5 Reasons Why You Should Go To The Manila Mini Maker Faire!

The Greatest Show (& Tell) on Earth is coming to The Philippines on June 10-11 at The Mind Museum!

Maker Faire is an international festival for all ages that celebrates the Do-It-Yourself mindset. Created by Make magazine, Maker Faire showcases the ingenuity of the Maker community all around the world: tech enthusiasts, hobbyists, engineers, tinkerers, and artists who create and innovate technology, and share what they've made. 


Why should you join this one? 

1. Manila Mini Maker Faire is the first-ever Maker Faire in the country!

Launched in 2006, the first Maker Faires were held in the United States; later, more than a hundred Mini and Featured Maker Faires were held around the world, such as in Tokyo, Rome, Detroit, Oslo, and Shenzhen. 

In Asia, upcoming Maker Faires this year will be held in Singapore, Malaysia, Japan, Taiwan and China - and the very first in the Philippines will be held in Manila this June 10-11!


Manila Mini Maker Faire will be held at The Canopy Plaza of The Mind Museum, and the program will run from 10 AM to 6 PM on both days. 

Because this will be the first of its kind in the country and tickets for registered attendees are FREE, the Manila Mini Maker Faire is a must-attend!

On-site registration booths will be available at the venue, but to avoid the long lines, you can pre-register at this link.

2. Meet over 60 different creative Makers!

The event will feature an exciting line-up of exhibits from over 60 different Makers and Maker groups, with delegates from all around the Philippines. 

Examples of these projects include (but are not limited to) the Smart Tix Clock and the Smart Portable Weather Station of FABLAB Mindanao, the 3D Printing projects of Puzzlebox 3D Philippines, the handmade musical instruments of MUSIKO IMBENTO, the solar lighting technology of Liter of Light, the drones of Flying Promotions and SkyEye Analytics, and the manned Hoverboard of Kyxz Mendiola.

Image credit: SkyEye Drone Race Series Facebook page
Image credit: SkyEye Drone Race Series Facebook page
The Faire will also feature the works of designers such as the toy and game design collective of Playing Mantis Philippines, creative wood crafts of Woodworks: Man x Machine, the handmade and custom folding bikes of Nyfti Lifestyle Bicycles, the upcycling projects of Roderick Banares and Ian Sarra, and many, many more!

To view the complete list of Makers and learn more about them, visit this link.

3. Attend free workshops to jumpstart your Maker journey!

The Maker Faire will also feature FREE scheduled workshops on the first day of the Faire (Saturday, June 10), that will take place at the Special Exhibition Hall of The Mind Museum. 

These workshops include "Introduction to Arduino" by SparkLab Innovation Center, a hands-on workshop on microcontroller programming, "Fusion 360 Workshop" by Mechaniweb, a workshop on how to design your own products using Autodesk's latest CAD software, and the "Machibox Robotics Workshop" by Machibox Inc., where you can learn how to program Machibot and race it through a series of obstacles and mazes. 

Since there are limited slots (offered on a first-come, first-serve basis), you can register now at these links: 


You may also view the schedules of the workshops at this link.

4. Watch free talks, demonstrations, and musical performances from the Makers themselves!

Aside from exploring the exhibits at the event, you can also watch free lectures given by Microsoft Philippines, Team Manila, Benilde Industrial Design, 3M Philippines, SparkLab Innovation Center, FABLAB Mindanao, Raimund Marasigan, Sebastian's Ice Cream, and the BS-HRM students of MSU-IIT, and learn about a variety of topics from innovative food packaging, to music production, and even the science of ice cream!

Image credit: Scienkidfic Xplorers
Image credit: Scienkidfic Xplorers

You can also catch the science experiments performed by Scienkidfic Xplorers, and listen to musical performances by Rayvan Beat and Ram Millisec. 

5. Find the inner Maker in you!

Yes, even you can be a Maker too! While the origins of the Maker movement are related to technology, you don't need to be an engineer or a programmer to become one.



Anyone who creates, builds, designs - whether it's coding software, designing experimental fashion, making music and art, baking cakes and cookies, fabricating costumes and props - is a Maker, and the Manila Mini Maker Faire is a celebration of the innovative and collaborative human spirit.


