by the apoapsis radius along the x-axis; and the periapsis radius on the y-axis. To make use of the Tisserand Graph, it is helpful to calculate many of the quantities associated with elliptical orbits as functions of these two parameters, the apoapsis radius and the periapsis radius.
In this note, I am going to work with standard Keplerian elliptical orbits. The archetypal image that one should have in mind is that of a spacecraft in orbit around the Sun (with no significant perturbations from any other bodies in the Solar System). Let's call the periapsis radius [MATH]r_p[/MATH], and the apoapsis radius [MATH]r_a[/MATH]. For convenience let's consider a coordinate system in which the elliptical orbit lies entirely in the x-y plane with the central gravitating body (e.g., the Sun) located at the origin; and let's align the ellipse so that both the periapsis and the apoapsis lie on the x-axis with the periapsis to the right-hand side of the origin. Finally. let's suppose that the spacecraft moves around the elliptical orbit in an anti-clockwise direction (when looking 'down' onto the x-y axis from along the positive z direction).
With this in mind, we can immediately write down a few useful (and probably well known expressions):
1. The semi-major axis, [MATH]a[/MATH]
[MATH]a = \frac{1}{2}\,\left(r_{a}+r_{p}\right)[/MATH]
2. The orbital eccentricity, [MATH]e[/MATH]
[MATH]e = \frac{r_{a}+r_{p}}{r_{a}-r_{p}}[/MATH]
3. The specific orbital angular momentum, [MATH]h[/MATH]
[MATH]h = \sqrt{2\,\mu\,\frac{r_{a}\,r_{p}}{r_{a}+r_{p}}}[/MATH]
4. The specific orbital energy, [MATH]\epsilon[/MATH]
[MATH]\epsilon = -\frac{\mu}{r_{a}+r_{p}}[/MATH]
5. The orbital speed at apoapsis, [MATH]v_a[/MATH]
[MATH]v_{a} = \sqrt{\frac{r_{p}}{r_{a}}\,\frac{2\,\mu}{r_{a}+r_{p}}}[/MATH]
6. The orbital speed at periapsis, [MATH]v_p[/MATH]
[MATH]v_{p} = \sqrt{\frac{r_{a}}{r_{p}}\,\frac{2\,\mu}{r_{a}+r_{p}}}[/MATH]
Now, imagine that the spacecraft is at a point in its orbit where its orbital radius is [MATH]r[/MATH]. It is clear that if the periapsis is the point of closest approach to the central gravitating body; and the apoapsis is the point farthest away it must be true that:
[MATH]r_p \le r \le r_a [/MATH]
At this point in the orbit, we can define two directions - [MATH]\hat{r}[/MATH] and [MATH]\hat{\theta}[/MATH]. The first of these, [MATH]\hat{r}[/MATH], being in the direction along the 'outward' pointing radius vector away from the gravitating body along a line connecting the origin to the point on the orbit. The second, [MATH]\hat{\theta}[/MATH], is at right-angles to this, pointing roughly in the direction of the spacecraft's motion. (N.B. [MATH]\hat{\theta}[/MATH] points roughly in the prograde direction, but most of the time, the prograde direction and the [MATH]\hat{\theta}[/MATH] direction are NOT the same). Then, with these comments in mind, we can write some less well-known equations:
7. The speed of the craft in the [MATH]\hat{\theta}[/MATH]-direction, [MATH]v_{\theta}[/MATH]
[MATH]v_{\theta} = \frac{1}{r}\,\sqrt{2\,\mu\,\frac{r_{a}\,r_{p}}{r_{a}+r_{p}}}[/MATH]
8. The speed of the craft in the [MATH]\hat{r}[/MATH]-direction, [MATH]v_{r}[/MATH]
[MATH]v_{r} = \pm\frac{1}{r}\,\sqrt{2\,\mu\,\frac{\left(r_{a}-r\right)\,\left(r-r_{p}\right)}{r_{a}+r_{p}}}[/MATH]
Here, in this last equation, we take the '+' sign if the craft is on the 'outward' leg of its elliptical orbit on its path from periapsis to apoapsis; and we take the '-' sign if the craft is on the 'inward' leg of its orbit from apoapsis back to periapsis again. To complete this section, we can write the the speed of the craft is:
9. The speed of the craft, [MATH]v[/MATH]
[MATH]v = \sqrt{v_{\theta}^{2}+v_{r}^{2}} = \sqrt{2\,\mu\,\left(\frac{1}{r}-\frac{1}{r_{a}+r_{p}}\right)}[/MATH]
10. The true anomaly, [MATH]\nu[/MATH]
