S.T.E.M. For the Classroom Magazine 2014 - 2015

VIRGIN GALACTICSuborbital Spaceflight

REACTION ENGINES, LTD.Orbital Payload

BIGELOW AEROSPACESpace Station Design

SPACEPORT AMERICASpacecraft Landing

HOHMANN TRANSFER EQUATIONSChange in Orbital Speed

Transfer Time

THE BOEING CO.Spacecraft Weight

The Rocket EquationLanding on the Moon

S.T.E.M. EDUCATIONThe Coming Paradigm Shift

All Students Have The Right Stuff

Volume 1 | Issue 1 | School Year 2014 - 2015

A Space Station in Low Earth Orbit Bigelow Aerospace makes inflatable space stations that can house six astronauts at a time, and cost many times less than the International Space Station (I.S.S.) to build and assemble. Each module generates electrical power using solar panels (in blue), and radiators expel waste heat (in gray). The number in the name of the module denotes the pressurized volume measured in cubic meters. This particular space station is comprised of two Bigelow BA-330 space station modules connected together along with a SpaceX Dragon docked at one end and a Boeing CST-100 docked at the other. It can accommodate 12 astronauts, and has a habitable pressurized volume of 660 m3. One of the topics in this magazine deals with the cost and specifications of building a Bigelow Space Station, then comparing the results to the I.S.S. currently in Earth orbit.

Image: Bigelow Aerospace

This prototype issue contains a collection of articles that are intended to be used as classroom S.T.E.M. projects, with the aerospace projects completed in the Fall Semester, and the astronautics projects completed in the Spring.

Educators have free use of all material presented in this magazine and accompanying website.

www.stemfortheclassroom.org

____________

Versions of these articles appear inRocketSTEM magazine

a S.T.E.M. advocacy publication.

www.rocketstem.org____________

S.T.E.M. for the Classroom

Editorial StaffManaging Editor: Dr. Rich Holtzin

Space Editor: Joe Maness

Board of DirectorsDr. Rich Holtzin

Dr. Steve RokickiDr. Harry E. CrossDr. Lonnie JuarezProf. Evan Davis

Joe ManessMike Maness

S.T.E.M. for the Classroom MagazineVolume 1 | Issue 1

School Year 2014 - 2015ISSN: pending

© 2014 Re-NewSpace Media

Re-NewSpace, LLC9200 Lagrima De Oro NEAlbuquerque, NM 87111

[email protected]

On the Cover: An R.E.L. Skylon spaceliner is towed to the propellant apron where the spacecraft will be loaded with rocket propellant for a trip into Low Earth Orbit (LEO) and back.

Image: Reaction Engines, Ltd.

S.T.E.M. Education AdvocacyCan we create a better society simply through S.T.E.M. education?

AEROSPACESuborbital SpaceflightHow high can the Virgin Galactic SpaceShipTwo go?

Orbital SpaceflightHow much payload can the R.E.L. Skylon carry to Low Earth Orbit?

Space Station DesignHow much money does it cost to build and assemble a Bigelow space station?

Unpowered Glide LandingWhat is the descent rate of a spacecraft landing at Spaceport America?

ASTRONAUTICSDelta V and Transfer TimeHow can a spacecraft go from one orbital altitude to another? How long will it take?

Spacecraft WeightHow much does a Boeing Crew Module weigh given the number of crew needed?

The Rocket EquationHow much propellant does it take to travel in outer space?

Landing on the MoonIs there a way to pay for the cost of placing humans on the lunar surface?

EditorialDo high school students have the Right Stuff to learn about rocket science?

www.stemfortheclassoom.org

Imagine yourself a high school student taking math courses entailing astronautics or aerospace projects. Most projects usually run about six weeks. You are part of a team of three or four other students and collaboration is imperative. Even though you may

have an inherent fear or dislike of higher math, the detailed lesson plans you’ll be working with are clear, concise, and cool. The projects also entail the best of the best astronautics and aerospace industries, like Boeing, R.E.L. (Skylon), Bigelow, and Virgin Galactic. There is one more thing about these projects geared mainly to Pre-Algebra, Algebra 1 and 2, and Pre-Calculus: each denotes an exclusive S.T.E.M. problem created for your high school.

Welcome to the world of astronautics and aerospace, where they don’t call it rocket science for nothing! What you’ll be experiencing is indeed the real McCoy in the guise of a tangible academic exercise. The above description applies to our S.T.E.M. for the Classroom program. While some advocates for S.T.E.M. projects think or assume students should eventually choose similarly related fields as future employment, we feel differently. In our view, students taking S.T.E.M. courses can choose any line of employment or academic field and still profit from the experience. The cognitive discipline and academics is that exceptional and far reaching! How did these imaginative projects come about? While most S.T.E.M. projects average just a few days, ours were designed for half-semesters or quarters. The intrinsic concept correlates to developing and implementing a robust, comprehensive, and sustainable New Space commercialization program. Moreover, the conviction that best describes our ideology utilizes reuse and commonality to achieve affordable and profitable spaceflight operations. Our approach to sustainable rocketry in all aspects was itself influenced by a movie, October Sky, released in 1999 (and based on the book, “Rocket Boys,” by Homer Hickam). The narrative was centered on a trio of high school students in a backwoods West Virginia coal-mining community, who became interested in launching rockets. One of the students, Homer, grew up to eventually become a NASA engineer, while the other two chose to work in non-S.T.E.M. fields. We were smitten with this film for many reasons. Primarily, we realized its greater potential for seeding minds with elemental constructs for all that follows in life and chosen vocations. As for the apt title of this article, given the recent successful prototype of our Algebra 2 class during the 2012-2013 school year, those students did ndeed boldly go where they never thought they could or would. The courses to follow, now well beyond the prototype phase, include the aforementioned