Interested yet?

Come join us at the first Mini Maker Faire in the Philippines! Who knows? You may be one of the next featured Makers to share your projects with the world. 

For updates on the Manila Mini Maker Faire and our upcoming programs and activities, follow Makerspace Pilipinas, Maker Faire Philippines, and The Mind Museum on Facebook, Twitter, and Instagram!

We Are 5! 5 Reasons To Visit The Mind Museum This Summer!


On March 16th, The Mind Museum turns five! Whether you're that Unlimited Science Pass holder who's at all our events, or someone who's never had an extraordinary science education experience at the museum, we have so much in store for you as we give back to our wonderful guests on our birthday.


We've grown a lot these past five years, and while some things are just as you remember them, we've come so far since AEDI the robot welcomed our first guests. Here are five things you can expect throughout our fifth birth year, and some just this week.

1. Over 250 hands-on, minds-on exhibits for kids 2 to 92!

You can still find your favorites from five years ago - like Stan the Tyrannosaurus rex, the Van de Graaff generator, and the Tunnelcraft - but you can also lose track of time at the Giant Orrery and its improved model of the solar system.



You can also learn about your sleeping habits with the Human Body Clock, or learn about the diversity of life in its shapes and forms with the Tree of Life. 

You can explore the planet-friendly technology of Future Cities, and explore your body's symmetry with the Anti-Gravity Mirror.



The Mind Museum's exhibit team regularly updates the permanent exhibits, so if you visited the first time we opened, we have new exhibits for you to see!

2. Free bonus events for museum ticket holders!

On Sunday mornings, you can watch Wild Conversations - a close encounter with animals, with wildlife trainer Isa Garchitorena. You can also catch Awesome Astronomy on Sunday afternoons with our resident astronomer Pecier Decierdo, and have a guided tour of the universe. See event schedules here!



Looking for something to do this weekend? On Saturday, (March 18) we'll be having our second session of Cafe Scientifique for the year, a free event where you can discuss timely scientific issues with the community in a round table discussion. This Saturday, we'll be discussing the science of addiction.

And of course, you can't miss the hourly Mind Moving Studios, where one of our resident scientists brings science up close and personal with various demonstrations.



The Mind Museum website is also updated with our regular roster of events, so check out our schedule for anything that might interest you!

3. Traveling Exhibitions and Special Exhibits!

This summer is also a great time to visit, with the traveling exhibitions Dino Play and Science Circus parked right in our Canopy Plaza, and Teenage Brain in our Technology Gallery. You can also visit our special exhibition, Planet Story, where you can create and star in your own digital story.

Guests in one of the rooms of the Teenage Brain Exhibition


Guests playing in the Dino Den of the Dino Play Exhibition

Guests playing with one of the illusions in Science Circus

Our guests enjoying one of the rooms in Planet Story
Since our traveling exhibitions visit the other parts of the Philippines, make sure to catch the exhibitions while they're still here! For a list of our current traveling exhibitions and ticket rates, you can also visit our website.

4. Beat the Heat with our Summer Programs!

Summer's coming up, and there's no better place to beat the heat than at The Mind Museum. One day is not enough when you're learning and having fun, so we made summer science programs just for you!

If you think you have what it takes to be like our Mind Movers, you can join the Junior Mind Mover program. The Junior Mind Mover program will offer three kinds of classes: the Tots class (for ages 4-6, from May 2 to 13), the Primes class (for ages 7-9, from May 3 to 27), and the Tweens class (for ages 10-13, from May 2 to 20).



If you're more inclined to make stuff with your own hands, the Maker Camp is there to make your imagination reality. Open to everyone ages 14 and above, Maker Camp will take place on April 21-23 (from 1:00 - 5:00 PM each day) at The Mind Museum.



To register for our programs, visit this link.

5. Avail of our 5th birthday promo!

What's a birthday, without a birthday treat? March 15th is the last day of our promo rates, so better catch it before it's gone:


Celebrate our fifth year of making science come alive and fall in love with science all over again!

Follow us on Facebook for more treats and upcoming events this summer!

