# Tag Archives: Kunal Nayyar

## The Big Bang Theory of Trojan Asteroids

Season 06, Episode 03: “The Higgs Boson Observation”

In “The Higgs Boson Observation”, we learn that the subject of graduate student and Sheldon’s (Jim Parsons) new assistant, Alex Jensen (Margo Harshman), is on Trojan asteroids at the Earth-Sun L5 Lagrange point. This also happens to be Raj’s (Kunal Nayyar) area of expertise. But what exactly are Trojan asteroids and why are they of interest to astronomers and astrophysicists?

Trojans are minor planets or satellites that share an orbit with a planet but does not collide with it because it orbits around one of the two stable Lagrangian points (trojan points). The L4 and L5 lie approximately 60° ahead of and behind the larger body, respectively .

# Area of Research

These are the five Lagrangian points in a two-body system with one body far more massive than the other (e.g. the Sun and the Earth)

There are five points on or near Earth’s orbit known as Lagrange points where an asteroid will remain stationary with respect to Earth as it orbits around the Sun. These points mark the positions where the combined gravitational pull of two large bodies, such as the Earth and the Sun, provides the centripetal force needed to orbit with them.

The Lagrangian points are approximate solutions to what is known as the three-body problem, a famous problem that attempts to model the motion of three bodies as they gravitationally affect each other. In essence, this problem dates back to 1687 when Sir Isaac Newton first published his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), often referred to as simply the “Principia”. Several scientists and mathematicians have all attempted to find an analytical solution to the problem or a simple equation that would describe the motions of all three bodies.

The three-body problem is difficult to solve because the three bodies tug against each other in chaotic and unpredictable ways. Only by simulating the problem on a computer can we see the objects paths. In comparison, the quantum three-body problem is relatively easy to solve and understand. However, approximate solutions can be useful. If the third mass is taken to be small enough that it does not affect the other two masses some interesting solutions crop up.

# Celestial Mechanics

To better understand how celestial bodies move in our solar system, we turn to Newton’s law of universal gravitation. This law states that every mass in the Universe attracts every other mass with a force that is proportional to the product of their masses. If we had two masses, $$M_{1}$$ and $$M_{2}$$, then the force between them would be proportional to the product of the two masses.
$F \propto M_{1}M_{2}$
This law also states that the force decreases with the square of the distance between them.
$F \propto \frac{1}{r^{2}}$
We can combine these into one equation that will help is analyze and solve the movement of celestial bodies anywhere in the Universe.
$F=G\frac{M_{1}M_{2}}{r^{2}}$
where $$G$$ is the gravitational constant. This law was deduced by Newton from observations and was formulated in the Principia.

If we were to look at the Sun and Earth system, we would have what is known as the two-body problem. While we may think this is a simple problem but in reality it isn’t. The Earth does not revolve around the Sun as if the more massive Sun was a stationary body but rather both the Earth and Sun revolve around a common point — the center of mass or barycenter. Though makes the problem a little more difficult, as this point is very close to the center of the Sun and we can make an approximation where the Sun is in the center and doesn’t move while the Earth moves around it.

If we use this technique, we can derive an equation that describes the Earth’s motion around the Sun. This is a classic “two-body” problem and its solutions describe the familiar elliptical orbits of the planets known since the time of Kepler. Unfortunately, when we add a third body to our equations of motion, such as a spacecraft lost in space between the Earth and the sun, we can no longer find an analytical solution; the equations become unsolvable.

While the three-body problem has no analytical solution, Joseph-Louis Lagrange in 1772 showed that if we restricted the mass of the third body to be so small that it couldn’t affect the other two, there were some solutions to be found. The solutions to this “restricted three-body problem” finds that the three bodies move in unison and always maintain the same position relative to each other.

These five points, where the third body stays in the same relative point in space, are known as Lagrange points and mark the positions where the combined gravitational pull of the two large masses precisely provides the force to orbit with them. They are labelled L1, L2, L3, L4 and L5 and also known as L-points or libration points.

## The Trojan Asteroids

The L4 and L5 libration points lie on the corners of two equilateral triangles. The points are balanced because the gravitational forces between the Earth and Sun keep any object in orbital equilibrium with the rest of the system. These points are sometimes referred to as “triangular Lagrange points” or “Trojan points” and come from the Trojan asteroids at the Sun-Jupiter L4 and L5 points. These asteroids were named after characters from Homer’s Iliad. Asteroids at the L4 point leads Jupiter and is referred to as the “Greek camp” while those are the L5 are referred to as the “Trojan camp”.

