OF ROCKS & ROCKETS: Launching rockets from PH

By Lanz Lagman

The Philippines will soon have its own space agency.

While one of its more urgent objectives would be to design and make satellites that help us in various ways – such as improving the Internet speed, enhancing territorial defense, and monitoring different environmental activities – one of its long-term objectives would be the construction of several rocket launching sites for our own rockets or have other countries launch theirs at our sites.

This would prove economical for us due to our strategic location. Our country is located near the equator, and launching rockets near or at it is cost-efficient. But why is this the case?

First of all, sending rockets to space is no easy task; this is why the term ‘rocket science’ is used to describe anything that’s hard to understand.

Aside from the specifics of the mission required, scientists have to consider a lot of factors in the design of the rocket: such as the materials used in its construction, how much fuel it carries, how efficiently the fuel is burned, and how efficient the engines are. These are just some of them.

But what could be potentially more difficult than designing rockets? It would be making rocket launches cheaper.

We may consider making more cost-efficient fuel or modifications for engines and fuel tanks, but these would surely take more time. Coming up with improvements for these two could not be done overnight. Why not launch rockets near or from the equator? And how does that work?

We have to look at two scenarios: the velocity of the rocket before the launch, and the final velocity as the rocket gets sent to its designated orbit.

By finding the change in velocity due to this maneuver or delta V, we could see how much rocket fuel is needed depending on where the rocket is located, with respect to the equator.

Rocket at Rest: Initial Velocity

It might seem that the rocket has zero initial velocity, since it sits idly with its fuel on its launch pad. However, since that launch pad is also sitting on a spinning Earth, the rocket is already moving about the Earth’s axis. The way we could describe how it moves depends on its latitude and its distance from the Earth’s equator and axis.

2.jpg

As we can see from this illustration, Φ is the latitude and RE is the Earth’s radius while ρ is the general distance from the axis of rotation. It’s easily seen that ρ is at its maximum at RE. Depending on what altitude the rocket sits, its velocity (tangential velocity) would be:

3

In this equation, ω is the angular velocity or how much it takes for a spinning object to complete a full spin. Our planet is a big ball of rock with some water on it, and since it takes around 24 hours to complete a full spin, ω would have a very small value, as we will see later. Together with ρ, the tangential velocity vt expands to:

4

Now that we know how the rocket moves with respect to its latitude while resting on its launch pad, let’s look at how it moves as it reaches its designated altitude where it will proceed to orbit.

Rocket at Orbit: Orbital Velocity
 
Now that our rocket has launched, consumed its fuel, and has entered in this case a circular orbit, it now experiences centripetal force.
This is the force responsible for making the rocket move in a circular manner, now independent from its previous attachment on the launch pad. It also happens that this centripetal force is gravity, which we represent as:
In this equation, mr will be the rocket’s mass after it has consumed its fuel, a would be the altitude of the orbit, and the orbital radius is just equal to the Earth’s radius plus a. The orbital velocity vr would be the velocity at RE + a. Solving for vr, we get:
Now that we have the equations for the initial and the final velocity, the delta v would simply be the difference between them. Then what would be the use of delta v? Enter the rocket equation.
Rocket Equation
 
The rocket equation would be:
Where delta v is obviously Δv; ve would be the exhaust velocity, or the measure on how fast it is, mT is the rocket’s mass with fuel, and mr would be the fuel-less mass of the rocket.
The exhaust velocity is also the product of the Earth’s acceleration due to gravity, go and the specific impulse of the rocket, Isp, change in momentum per unit mass of fuel. Expanding the rocket equation using the previous equations lead to:
After simplifying this equation, solving for the fuel mass mf leads to the expression of Φ as a function of mf:
Let’s say we plan to launch a rocket, somewhat similar to Space X’s Falcon 9 v1.1, weighing 105,000 kg without its first stage fuel, and with a specific impulse of 280 s at sea level towards an altitude of 400 km, a typical Low-Earth orbit and from a launch pad with a latitude of 30.0000°N, near Japan’s Tanegashima Space Center’s latitude.
Together with these important values:
Radius of the Earth, RE = 6,371,000 m;
Gravitational constant, G = 6.674 x 10-11;
Mass of the Earth, ME = 5.972 x 1024kg;
Angular velocity of Earth, ω = 7.292 x 10-5 rad/s,
The amount of first-stage fuel that would be burned would be 1,352,675.80 kg.
If we launch it from a hypothetical launch pad from Paulau in Sarangani with the coordinates of 5.4431°N, 128.4859°E; the amount of fuel burned would only be 1,322,293.30 kg.
 