Why It Might Be A Good Idea To Launch Rockets From The Philippines 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.



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: 



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:


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

5 Reasons Why You Should Join CSI: Crime of Passion


Are you looking for something to do this Valentine's that isn't overly sentimental or cheesy? Do crime dramas, detective role-playing games, and unsolved cases tickle your fancy? 

Do you want a quick and fun way to spend time with your friends (or your loved one) before going out on the town on a Saturday night?

If so, then CSI: Crime of Passion might just be the ideal way to spend your Valentine's weekend!



What is CSI: Crime of Passion, anyway? 

Crime of Passion is a competitive crime-solving activity where you and your friends play rookie agents on your first murder case: a tale of romance gone wrong. You must work together with your team to investigate the crime scene and be the first to uncover the killer!

If you need more convincing, here are 5 reasons why you should join: 

1. You get to solve a new mystery!

Every CSI 101 program provides a new mystery and storyline, and this year's Crime of Passion is no exception. Based on real-life cases of notorious murderers and serial killers, the CSI 101 program offers fun for both casual fans of the detective genre, and for forensic science geeks. 

As the name suggests, Crime of Passion will tackle a case of love and obsession (perfect for Valentine's!).


Because every storyline is unique, even participants who enjoyed their experience last year can join and enjoy this year's event. 

2. It's affordable!

At only P400 per head for groups of 4 and below that register, and P350 per head for groups of 5-8Crime of Passion is an affordable way to spend an afternoon in the Museum with your friends. 

You can even combine it with an all-day visit to the Museum for only P1000 per head, with a 10% discount for holders of the Unlimited Science Pass. 

3. You can bond with your friends, and compete with other teams!

When you register, you'll compete with your friends against other teams in a race to uncover the identity of the culprit and bring him or her to justice.



Because the activity is competitive, it can be a bonding experience where you get to strategize and bring each other's strengths to the table. You can also meet and befriend other like-minded people (before you defeat them, of course)!

4. It's quick!

Are you interested, but don't want an activity that takes up the entire afternoon? Don't worry! Because you will be under time pressure, the entire program will only take 1 and 1/2 hours.

Upon registering, you can choose to play from one of two time slots, depending on which is more convenient for you:

February 18 (Saturday): 1:30 - 3:00 PM or 3:30 - 5:00 PM

After registration, you can then pay through bank deposit, or at The Mind Museum ticket booth. 

5. You can win prizes!

The winning team from last year's Crime of Passion won tickets to The Mind Museum, and gift certificates for dinner from Balkan Yugoslavian Home Cooking Restaurant in BGC. 

The winning team from last year's CSI: Looking for Substance, where the participants unlocked a crime-ring in a University, also won gift certificates for dinner from Shawarma Bros. 

With all these exciting prizes, what could this year's event have in store? 



If the above reasons interest you, register now, as slots are limited! And if you want more information on the event, you can also view the FAQs here.

See you there, detectives!

video

From Epicycles to Ellipses: Kepler's Laws of Planetary Motion, Part 1 by Lanz Lagman


This is the story of how an apprentice meticulously gathered and analyzed data for decades to lead us closer to the truth on how the heavens moved.

His groundbreaking work helped lead to the Scientific Revolution, allowing us to break free from the confines of our home planet and reach the once unreachable heavens.

"I demonstrate by means of philosophy that the earth is round, and is inhabited on all sides; 
that it is insignificantly small, and is borne through the stars"
Johannes Kepler. Image credit: ESA
That apprentice was Johannes Kepler (1571-1630).

He established the laws of planetary motion after years of hard work analyzing the records of his superior, Tycho Brahe, then struggling to fit them into the available models of the Solar System. 

There are three laws: the first states that a planet moves in an elliptical orbit around a Sun, with the Sun located at one focus.

An elliptical orbit.
The second law states that the area covered by the line connecting a planet to the Sun sweeps equal areas in equal intervals of time.

This implies that the planet will move faster as it approaches the Sun, and will move slower when it is much farther from it. Planets with circular orbits should only have a constant orbital speed.

