Physics

ISP220, spring 2018: In-Class Project #4; 15 pts

Quarks, Spacetime, and the Big Bang

Tuesday, February 6, 2018

Name: Student #

Figure 2: The top figure shows altitude and distances from the center of the earth. Bottom figure shows a blowup of the region near the earth.

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1 International Space Station, 8 pts

The figures on the previous page shows cartoons of distances measured away from the earth using the diameter or radius of the earth as a measure. The radius of the earth is 6,378 km and for these pictures I’ve pretended to use RE ⇠ 6, 000 km. The bottom figure is just a blowup of the top one.

The International Space Station (ISS) is at an altitude above sea level of 350 km.

Mark the ISS orbit on both pictures with a big “X.”

Are you surprised at how low it is? This is for two reasons, at least:

• It already takes hours for a Russian Soyuz spacecraft to get all the way to the ISS. So it would take much more energy and require more fuel.

• To go farther out would begin to get too close to the Van Allen Ra- diation belts, which are the shaded orange donut around the earth in the pictures.

The Van Allen belts (there are two) are regions created by the earth’s mag- netic field (thanks, Mr Gilbert) which actually trap charged particles, which in the near belt—the one pictured as orange—are mostly protons. They’re produced by cosmic rays and particles that come from the sun. They would damage electronics and be a hazard for humans...so best to stay under the belt, or above it.

Let’s look at some practical “orbital mechanics” of the 21st century. We’ve done this in class:

v2 = GME R

so:

v =

r GME R

(1)

I’ve plotted Eq. 1 in Fig. 4 where I’ve manipulated the units in order to get the velocity in km/h and the distance in km. Figure 5 shows two circular orbits and various distance relationships of use to us. Let’s figure out how fast the ISS is actually moving. First, remember that the ISS altitude is

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Figure 3: The Soyuz MS-01 spacecraft docked to the ISS in July of 2016. The crew consisted of Anatoli Ivanishin (Russia), Takuya Onishi (Japan), and Kathleen Rubins (USA). Scott Kelly had returned from his year on the ISS the previous March. There have been roughly 50 Soyuz missions to ISS.

A = 350 km. If the average radius of the earth is 6,378 km, how far is the ISS from the center of the earth, RL (for Radius of Low earth orbit)?

km

With that information, Eq. 1 tells us what the speed of the ISS is in general, and Fig. 4 shows us that function evaluated in km/h.

Mark a big X on the Fig. 4 for the ISS parameters.

Fast, right? How fast in ’merican units...The conversion from km/h to mph is about 5/8ths.1 Or, you can ask Mr Google. So what is the approximate speed in mph?

1 Look at your speedometer and you’ll see that when you’re driving 80 km/h in

Canada...it’s about the same speed as 50 mph in the US. That’s how I do the conver-

sion...I run out to my car and look at my speedometer. Now you can too.

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Figure 4: The orbital speed of a satellite of earth in a circular orbit versus the distance from the center of the earth.

Figure 5: The orbital speed of a satellite of earth in a circular orbit versus the distance from the center of the earth.

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v(ISS) = mph

Equation 1 points at a really funny thing about Newtonian orbital me- chanics that really caught the space program o↵-guard. If the only force that acts on a spacecraft (or an actual moon, or anything!) is that force of gravity, there are counter-intuitive maneuvers necessary in order to dock two spacecraft. What’s landing on the moon, or on mars? It’s “docking” a spacecraft with those objects—called a space rendezvous. So we had to figure out how to do that before going to the moon and much of the Gemini program was spent trying to get that right. There were failures!

