Creating a Colour-Magnitude-Diagram

In graduate school one of the things I spent a lot of my time on was creating colour-magnitude-diagrams (CMDs). These diagrams are extremely useful in studying stellar populations, making models of stellar evolution and for use as redundancy in cross checking other theories in astronomy. Below is a post of my first reddit post that was extremely well received and made the front page of reddit. The reddit post is here.

Rodgers to Rodgers

Aaron Rodgers is good, but how good is he? Well the has one of the best arms in the game and he threw two last second Hail Marys in one season to send the game to overtime. Below is the first one, a last second pass to Richard Rodgers against the Detroit Lions.

To throw the football that far you need some serious arm strength and accuracy. In the pass he threw the ball 67 yards and the hang time was 4.1 seconds. With this in mind its possible to calculate the horizontal and vertical velocities. The horizontal velocity is vx = 60.26m/4.1s = 14.9m/s. The vertical velocity is v = gt/2 = 20.1 m/s. This gives an actual velocity of 25m/s or 90km/hr. 

Next find the angle at which he threw the ball. From the above picture the angle I measured was 53 degrees. With all of this in mind now plug into kinematics equations to work out a graph of the height of the football and downfield distance. 

From here its clear that Rodgers had to throw the football at a 53 degree angle, if he miscalculated the angle by just 5 degrees the ball would have landed just 63 yards downfield; four yards outside of the endzone. If he had thrown the football at 45 degrees to get maximum distance the ball would have travelled 70 yards down the field and ended up in the back of the endzone where no receivers were present. The kinetic energy of the pass was about 135J which is the same as a baseball pitcher's fastball going at 100mph. 

But what if somehow Rodgers had magically ended up on the Moon as he was about to launch the football?

On the Moon the acceleration due to gravity is just 1/6th that on the Earth. If Rodgers threw the football with same force and angle the football would have ended up landing 410 yards away and reached a maximum height of 120 meters. So how good is Aaron Rodgers? Probably the best quarterback in the sport that can not only throw a football very far but also very accurately. 

Gravitational Orbit Decay

So the detection of gravitational waves was announced yesterday. Its quite an amazing detection and will usher in a new age of astronomy. The waves were detected from the collision of two black holes, as they spiraled in towards each other they emitted the energy equivalent of about 3 solar masses of gravitational radiation. We know that black holes and neutron stars in binary systems will one day spiral inwards and undergo a collision, but what about our solar system? Obviously the solar system's lifetime will not permit any planet or satellite of a planet to undergo orbital decay from gravitational forces. The lifetime of our solar system is on the order of 10 billion years. However, what if the Sun was a black hole and we ignored any external forces, how long would it take some of the planets to undergo gravitational orbital decay? From Einstein's field equations we can get an simple formula that predicts how long it takes for a two body system to collide. I calculated the lifetime of the orbits of Mercury, Venus and Earth as well as the power Each planet emits due to gravitational radiation. Currently the Earth emits just 200W of gravitational radiation compared to the 200 PW (Petawatts 10^15) which is emitted in the form of electromagnetic radiation. 

Orbital time decay and gravitational radiation radiated

Another note is that Mercury and Venus will cross paths likely leading to a collision with each other before they spiral into the Sun. But the most astounding aspect of this is the gargantuan timescales. The horizontal axis is in the order of trillions of billions of years. Basically for the Earth to collide with the Sun the equivalent time of the current age of the universe must pass a quadrillion times. Needless to say this is not going to happen, the only element stable enough in these timescales is Tellurium-128, which has a half life of 2.2*10^24 years. About 22x longer than the orbital decay timescale of the Earth.  This was an interesting calculation, got to see how weak gravity really is.