The area covered from p1 to p2 is equal to the area covered from p3 to p4.
The third law states that the cube of the mean distances is approximately equal to the square of the period of the planet's revolution. This means that the greater the distance of the planet from the Sun, the slower it revolves. 

In this article, we will discuss how the first law was established by Kepler based on the knowledge and technology available during his time. Afterward, we'll tackle how later scientific advancements, some long after his death, provided the grounds for justifying why orbits are elliptical. 

Renaissance Astronomy: A Background

Kepler lived during a time when three models described the motions of planets: the widely accepted but increasingly questioned Ptolemaic system, the obscure but more realistic Copernican system, and finally the forced combination of these two models, the Tychonic system.

From left to right: Ptolemaic, Copernican, and Tychonic System
These models tackle the position of the Sun, Mercury, Venus, Earth, Moon, Mars, Jupiter, Saturn, and the stars. The main conflict between these two models were: 

1. What's at the center of the universe, the Earth or the Sun?
2. What planet is orbited by the others? 

According to Ptolemy (100-168), the immovable Earth was at the apparent center of the universe, with the others surrounding and orbiting it. Nicolaus Copernicus (1473-1543) disagreed with Earth being the center, and replaced it with the Sun, the other planets revolving around it.

Tycho Brahe (1546-1601), the astronomer that Kepler assisted, saw the benefits of the Copernican system and combined it with the more philosophical and religiously aesthetic Ptolemaic system, proposing that the Sun orbits the Earth, and the other planets orbit around the Sun. 

Despite their fundamental differences, they were made for the same purpose: to predict how the planets moved throughout the sky. Each of these models were proposed and modified from data continuously accumulated from night sky observations that were conducted thousands of years before. 

Astronomers obtain these data by recording the positions of planets at a fixed time, every day (as much as possible). Remember that the Renaissance was an era where instead of cars, we had horse carriages; exclusive libraries instead of Google, and 'selfies' only being possible if you were a skilled male artist.

All of these models were made at the vantage point of the three astronomers, and they had to observe for decades while working on these, while using the data that the others have collected before. 

Astronomy was then fused with astrology, not physics. This means that the movement of stars and planets were only tracked, and their main concern was to make a model that recognized the pattern of planetary movements - not what made them move in that pattern.

Leonardo da Vinci's #selfie
Image credit: leonardodavinci.net
It's also important to mention how Ptolemy's work held so much influence; despite how the heliocentric model of Copernicus diverged from it, a lot of the Ptolemaic model's elements were still retained. These included the concept of nested celestial spheres, where the planets are embedded in rotating hollow spheres made of Aether.

The rotation of these celestial spheres was believed to cause the movement of the planets, causing them to move around in circles with unchanging speed. However, the stars are all embedded in the outermost celestial sphere. 

Contrary to popular belief, these models were actually similar in accuracy. The accuracy is dictated not by how they define what is at the center, and what orbits which, but how the models employ epicycles. 

Established Epicycles: What Are They For?

The Earth revolves around its axis as an example of uniform circular motion. The stars appear to stay in their place at all times, which explains why a picture taken of the night sky with long exposure (to reveal star trails), will look like this:

Star trails. Image credit: Jerry Lodriguss
The stars seem to revolve around the star at the axis of rotation, which is the North Star, Polaris. Since the models state that the planets move in a uniform circular motion, we would expect that tracing their paths would reveal curved streaks - expected of a segment of a circle - and when they are connected, will form a perfect circle.

But, what's this? 

Movement of Mars in 2003. Image credit: NASA
Soon, the notion of uniform circular motion and geocentrism started accumulating holes. Mercury and Venus are both unseen at midnight, and are only clearly visible when the Sun is either starting to rise or set.

Planets appear to speed up then slow down, and their brightness during night subtly varies over time. The illustration above is an example of retrograde motion. 

 Mars, Jupiter, and Saturn have been observed to suddenly move backward, then move forward again. Segments of perfect circles aren't supposed to look like that, right? Perhaps the motion is not circular, right? Nope. Enter the epicycle.

Ptolemy's epicycle.
Epicycles literally mean circles within circles. In this diagram of Ptolemy's model of an orbit, the red circle is called the deferent, and the blue circle is the epicycle.

