Sunday, July 2, 2017

Alternate ITS missions part 3

This post got too long again, so I'm splitting it into yet another section. This will cover how to calculate interplanetary trajectories.

All orbits are conic sections. circles, ellipses, parabolas, or hyperbolas. Circles and ellipses are the orbits that a spacecraft is in when it is in a stable orbit around a planet. Parabolas are when the spacecraft is going at exactly escape velocity, and hyperbolas are when the spacecraft is going faster than escape velocity.

For ellipses (assuming circles are a type of ellipse) the planet the spacecraft is orbiting (or the primary) is always at one of the two foci. The major and minor axes are the longest and the shortest lines that can be drawn through the ellipse, respectively. The semi-major or minor axes are half of the major or minor axes.

The periapsis is the lowest point on an orbit, and the apoapsis is the highest. They are always 180 degrees apart. The closer to the periapsis, the faster the satellite travels, and the closer to apoapsis, the slower.

A quick aside on nodes: nodes are the points at which the orbit passes the plane of the equator or ecliptic. The ascending node is the one the spacecraft passes travelling from south to north, and the descending node is the one the spacecraft passes travelling from north to south.

Eccentricity is the distance between the foci divided by the length of the major axis (always between 0 and 1 except for hyperbolas), and is essentially a measure of how "squished" the ellipse is. 0 is a circle, 1 is parabola, > 1 is a hyperbola.

Inclination is how tilted the orbit is from the equatorial plane (or the ecliptic) of the primary. An inclination of less than 90 degrees indicates a orbit in the same direction as the spin of the primary, or prograde, an inclination of greater than 90 degrees means that the orbit is in the opposite direction of the spin, or retrograde.

I'm ignoring how to calculate these, as that would make this post much longer.

So, let's imagine you want to go from one circular orbit to another. What is the most efficient way of doing this? The answer is a Hohmann transfer.

Image from Leafnode

A Hohmann transfer is the most efficient way to go from a coplanar orbit to another. The image shows a transfer with two circular orbits, however it is the most efficient for any two coplanar orbits where one cannot be reached from the other with only a single burn. For example, in the image, 1 and 3 require a hohmann transfer, while 1 and 2 or 2 and 3 only need a single burn.

The Delta-V required for a hohmann transfer is the sum of the Delta-V required for the burn to go from 1 to 2, and the Delta-V required to go from 2 to 3. It is the same in the opposite direction (3 to 1 = 3 to 2 + 2 to 1), and the burns are the same, except with the spacecraft burning its engines in the opposite direction.

The transfer orbit between two planets is the same thing, with the transfer orbit going from one planet to another, except for the fact that you start in orbit around one of those planets.

Transfer windows are times when the orbits of the planets align such that a spacecraft on a transfer orbit reaches apoapsis (or periapsis) at the same time as the destination planet moves through the part of the orbit that intersects the transfer orbit.

It isn't very important to calculate transfer windows for this post, as they do occur regularly for most planets.

Delta-V for an interplanetary transfer is less than you might expect, as the Oberth effect comes into play on the escape burn. The Oberth effect is, in short, an effect that makes Delta-V count for more when near a large mass. It is why the Delta-V required for a hohmann transfer to GEO is more than the Delta-V required for escape velocity.

The Oberth effect is the effect of the extra velocity of a circular orbit around earth has compared to a circular orbit around the sun at earth's altitude. As the satellite orbits around the earth, it traces out a spiral pattern relative to the sun. When it is orbiting in the same direction around the earth that the earth is orbiting around the sun, the velocity relative to the sun is increased. It isn't as simple as adding the velocities, but we'll get into that later.

On arrival at the destination planet, the Delta-V can be further reduced with aerobraking or aerocapture. This is where the you use the atmosphere of the planet you are arriving at to slow down or speed up without using any fuel. This only works on planets with atmospheres, though, and you cannot capture into a circular orbit without a burn, although it is much less with most of the velocity bled off already.