The Earth-Sun system, like the Jupiter-Sun system, also uses the same terms of reference. There is one known Greek asteroid in the Earth-Sun system, 2010 TK7, first detected in October 2010 by Dr. Martin Connors using the Wide-field Infrared Survey Explorer (WISE) . The asteroid’s designation come from the provisional naming system for minor planets. This asteroid which has a diameter of about 300 meters oscillates about the L4 Lagrangian point. The region around these points is known to contain interplanetary dust. As people tend to search for asteroids at much greater elongations, very few searches have been done at these locations. Currently, there are no confirmed or suspected L5 Trojans of Earth which makes both Raj’s and Alex’s work interesting and something to talk about.

## The Big Bang Theory of Trans-Neptunian objects

Season 02, Episode 04: “The Griffin Equivalency”

Rajesh Koothrappali (Kunal Nayyar) looks pleased as he talks about his inclusion into People magazine’s “30 Under 30 to watch” list for his discovery of the fictional trans-Neptunian object, 2008 NQ17 which exists beyond the Kuiper belt

In “The Griffin Equivalency” episode of The Big Bang Theory, Raj (Kunal Nayyar) announces that he has been chosen as one of People Magazine’s “30 visionaries under 30 years to watch” list for his discovery of the trans-Neptunian object 2008 NQ17. This fictional astronomical body exists just beyond the Kuiper Belt, a region of the Solar System that extends from the orbit of Neptune (about 30 Astronomical units(AU) away), to a distance of 50AU from the Sun. (1AU is roughly the distance of the Earth from the Sun).

The distribution and classification of the trans-Neptunian Objects in our solar system

The first trans-Neptunian object discovered was Pluto in 1930 and is the second most massive known dwarf planet in the Solar System. These objects are so small and distant that it wasn’t until 1992 that the second trans-Neptunian object orbiting the Sun, (15760) 1992 QB1, was discovered. As cryptic as these names sound, i.e 2008 NQ17 and 1992 QB1, the nomenclature comes from a provisional designation in astronomy that names objects immediately following their discovery. Once a proper orbit has been calculated, a permanent designation consisting of a number-name combination is given by the Minor Planet Center. Unfortunately for Raj, so many of these minor planets are discovered that most won’t be named for their discoverers.

# Provisional designation of Minor Planets

The provisional naming system for minor planets, of which dwarf planets, asteroids, trojans, centaurs, Kuiper belt objects, and other trans-Neptunian objects are a part of, have been in place since 1925. The first element in a minor planet’s provisional designation is the year of discovery, followed by two letters and sometimes a number.

The first letter indicates the “half-month” of the object’s discovery. “A” denotes a discovery in the first half of January (between January 1st and January 15th) while “B” denotes a discovery in the second half of January (between January 16th and January 31st). The letter “I” is not used so the letters go all the way up to “Y”. The first half is always the 1st through to the 15th of the month, regardless of the numbers of days in the second “half”.

First Letter (A-G)
A B C D E F G
Jan 1 Jan 16 Feb 1 Feb 16 Mar 1 Mar 16 Apr 1

First Letter (H-O)
H J K L M N O
Apr 16 May 1 May 16 Jun 1 Jun 16 Jul 1 Jul 16

First Letter (P-V)
P Q R S T U V
Aug 1 Aug 16 Sep 1 Sep 16 Oct 1 Oct 16 Nov 1

First Letter (W-Y)
W X Y
Nov 16 Dec 1 Dec 16

The second letter and the number indicate the order or discovery within that half-month.

Second Letter (A-K)
A B C D E F G H J K
1 2 3 4 5 6 7 8 9 10

Second Letter (L-U)
L M N O P Q R S T U
11 12 13 14 15 16 17 18 19 20

Second Letter (V-Z)
V W X Y Z
21 22 23 24 25

So the 14th minor planet discovered in the first half of June 2013 will get the provisional planetary designation 2013 LO. As modern techniques has made it (relatively) easier to detect these objects, there can be more than 25 discoveries in a given half-month. To solve this a subscript number is added to indicate the number of times that the letters have cycled through. So the 40th minor planet to be discovered in the first half of June 2013 will gain the designation 2013 LP1.
\begin{aligned} 40 & = 25 + 15 \\ \therefore P & = 15 \end{aligned}

# Planet Bollywood

Using this information we can calculate when Raj discovered “Planet Bollywood”. It was discovered some time between July 1-15 and it was the 441st minor planet to be discovered in that half month.
\begin{aligned} Q & = 16 \\ 25 \times 17 & = 425 \end{aligned}
And $$425 + 16 = 441$$.

It must have been a particularly busy month in the Big Bang Theory Universe. A quick check at the Minor Planet Center’s Provisional Designations webpage indicates that there were 137 astronomical bodies discovered in that half-month. That would make that last trans-Neptunian object to be discovered to receive the provisional designation 2008 NM4.

Long live Planet Bollywood!