The planet revolves around the center of the epicycle, then the center moves through the deferent. Now this is the confusing part, because these seemingly contradict the notion of circular motion. 

The center of the epicycle revolves around the center of the deferent called the eccentric, at a constant distance; that's why the deferent is circular. However, with respect to the point called equant, the center of the epicycle revolves at a constant rate. 

In this way, the planets seem to move faster when they're approaching our Earth, then slowing down when they're moving away.

This was Ptolemy's clever way of devising his epicycles not only to explain why the planets sometimes appear to speed up, but also why retrograde motion occurs. However, when the path of the planets are traced, they'll look like this:

Orbits of the Ptolemaic System. Image credit: James Ferguson
The battle between the Ptolemaic and Copernican models are interesting in how they used epicycles.

Ptolemy altered the size of the epicycles of Mercury and Venus so that his model would explain why they are invisible at night. Ptolemy's epicycles are more complicated, however, because of the equants. 

Copernicus sought a simpler model, leading to the heliocentric model. Mercury and Venus now revolved around the Sun, a simpler explanation for why they can't be seen at night. Smaller epicycles then replace the idea of equants.

The Copernican model might have simpler epicycles, but it also employs more epicycles than the Ptolemaic system. 

In conclusion, both systems have their own complexities, but are almost the same in terms of accuracy. Inevitably, these models proved to be inadequate over time upon comparison to thousands of years of compiled data; but everything changed when Kepler attacked. 

Enter Kepler


Tycho Brahe at Uraniborg.
 Image credit: Astronomiae instauratae mechanica

Johannes Kepler was employed in 1600 by Tycho Brahe as an assistant in his observatory in Castle Uraniborg, in Sweden. Of all the tasks assigned to him, the most important one came when Brahe unexpectedly contracted illness.

On his deathbed, he instructed his assistant to complete their work on the Rudolphine Tables, and use his formerly restricted, but accurate data to prove his Tychonic system.

Kepler was earlier assigned to analyze Brahe's observational data on Mars since there were a lot of errors discovered. Kepler was a known adherent of the Copernican system, and sought to analyze Brahe's data in order to prove and modify it.

He began his study of the data in 1601, and fortunately concentrated on studying Mars's orbit. Years passed, and he realized he had to bring back Ptolemy's equant and modify the Copernican model. This led him to come up with the idea of an ovoid-shaped orbit (resembling an egg, or a teardrop).

He then set out to calculate the ovoid orbit using trial-and-error. After many mistakes, he realized that perhaps, the orbit was elliptical instead!

After repeating his calculations with this new idea in mind and discovering that the orbit was indeed elliptical, Kepler finally declared that orbits in general are elliptical in shape.

An egg-shaped orbit versus an elliptical orbit.

Kepler's second law was actually discovered before the first. The result of his hard work was the publication of Astronomia Nova (New Astronomy) in 1609.

Additionally, geocentrism and the epicycles slowly died in Europe, partially due to Kepler's works, but also because of the discoveries made by Galileo Galilei through his telescopes of the phases of Venus and the moons of Jupiter.

What he observed of the phases of Venus could not be justified by any geocentric model or the modified epicycles, and the moons orbiting around Jupiter should be orbiting around the Earth.

The appearance of comets during the Renaissance and subsequent studies also revealed that the heavenly bodies have highly elliptical orbits, refuting the notion of uniform circular motion.

While Kepler could show that planetary orbits are ellipses, he couldn't exactly determine why - like learning from a friend that a new flavor of pizza tastes great, but neither of you could figure out why.

This would be proven decades later by none other than Sir Isaac Newton (1642-1727), who was so badass that he almost single-handedly invented his own math (calculus) and his own physics (Newtonian mechanics).

Now why would an orbit be an ellipse instead of a circle? 

Sir Isaac Newton.
Image credit: Royal Society Print Shop
It was Kepler's laws of planetary motion that served as one of Newton's inspirations in searching for a rational basis on why and how things move in general.

In his most famous work, the Principia, Newton established his well-known laws of motion, theory of gravitation, and the beginnings of every math-hater's worst nightmare, calculus.