That’s around 30,382.49 kg less fuel!

Paulau, Sarangani.
Image credit: Google Earth.
SpaceX’s Falcon 9 launches.
Image credit: SpaceX

Additionally, the minimum inclination of the orbit is equal to the latitude of the launch site. This means that launching from a latitude nearer the equator means the rocket needs to turn less in order to change how inclined its orbit is with respect to the equator, therefore less fuel would be used.

Take note that for the sake of simplicity, we didn’t consider several factors, such as atmospheric drag and overall propulsive efficiency.

This means that for our scenario, our rocket traveled unhindered in its journey, and all the consumed fuel was used to propel the rocket. Nothing was wasted as heat energy.
Finally, we also assumed that after the separation of its first-stage component, it would immediately orbit. in reality, rockets break all their stages until only the payload remains, and is therefore the only one that proceeds to orbit.
Using the same example, here’s a graph for a better representation of the relationship between the latitude of the launch pad and the required fuel.
Due to the logarithmic nature of our equation, we can see that there’s very little difference in the fuel spent between rockets launched near the equator, compared to when they’re located much farther.
Here’s a map of several launch sites. Indonesia, Maldives, and Brazil have the positional advantage when it comes to their proximity to the equator.
Image credit: Reddit
What if we launch the same rocket towards the same altitude, but from different launch sites? We could compare how much fuel would be spent using this graph:

Now, regarding the other significant advantage of launching from the equator: the minimum inclination of an orbit is equal to the latitude of the launch site. If a rocket is launched from north of the equator, it has to go southward until it reaches the same distance south of the equator as it started north of it. Only then will it arc back north.

This inclination can be reduced in-flight with a plane-change maneuver or in simpler words, turning the rocket. However, it’s terribly expensive in fuel expenditure. If we want an equatorial orbit (zero inclination), we could either launch from the equator or burn a lot of extra fuel for a plane change maneuver.

If we want a higher inclination, that’s easy; we just point the rocket away from the equator into the desired orbital plane. This makes launching as near as possible to the equator favorable.

Together with the type 4 climate prevalent in the Southern Philippines (with rain being evenly distributed and the weather being stable overall), we do have potential launch sites.

As the days of the Philippine Space Agency nears, we could construct several of these sites, modify them, and invite note only our ASEAN neighbours, but also Japan, China and private companies like SpaceX to launch their rockets here.
Perhaps private space companies could build their facilities here, and the income generated could be used to improve our infrastructure and technology.
These are just some of the benefits of having a homegrown space program. Great times ahead, indeed.

REFERENCES:

1. SpaceX Falcon 9 v1.1 Data Sheet. (2016, January 17). From http://www.spacelaunchreport.com/falcon9v1-1.html
2. Falcon 9 Launch Vehicle Payload User’s Guide (2015, October 21). From http://www.spacex.com/sites/spacex/files/falcon_9_users_guide_rev_2.0.pdf
3. List of Rocket Launch Sites, from http://en.wikipedia.org/wiki/List_of_rocket_launch_sites

Featured photo by NASA
The above article is originally titled “Why It Might Be A Good Idea To Launch Rockets From The Philippines” and was first published in The Mind Museum official blog.
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