A very useful equation is the Vis-visa equation:
v^2 = GM(2/r - 1/a)
Where v is the speed of the spacecraft at any point in a orbit, G is the gravitational constant, M is the mass of the primary, r is the distance between the spacecraft and the primary at the point at which you wish to know velocity, and a is the semi-major axis.

The calculations for hohmann transfer orbits assume instantaneous impulses, which lessens delta-v requirements. To compensate, I will multiply the estimate by 1.1.

To calculate Delta-V requirements, you simply need to get the difference in velocities between the orbits at the correct points. This is why the Vis-visa equation is so useful.

We apply this to a hohmann transfer from one planet to another, getting the velocities of the transfer at apoapsis and periapsis. Then, we calculate the velocity of the starting orbit, and calculate the required Delta-V to go to the transfer orbit, accounting for the Oberth effect.

Reversing the rocket equation will then let us calculate payload.

Earth to Venus example

Earth has a mass of 5.9723 * 10^24 kg. It has a semi-major axis of 149.6 * 10^9 meters. Its orbital radius goes from 147.09 * 10^9 m to 152.1 * 10^9 m.

Venus has a mass of 4.8675 * 10^24 kg. It has a semi-major axis of 108.21 * 10^9 m. Its orbital radius goes from 107.48 * 10^9 m to 108.94 * 10^9 m.

The sun has a mass of 1988500 * 10^24 kg.

A transfer orbit between them depends on the position of the planets at departure and arrival. For this example, the position of Venus does not matter, as aerobraking will work at either of those speeds.

Earth should be at apoapsis, because in this case we want to lose, rather than gain, velocity as Venus is in a lower orbit than Earth.

Let's begin. First, the velocity of Earth around the Sun at apoapsis. We don't really need to calculate this, as it is well known: 29290 m/s.

The transfer orbit is an elliptical orbit with its apoapsis at Earth, and its periapsis at Venus. This gives a semi-major axis of:
(Radius of Earth's orbit at departure + Radius of Venus' orbit at arrival)/2.

Or (152.1 * 10^9 m + 108.94 * 10^9 m)/2 = 130.52 * 10^9 m.

Velocity at apoapsis:
v^2 = 1.327 * 10^20 (2/152.1 * 10^9 - 1/130.52 * 10^9)
v = 26985 m/s

At periapsis:
v^2 = 1.327 * 10^20 (2/108.94 * 10^9 - 1/130.52 * 10^9)
v = 37676 m/s

The velocity of the spaceship in Earth orbit can be calculated from the altitude, and the altitude is as low as possible to take maximum advantage of the Oberth effect. I'm going to guess around 200 km.

v^2 = 3.986 * 10^14 (2/6578000 - 1/6578000)
v = 7784 m/s.

To figure out how much of an advantage the Oberth effect gives us, we use something called hyperbolic excess velocity. This is how much extra velocity the spaceship will have when it reaches infinite distance on a hyperbolic escape orbit. If you reach exactly escape velocity, HEV will be zero. The desired HEV is equal to the orbital velocity of the primary - the velocity of the transfer orbit at the point where it begins so when the spacecraft escapes the primary's gravity, the remaining velocity puts it on the transfer orbit.

The equation for HEV is HEV^2 = v^2 - ve^2

Our desired HEV is 29290 - 26985 = 2305.

Rearranging the equation:
v^2 =2305^2 + 11008^2.
(as escape velocity is equal to the velocity of a circular orbit at that altitude times 2)

So v = 11246 m/s.

11246 - 7784 = 3462.

Here you can see the clear advantage of this over a escape burn (3224) + a burn outside of Earth's sphere of influence to put the spacecraft on a transfer orbit (2305), which would be 2067 m/s more.

Venus is inclined 3.39 degrees off of Earth, however this amounts to at most a few m/s if it is corrected for at the very start of the transfer. For this kind of loose approximation, we can ignore this.

So, multiplying by 1.1, we have a Delta-V of 3808 m/s.

Next post, we will calculate transfers for other missions, including approximate landing Delta-V, and use the reverse rocket equation to figure out how much payload can be carried to them.