[MATH]\cos\nu = \frac{2\,r_{a}\,r_{p}-r\,\left(r_{a}+r_{p}\right)}{r\,\left(r_{a}-r_{p}\right)}[/MATH]
[MATH]\sin\nu = \pm2\,\frac{\sqrt{r_{a}\,r_{p}\,\left(r_{a}-r\right)\,\left(r-r_{p}\right)}}{r_{c}\,\left(r_{a}-r_{p}\right)}[/MATH]
[MATH]\nu = \pm2\tan^{-1}\left(\sqrt{\frac{r_{a}}{r_{p}}\,\frac{r-r_{p}}{r_{a}-r}}\right)[/MATH]
11. The eccentric anomaly, [MATH]E[/MATH]
[MATH]\cos E = \frac{r_{a}+r_{p}-2\,r}{r_{a}-r_{p}}[/MATH]
[MATH]\sin E = \pm2\,\frac{\sqrt{\left(r_{a}-r\right)\,\left(r-r_{p}\right)}}{r_{a}-r_{p}}[/MATH]
[MATH]E = \pm2\,\tan^{-1}\left(\sqrt{\frac{r-r_{p}}{r_{a}-r}}\right)[/MATH]
12. The mean anomaly, [MATH]M[/MATH]
[MATH]M = \pm2\left(\tan^{-1}\left(\sqrt{\frac{r-r_{p}}{r_{a}-r}}\right)-\frac{\sqrt{\left(r_{a}-r\right)\left(r-r_{p}\right)}}{r_{a}+r_{p}}\right)[/MATH]
To this point, the equations are valid for all elliptical orbits. And as 'interesting' as these expressions are, the purpose of expressing them in this fashion (as functions of [MATH]r_a[/MATH] and [MATH]r_p[/MATH]) is to use them to design gravity assist and de-assist sequences, we now consider the special case where the spacecraft encounters a planet (or a moon) at some point in its elliptical orbit. To simplify things, we assume that the encountered planet (or moon) is in a circular orbit at radius [MATH]r_c[/MATH] around the primary and that the spacecraft is coplanar with that body's orbit. The we can write:
13. The orbital speed of the planet/moon about the primary, [MATH]v_c[/MATH]
[MATH]v_{c} = \sqrt{\frac{\mu}{r_{c}}}[/MATH]
where [MATH]\mu[/MATH] is the gravitational parameter 'GM' for the primary gravitating body.
14. The speed of encounter (hyperbolic excess velocity) of the craft with the planet/moon, [MATH]v_{\infty}[/MATH]
[MATH]v_{\infty} = v_{c}\,\sqrt{3-\frac{2\,r_{c}}{r_{a}+r_{p}}-2\,\sqrt{\frac{2\,r_{a}\,r_{p}}{r_{c}\left(r_{a}+r_{p}\right)}}}[/MATH]
15. The angular rotation of the planet/moon-centric velocity vector of the spacecraft because of the gravitational enounetr with the planet/moon, [MATH]\delta[/MATH]
[MATH]\delta = \pi - 2\,\cos ^{-1}\left(\frac{\mu_p }{\mu_p +\rho\,v_{\infty }^2}\right)[/MATH]
where [MATH]\mu_p[/MATH] is the gravitation constant or the planet/moon; and [MATH]\rho[/MATH] is the periapsis radius of the planet/moon hyperbolic encounter.
Taken together, these equations furnish us with the basic mathematical tools needed to calculate VILTs (at least using the linked-conics model). In the next post in this series, we will work out how these equations can be used to do those calculations.
a. the 'Begin Game' problem - i.e., using gravity assists to boost the spacecraft's kinetic energy so as to reduce the overall mission delta-V requirements (e.g. leaving Earth and setting up a transfer trajectory to, say, Jupiter or Saturn);
b. the 'End Game' problem - i.e, using gravity de-assist to shed kinetic energy upon arrival at the target planetary system in order to slow the spacecraft sufficiently to permit final orbit insertion around one of the moons).
Such transfer manoeuvres are generally called VILTs ("V-ininity Leveraged Transfers"). They form the basis of most modern real-world mission planning. Without them, many of the missions currently underway (or on the drawing board) simply wouldn't be possible.
This post will have a sequel which will apply the equations presented in this post to perform a 'first principles' calculation of a real VILT sequence that can be implemented in Orbiter 2010 using TransX and IMFD.
And there will be a sequel to that post where a VILT sequence to reduce the delta-V requirements of getting to Jupiter will be designed. Moreover, this first principles design of a VILT will be shown to replicate the essential features of the Begin Game sequence of the current Juno mission to Jupiter.
But, to begin....
