The day the Earth changed its rotation-rate

This week I saw a great article in Wired.com about the static test of one of the solid boosters for the SLS rocket. So I tweeted: "The Earth rotated a little bit faster that day! #rocketengine". I should have known that I cannot make these kind of statements without proving them first. One of my friends called my bluff.

So, I replied with "Challenge accepted!". The blog post will report on my investigation and show how much the Earth speed up/slowed down by the QM-2 test of the SLS booster rocket. My investigation consists of three parts: the model, the search for reliable information, and my conclusions.

Model of the Earth with an attached booster

By igniting their propellant, rockets can produce thrust that normally propel them upwards towards space. If you fix them on the Earth they could speed up or slow done the rotation of the Earth, depending on  the direction of its thrust. Ok, hold your horses! Of course this will have a small effect, but I still need to solve the challenge proposed by my friend. So, lets sketch the situation:

Sketch of the situation of the tied down rocket on a sphere, while letting it burn. top view: Blue circle represents the Blue Marble, or Earth, with its rotation illustrated by 𝝎. The thrust is represented by F and the arm of the torque due to the thrust is r. The moment of inertia of Earth is depicted by IEarth. side view: the latitude of the test location is given by 𝜃.

The thrust or force, F, that is pointing in the direction of the rotation can speed up the rotation, and forces pointing away from the rotation can slow down the rotation. Forces parallel to the rotation vector (the rotation vector is seen in the side view, sticking out the north pole) will not contribute to the change of rotation. So, we need to determine the direction and magnitude of the booster's thrust and its latitude on Earth to determine F and r. In this way we can calculate the torque that is responsible for changing the rotation-rate of Earth:

After we have determined the torque, it is possible to calculate the angular acceleration, 𝞪, with help of the moment of inertia of Earth, IEarth.

We're almost there! If the angular acceleration is obtained, we can calculate the new rotational velocity by using the following equation:

I have placed a minus sign, because in my sketch the force due to the rocket counteract the rotation of Earth. The angular acceleration is multiplied by the period of the rocket burn. The rotation rate of the Earth is around 7.29 ×10⁻⁵ rad/sec. With these equations we can find the change in rotation due to the QM-2 booster test.

Information investigation

From the model we can see that there are some numerical parameters needed to calculate the actual change in rotational velocity of the Earth. They are listed bellow:

• location of test (latitude), 𝜃
• direction of firing
• thrust of the booster, F
• period of firing during test, Δt
• moment of inertia of the Earth, IEarth

So, lets find them on the internet. We have the Wired.com article with the movie of the firing: link. In this article, I found out that the test is conducted by Orbital ATK. Googling "Orbital ATK QM-2 ground test" gave me some more information. First of all, the expected thrust is reported to be 3.6 million pounds of thrust, which is 16 million Newton of thrust (lets keep it metric). In one of the documents on the website of Orbital ATK I found some more information:

Here, also the operational time of the test was listed as 126 seconds. Somewhere else they wrote about a two minute test, so I am confident this value is ok. This means, two down, three to go. 

The moment of inertia of the Earth can be found anywhere on the web, we will use 8.0 ×10³⁴ kg m². This is the integrated moment of inertia around the rotation axis of a differentiate sphere. My students need to calculate this for Earth, Moon, and Jupiter. Its a nice mathematical exercise, maybe for a different post.

The last two parameters were a little bit more difficult to find and made me really feel like the Sherlock Holmes of the internet. On the internet webpage of Orbital ATK, I found that their Rocket Testing Facility was in Promontory, Utah. If you type this in Google maps, you still need to go to the northeast towards the road 83. There the facility is clearly visible at latitude 41.62 degrees, which is extremely huge:

The Orbital ATK rocket test facility. The red circle shows the two test areas I found, showing clear exhaust-plume evidence

After half an hour inspecting the facility, I found two sites where the evidence of rocket tests were clearly visible. In the movie of Wired.com a large exhaust plume is visible, which affects the terrain, this must be visible on Google maps. Unfortunately, there are two sites on the Promontory grounds, so I still did not know exactly the location and direction of the thrust vector. Lets zoom in:

Zoom in of the testing sites of Orbital ATK. The red lines indicate visual evidence. I indicated the terrain that is affected by the exhaust plumes by 1. Sites 2 and 3 give more visual information. 

How to pick between the two sites? I went back to the movie posted on Wired.com and made stills that gave me confidence about the correct site where the test took place. In the beginning of the movie the rocket is ignited and the background is visible. Here, we see visual evidence number 2. 

An indent in the hill is clearly visible in the video, which can be linked to a geological structure in the southern test site. However, there is also an indent in the northern test site, so you could interpret this evidence in favour of the northern test site. Luckily, several seconds later in the video a larger overview of the test was shot, both showing evidence 2 and 3.

Here, both the indent behind the test location is seen and a straight road and geologic structure in the right bottom of the picture. These visual objects can only be linked to the south test site in the map and are not seen on the northern test site. This make me confident that the location of the QM-2 test is the southern test site. Finally, this allows me to determine the direction of thrust and calculate the east-west component of the thrust vector that is responsible for change in rotation rate.

The thrust of the rocket is towards the north-ish-west, and therefore it will slow down the Earth instead of speed up. I was wrong in my tweet (always factcheck, boys and girls), but I have answered part of my friends challenge! Now for the last part, we have all the information to calculate the amount of slow-down of Earth's rotation.

Calculating the rotation-rate change
In the last figure, I made a decomposition of the thrust vector, because we only need the part in the direction of the rotation (east-west). By using the cosine rule we determine the magnitude of the thrust vector in the east-west direction:

I measured the angle (16.4 degree) and this results in an east-west force of 15.35 million Newton, slightly less than the full thrust of the rocket. Continuing, the arm of the torque, r, we also have to decompose:

With the averaged radius of the Earth, R = 6371 km, the arm of the torque becomes r = 4762.7 km, together the torque is calculated to be 7.31 ×10¹³ Nm. By dividing the calculated torque by Earth's moment of inertia will result in an angular acceleration of α = 9.14 ×10^-22 rad/sec². The angular acceleration is very small due to the huge moment of inertia of the Earth. Still, I am on a quest to answer my friend's last question.

From Orbital ATK's website, I found the duration of the test (Δt = 126 secondes), which results in a rotation-rate change of 𝝎 = 1.15 ×10^-19 rad/sec, or 6.6 ×10^-18 deg/sec. This tiny slowing-down of the Earth (14 orders smaller than its current rotation-rate) will only have a one degree difference in position after a billion years. So no, we will not have observed the change in rotation-rate due to the QM-2 testing of Orbital ATK. It did teach me to always fact check my statements on twitter, somebody could call your bluff.