What happens is that the Soyuz takes o↵ from Kazakhstan riding on top of the most reliable launch vehicle in history, the Soyuz Launcher. The space- craft is boosted into a low, “parking” orbit and then begins the docking process. Look at Eqn. 1 and Fig. 5. If the lower orbit, RL is the Soyuz spacecraft’s parking orbit and RH is the ISS’s orbit, is the Soyuz going faster or slower than the ISS? (circle)

faster slower

(Hint...it’s faster.) (We will have talked about this just before the project.) Somehow these two objects feeling only Newton’s Universal Gravitation have to come together and that’s remarkably di�cult. Suppose you’re in a lower orbit than the space station...wouldn’t you just point your spacecraft at it and briefly fire your booster rockets like shooting a bow and arrow? That impulse of force would instantly put you into an elliptical orbit and cause you to gain so much altitude that you’d move away from the ISS...and then because your R is larger, you’d be going slower than ISS at that higher altitude! That’s right: aiming at the target moves you further away from the target. That, and a lack of depth perception caused no end of trouble until NASA engineers (yup, including the ladies from the great movie, Hidden Figures) figured it out.

(We just watched a movie about this.) They dock using the “Hohmann transfer” ...that’s what the dotted line in Fig. 5 represents. In order to dock at point B, the boosters need to be fired exactly on the other side of the earth at point A in order to precisely put the spacecraft into a particular elliptical

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orbit to match up with the ISS at about that point...but just ahead of it. Then there’s a careful series of burns to let the ISS come up from behind and dock to the waiting Soyuz...all of which are traveling at more than 15,000 mph.

Back to work.

2 Kepler’s 3rd Law & Newton’s Gravitation, 7 pts

Just before this we looked Kepler’s Third Law and found:

T 2 = 4⇡2

GME R3

T =

s 4⇡2

GME R3 (2)

I’ve plotted this function in Fig. 6 for earth satellites. Here’s an interesting question: how long does it take for the ISS to circle the earth? You’ll need RL from before and you can then mark a big X on Fig. 6 and transfer your answer here:

Roughly, it takes minutes for ISS to go around the earth once.

There is a particularly famous kind of orbit, called “geosynchronous” in which a satellite has a period that is exactly 1 earth day, 24 hours. These satellites can be viewed from earth without the ground equipment having to track the its movement: it’s always at in the same place in the sky. If you have DirecTV or Dish Network, then that’s what your dish is looking at. The satellite has a big “footprint” in its down-broadcast so many customers view the same satellite. DirecTV has 13 satellites (made by Hughes Aircraft, Boeing, and Space Systems/Loral) and Dish Network has 16 (mostly owned by EchoStar).

Our question is: how far up are these satellites? We’ll pick on a particular kind of geosynchronous orbit called a geostationary orbit, which is one that’s directly over the earth’s equator. I’ll tell you that there are 1440 minutes in a day.

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Figure 6: Kepler’s 3rd Law represented by the Period (T) in hours plotted against the distance a satellite is from the center of the earth in km.

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Put a big dot on Fig. 6 at the geostationary point and put a big dot on top cartoon at the beginning of the worksheet.

Tell me how many kilometers from the center of the earth are those satellites.

RL = km

Just to complete the story of satellites....there are a half million pieces of stu↵ orbiting the earth, from actual satellites to man-made debris. Goddard Space Flight Center in Maryland catalogs 2,271 actual functioning satellites now. Of course the satellites that we all deal with throughout our waking days are the Global Positioning Satellites (also called Navstar GPS), the US-funded GPS system. There are 32 GPS satellites in orbit in a medium earth orbit of around 20,000 km.

Put a big circle on the first page cartoons at that GPS altitude.

Ah...you already did that in the quiz, didn’t you. Do it again.

The Hubble Space Telescope’s distance from the center of the earth is about 6,919 km.

Put a big H on the cartoon for its position.

The James Webb Space Telescope is the next generation, and it will be spec- tacular and will launch in October of 2018. It’s orbit will be...sit down...1.5 million km. Can’t plot that on the cartoons...rather you’d need 22 more pages side by side in order to get to that altitude!

Among many cool websites for satellite inventory is http://stu�n.space. You’ll be surprised at the amount of ”stu↵” in especially low earth orbit.

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