A very quick background to VILTs
VILTS are widely used by NASA and ESA in designing missions to planetary systems - e.g., Jupiter, Saturn and Mercury. They employ a specialised a specialised class of manoeuvre whose purpose is to efficiently increase (or decrease) the encounter velocity of a spacecraft during a fly-by of a gravitating body.
A VILT trajectory usually consists of splicing two half ellipses together connecting an encounter with one gravitating body with another gravitating body. By performing a small burn at the apoapsis of those two half-ellipses, the encounter velocity of the craft at the next encounter with than body can be made to be substantially larger or smaller. By a judicious choice of orbits, one can use the gravity assists to add energy to the craft (a gravity assist), or to slow it down (gravity de-assist).
The equations
Although the focus of this work is on the design and implementation of a single VILT, the broader motivation for this post is to develop the mathematical machinery for designing more complicated VILT sequences - e.g., VEEGA-like (Venus-Earth-Earth-Gravity-Assists) sequences; or the gravity de-assist sequences for approaching Jovian or Saturnian moons.
In this respect, a useful tool for designing a complicated sequence of VILTs is something called the Tisserand Graph. Without going into a lot of unnecessary detail at this stage, all that one needs to know about the Tisserand Graph for now is that all elliptical orbits around the Sun can be represented as points on a 2 dimensional graph - spanned by the apoapsis radius along the x-axis; and the periapsis radius on the y-axis. To make use of the Tisserand Graph, it is helpful to calculate many of the quantities associated with elliptical orbits as functions of these two parameters, the apoapsis radius and the periapsis radius.
In this note, I am going to work with standard Keplerian elliptical orbits. The archetypal image that one should have in mind is that of a spacecraft in orbit around the Sun (with no significant perturbations from any other bodies in the Solar System). Let's call the periapsis radius [MATH]r_p[/MATH], and the apoapsis radius [MATH]r_a[/MATH]. For convenience let's consider a coordinate system in which the elliptical orbit lies entirely in the x-y plane with the central gravitating body (e.g., the Sun) located at the origin; and let's align the ellipse so that both the periapsis and the apoapsis lie on the x-axis with the periapsis to the right-hand side of the origin. Finally. let's suppose that the spacecraft moves around the elliptical orbit in an anti-clockwise direction (when looking 'down' onto the x-y axis from along the positive z direction).
With this in mind, we can immediately write down a few useful (and probably well known expressions):
1. The semi-major axis, [MATH]a[/MATH]
[MATH]a = \frac{1}{2}\,\left(r_{a}+r_{p}\right)[/MATH]
2. The orbital eccentricity, [MATH]e[/MATH]
[MATH]e = \frac{r_{a}+r_{p}}{r_{a}-r_{p}}[/MATH]
3. The specific orbital angular momentum, [MATH]h[/MATH]
[MATH]h = \sqrt{2\,\mu\,\frac{r_{a}\,r_{p}}{r_{a}+r_{p}}}[/MATH]
4. The specific orbital energy, [MATH]\epsilon[/MATH]
[MATH]\epsilon = -\frac{\mu}{r_{a}+r_{p}}[/MATH]
5. The orbital speed at apoapsis, [MATH]v_a[/MATH]
[MATH]v_{a} = \sqrt{\frac{r_{p}}{r_{a}}\,\frac{2\,\mu}{r_{a}+r_{p}}}[/MATH]
6. The orbital speed at periapsis, [MATH]v_p[/MATH]
[MATH]v_{p} = \sqrt{\frac{r_{a}}{r_{p}}\,\frac{2\,\mu}{r_{a}+r_{p}}}[/MATH]
Now, imagine that the spacecraft is at a point in its orbit where its orbital radius is [MATH]r[/MATH]. It is clear that if the periapsis is the point of closest approach to the central gravitating body; and the apoapsis is the point farthest away it must be true that:
[MATH]r_p \le r \le r_a [/MATH]