Together, these three were used to justify Kepler's laws and in his section, we'll discuss how these concepts lead to the elliptical shape of planetary orbits.

The three could be summarized as follows. Newton's three laws of motion state the following in order: things will stay at rest unless they are moved (law of inertia); moving objects would have greater acceleration if more force is applied (law of acceleration); and for every action, there is an equal and opposite reaction (law of action and reaction).

The second law could be summarized as the force applied being equal to the mass multiplied by its acceleration:
The gravitational theory states that everything with mass gets attracted to each other by the gravitational force in an inverse-square distance relationship. In simpler terms, the nearer two objects are to one another, the stronger they get attracted to each other.

Objects with greater masses are more attractive, but that may not be a good way to describe how attractive your crush is. The equation for the gravitational force is expressed as:


In this equation, Fg is the gravitational force, G is the gravitational constant, M and m are bigger and smaller masses of the two objects respectively, and r is the distance between them.

It's now easier to understand the concept of inverse square law; since the divisor is the square of r, that means doubling the distance of these two objects would mean the gravitational force between them would be reduced to a quarter of its former value.

Combining the two equations, we produce an important mathematical expression: acceleration due to gravity.

For now, do not mind the weird markings, for they are the result of deeper applications of Newtonian mechanics that we will not discuss.

Calculus is the mechanical study of change. Newton and Gottfried Leibniz (1646 - 1716) both independently invented it. This new form of math is perfect for accurately describing how slopes become steep and flat at some sections, and in this case, how the variations in the speed of planetary orbits influence the shape of their orbits.

The full use of these principles leads to the following expression:


This dictates the distance of the orbiting body from its parent body (r) with respect to time (t). It turns out, if you draw a graph of the expression above, what you get is a special family of shapes called conic sections. Conic sections are the shapes you get when you slice a cone, hence the name.

There are several kinds of conic sections. The kind will depend on the numerical value of the letter ε. It's the eccentricity of the conic section and this tells us how much the orbit deviates from a circle!

 For a circle, ε = 0; for an ellipse, 1 > ε > 0 or in other words, an ellipse has an eccentricity greater than zero but less than one, while a parabola has ε = 1, and a hyperbola has ε > 1.

Possible shape of orbits as dictated by their eccentricities.

In fact, a circle is a special type of ellipse; this means that perfectly circular planetary orbits are impossible in nature. For example, the orbits of the planets Mercury, Venus, Earth and Mars are the following: 0.205, 0.007, 0.017, and 0.094.

Despite Mercury being the most eccentric, astronomers failed to notice this at all, because Mercury is less-studied. After all, it moves quickly and could not be seen at night when planetary movements are easily analyzed.

The Orbits of the Inner Planets.
Image credit: NASA's Eyes

Notice that all planetary orbits are closed; this means they are not parabolic or hyperbolic. This means a huge amount of force must be exerted upon them to have an open orbit, since as per the laws of motion, they are massive and harder to move because of more inertia.

Spacecraft and comets however, have significantly less mass and are therefore easier to accelerate, making it easier for them to achieve an open orbit.

We already knew what happened next: Newtonian mechanics dethroned the method of thinking that resulted in the geocentric models, and became the most authoritative tool for hundreds of years for scientists in explaining how everything moves and interacts with one another.

Soon, Uranus was discovered, a planet too dim for the eyes of our ancestors. Then followed Neptune, whose movements were predicted by Newtonian mechanics, and thus solidifying its authority.

Who would have thought that what we discovered about the motions of these planets would be used as a noble attempt to understand one of the most basic building blocks of everything?

The atoms, once considered the smallest unit of matter, were once thought to be miniature solar systems themselves: with the nucleus as the Sun and the electrons as planets.

We as humans must marvel and appreciate the courage, curiosity, diligence and passion in the search for truth displayed by Kepler, Newton and all other scientists that brought us the Scientific Revolution.

In just a several hundred years, we may now reach the heavens through our rockets and spacecraft. We may now finally fulfill what our ancestors thousands of years ago could only dream about and marvel for the rest of their lives.

For the advanced readers who can stomach some math, we will derive Kepler's first law using the mathematical expressions of Newton's principles.


For the Advanced Reader

In order to derive Kepler's first law using Newtonian dynamics, we have to use vectors, and it is expected that the readers of this section would be familiar with basic vector calculus. They are entities with a magnitude and direction.

In the case of planetary orbits, the central mass providing the central force is the Sun and the planet with a much smaller mass moves around it because of the Sun's gravitational force. We will study this system in three dimensions and therefore, we have to observe the system at a viewpoint perpendicular to the plane where the Sun and the planet's initial positions are located.

That viewpoint would be the z-axis. The mathematical expression using vectors would be:



In the following diagram and equations, vector L, the orbital angular momentum, is the cross product of the radius vector r and the linear momentum p. The masses of the Sun and the planet are denoted by M and m respectively.

We could think of p as something that causes the planet to go straight forward with respect to the Sun but r keeps it from going further away, just like a string attached to a yo-yo. The orbital angular momentum dictates how hard it would be for the planet to keep orbiting the Sun.

Since the system is moving, the position of mass m changes over time, and it leaves its  initial position. To describe this mathematically, we take the time derivative of equation (2).






Vectors that are parallel and antiparallel to each other cancel out. Therefore, equation (3) is reduced to:



This means that vector L is constant and the orbital angular momentum is conserved. Any force whose direction is toward the center or away from the center has a zero cross product with r. 

As long as the system is conserved and no moving stars or black holes go near the system, the movement of the planet is confined on the plane (gray area) perpendicular to L, as if it only glides through it and therefore does not jump up and down with respect to it.



We have to get back to equation (2) and expand it; we will see later how practical these steps would be. Let's expand the vectors r and v as products of their magnitudes and direction vectors. Since the time derivative of the position r is velocity v, we get:



Through product rule, equation (5) now expands to:



Let's leave first equation (7) as it is and introduce the child of Newton's second law and the law of gravity, the acceleration due to the gravity of the planet. Since the Sun would be the point of reference and we're looking at how the planet is attracted toward it, the vector form would be negative. This is expressed as:



We can clearly see that the acceleration toward the center is an inverse square. This means that the acceleration becomes greater as the planet approaches the Sun.

We will now get to the cross product of acceleration due to gravity and angular momentum. This may seem weird but again will pay off later because of vector identities.



Notice that at equation (11), the common factors are the time identities. Remember that the time derivative of vector L is 0, just as we've shown in equation (4).



By integrating both sides, we get:



This may seem a dead end, but the BAC-CAB rule saves the day for us. Thus, we transform the left-hand side of equation (14) to:



Notice that the left-hand side is the dot product of vector L and the cross product of vectors r and v. Remember that the dot product of the same vector is equivalent to its magnitude, by using equation (2), we could now transform equation (15) to:



This equation seems familiar already. Let's isolate r to the left side so that we would know the present distance of the planet from the Sun at a particular time.



Equation (17) looks very similar to some well-known mathematical equation. Maybe making the following substitutions would do the trick.



We now get the equation for a conic section, with the eccentricity dictating its type. This is a much more general statement for all orbiting bodies.

Spacecraft that orbit the Sun have so little mass compared to planets, and therefore orbital angular momenta. This means that spacecraft orbits are easier to alter, and why we don't see planets with parabolic and hyperbolic orbits.



It's just almost impossible to find a perfectly circular orbit, but also impossible for planets to perform epicycles.

REFERENCES: 

1. Caroll, B.W., & Ostlie, D.A. (2009). An introduction to modern astrophysics. San Francisco, Calif; Munich: Pearson/Addison Wesley.
2. Caspar, M. (1959). Kepler. London: Abelard-Schuman.
3. Riebeek, H. (2009, June 7). Planetary Motion: The History of an Idea That Launched the Scientific Revolution. Retrieved from http://earthobservatory.nasa.gov/Features/OrbitsHistory/
4. Johannes Kepler: His Life, His Laws and Times. (n.d.). Retrieved from http://earthobservatory.nasa.gov/Features/OrbitsHistory/
5. Williams, D.R. (n.d.). Planetary Fact Sheet - Metric. Retrieved from http://nssdc.gsfc.nasa.gov/planetary/factsheet