At this point in the orbit, we can define two directions - [MATH]\hat{r}[/MATH] and [MATH]\hat{\theta}[/MATH]. The first of these, [MATH]\hat{r}[/MATH], being in the direction along the 'outward' pointing radius vector away from the gravitating body along a line connecting the origin to the point on the orbit. The second, [MATH]\hat{\theta}[/MATH], is at right-angles to this, pointing roughly in the direction of the spacecraft's motion. (N.B. [MATH]\hat{\theta}[/MATH] points roughly in the prograde direction, but most of the time, the prograde direction and the [MATH]\hat{\theta}[/MATH] direction are NOT the same). Then, with these comments in mind, we can write some less well-known equations:
7. The speed of the craft in the [MATH]\hat{\theta}[/MATH]-direction, [MATH]v_{\theta}[/MATH]
[MATH]v_{\theta} = \frac{1}{r}\,\sqrt{2\,\mu\,\frac{r_{a}\,r_{p}}{r_{a}+r_{p}}}[/MATH]
8. The speed of the craft in the [MATH]\hat{r}[/MATH]-direction, [MATH]v_{r}[/MATH]
[MATH]v_{r} = \pm\frac{1}{r}\,\sqrt{2\,\mu\,\frac{\left(r_{a}-r\right)\,\left(r-r_{p}\right)}{r_{a}+r_{p}}}[/MATH]
Here, in this last equation, we take the '+' sign if the craft is on the 'outward' leg of its elliptical orbit on its path from periapsis to apoapsis; and we take the '-' sign if the craft is on the 'inward' leg of its orbit from apoapsis back to periapsis again. To complete this section, we can write the the speed of the craft is:
9. The speed of the craft, [MATH]v[/MATH]
[MATH]v = \sqrt{v_{\theta}^{2}+v_{r}^{2}} = \sqrt{2\,\mu\,\left(\frac{1}{r}-\frac{1}{r_{a}+r_{p}}\right)}[/MATH]
10. The true anomaly, [MATH]\nu[/MATH]
[MATH]\cos\nu = \frac{2\,r_{a}\,r_{p}-r\,\left(r_{a}+r_{p}\right)}{r\,\left(r_{a}-r_{p}\right)}[/MATH]
[MATH]\sin\nu = \pm2\,\frac{\sqrt{r_{a}\,r_{p}\,\left(r_{a}-r\right)\,\left(r-r_{p}\right)}}{r_{c}\,\left(r_{a}-r_{p}\right)}[/MATH]
[MATH]\nu = \pm2\tan^{-1}\left(\sqrt{\frac{r_{a}}{r_{p}}\,\frac{r-r_{p}}{r_{a}-r}}\right)[/MATH]
11. The eccentric anomaly, [MATH]E[/MATH]
[MATH]\cos E = \frac{r_{a}+r_{p}-2\,r}{r_{a}-r_{p}}[/MATH]
[MATH]\sin E = \pm2\,\frac{\sqrt{\left(r_{a}-r\right)\,\left(r-r_{p}\right)}}{r_{a}-r_{p}}[/MATH]
[MATH]E = \pm2\,\tan^{-1}\left(\sqrt{\frac{r-r_{p}}{r_{a}-r}}\right)[/MATH]
12. The mean anomaly, [MATH]M[/MATH]
[MATH]M = \pm2\left(\tan^{-1}\left(\sqrt{\frac{r-r_{p}}{r_{a}-r}}\right)-\frac{\sqrt{\left(r_{a}-r\right)\left(r-r_{p}\right)}}{r_{a}+r_{p}}\right)[/MATH]
To this point, the equations are valid for all elliptical orbits. And as 'interesting' as these expressions are, the purpose of expressing them in this fashion (as functions of [MATH]r_a[/MATH] and [MATH]r_p[/MATH]) is to use them to design gravity assist and de-assist sequences, we now consider the special case where the spacecraft encounters a planet (or a moon) at some point in its elliptical orbit. To simplify things, we assume that the encountered planet (or moon) is in a circular orbit at radius [MATH]r_c[/MATH] around the primary and that the spacecraft is coplanar with that body's orbit. The we can write:
13. The orbital speed of the planet/moon about the primary, [MATH]v_c[/MATH]
[MATH]v_{c} = \sqrt{\frac{\mu}{r_{c}}}[/MATH]
where [MATH]\mu[/MATH] is the gravitational parameter 'GM' for the primary gravitating body.
14. The speed of encounter (hyperbolic excess velocity) of the craft with the planet/moon, [MATH]v_{\infty}[/MATH]
[MATH]v_{\infty} = v_{c}\,\sqrt{3-\frac{2\,r_{c}}{r_{a}+r_{p}}-2\,\sqrt{\frac{2\,r_{a}\,r_{p}}{r_{c}\left(r_{a}+r_{p}\right)}}}[/MATH]
15. The angular rotation of the planet/moon-centric velocity vector of the spacecraft because of the gravitational enounetr with the planet/moon, [MATH]\delta[/MATH]
[MATH]\delta = \pi - 2\,\cos ^{-1}\left(\frac{\mu_p }{\mu_p +\rho\,v_{\infty }^2}\right)[/MATH]
where [MATH]\mu_p[/MATH] is the gravitation constant or the planet/moon; and [MATH]\rho[/MATH] is the periapsis radius of the planet/moon hyperbolic encounter.
Taken together, these equations furnish us with the basic mathematical tools needed to calculate VILTs (at least using the linked-conics model). In the next post in this series, we will work out how these equations can be used to do those calculations.
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