S.T.E.M. projects for launching student minds into space

While we have no answer for the last statement, a legitimate explanation does exist for the other two questions we struck upon while conducting our research: that the more S.T.E.M. projects that students can be exposed to, the better their earning potential, regardless the education level or occupation. We revere the value of practical education on all levels. This is why we press on with myriad and distinctive learning convictions along such lines, and not just reliance on continued testing that we feel is too rampant in our schools today. For those who say that our society focuses too much on education as a way to get a high paying job instead of a viable reward unto itself, we wholeheartedly concur. The focus on money in our society has also led to questionable behavior on the part of the few that affect the many. Likewise, we believe in the Star Trek scenario, where you do a job because you like the job and because you can grow as a human being, not because of how much it pays. This is why we try to make our S.T.E.M. projects stimulating even if we do have a warped sense of what’s fun (Star Trek pun intended). Besides, most of the students who completed the prototype courses said they had fun working out the parameters of each project. Imagine that: students actually enjoying doing math! Live long and work the problems, indeed! One final note worth repeating: we are not trying to encourage every student to go into S.T.E.M. related fields. On the contrary, we feel that exposing students to empirically-based projects provides for a well-rounded education, regardless what direction they go after High School. We want to help the teacher to encourage students to look beyond the textbook and to achieve something real-world that often lies outside a student’s comfort zone. This way students will always bring their particular talent to the projects, whether it involves art, writing, history, or any other academic subject. In short, we worked out everything for any math teacher who is willing to put forth a little effort to take his or her students higher, literally. Other future S.T.E.M. projects are also listed, including plans to introduce Elementary and Secondary levels of a similar nature.

Pre-Calculus project about to be launched in the fall which will extend through 2014. Which brings us to our thesis of teaching: To offer high school students at all Socioeconomic Status (S.E.S) levels engaging S.T.E.M. projects at no cost. All that’s needed is an Internet connection, which most schools already provide. Our hands-on projects challenge students to step out of their comfort zone by designing real-world space missions using real-world spacecraft data, thereby gaining a better understanding of all four S.T.E.M. facets. As it turns out, designing and planning a space mission for the projects entails the use of the various mathematical concepts and equations students typically learn in high school classes. We also boldly set out on this path and wanted to find a way to give back to the community something totally innovative and highly stimulating. Ergo, a pragmatic approach to education that made better sense and would maintain a student’s interest. Now that we know the classes thus far taught were so well received by our students, we believe even more in the synergy of our S.T.E.M. concept. Additionally, there can be a 100% success rate for all those students to follow. This claim cannot be fostered by typical testing methods mandated for most school programs! What is our pragmatic approach given our version of S.T.E.M. projects? While the abstractions of mathematics involved in astronautics and aerospace can be daunting, only the basics of rocketry equations and mathematics are applied (i.e., algebra, geometry, linear equations, quadratics, square roots, natural logarithms, trigonometry, and pre-calculus). Students will also be required to develop a space mission app using a spreadsheet, develop a slide show using presentation software, embed said documents in a website that they build, and then demonstrate everything in a presentation done in front of their class. But let’s consider where bold students are indeed going and what they think about these projects ahead of them. In short, how do they feel about undertaking such projects? The usual and invariable questions from most students are, “Why do we have to do all this extra work?” and “Why do we have to do all of this S.T.E.M. stuff anyway?” Some might even exclaim, “This is so lame; so not me!”



Sir Richard Branson (opposite page) and crew conduct a practice preflight check inside the cabin of the Virgin Galactic SpaceShipTwo suborbital spacecraft.

Image: Virgin Galactic


For a more in-depth treatment of this High School S.T.E.M. project by Joe Maness and Rich Holtzin visit www.stemfortheclassroom.org.

Plugging that information into the quadratic equation allows us to calculate the time the spacecraft reached maximum altitude, and the maximum altitude itself. All we have to do is find the vertex of the parabola, since that is the point of maximum height.

::

Analysis The SS2 parabolic spaceflight profile shows that the rocket engine cutoff is at a certain Mission Elapsed Time (MET) and height above Mean Sea Level (MSL). This is equivalent to the state a baseball is in at the moment a player releases it into the air.


Vocabulary• Parabola: The graph of a quadratic equation, which for this project is in the shape of an upside-down capital “U”• Quadratic Equation: The equation that creates a parabola when graphed• Vertex: The maximum (or Minimum) point on a parabola

Narrative If a baseball is thrown into the air to another ball player, it will continue moving upward after it leaves the ball player’s hand. The ball will eventually reach a maximum height (which we can calculate), and then drop back down to the other player’s glove. Virgin Galactic’s (http://www.virgingalactic.com) SpaceShipTwo (SS2) follows a flight profile very similar in nature. Can we figure out how high the spacecraft went? Why, I’m glad you asked! That baseball (spacecraft) follows a nice parabolic curve, and can be described using a Quadratic Equation:

h = at2 + v0t + h0

where

• a is a constant = -4.9• v0 is the initial velocity• h0 is the initial height


Image: S.T.E.M. For the Classroom

http://www.virgingalactic.com

Once we find the time at maximum altitude, we can finally use that calculate the maximum altitude.

::

Example Let’s suppose that the SS2 rocket burnout time is at 110 min MET at an altitude of 135,000 ft MSL with a velocity of 2,600 mph. Will it reach space, which is to say, will it go above 62 miles? The students at The Learning Community Charter School (www.tlcnm.net) surely wanted to know! First, as always, we must convert our input into S.I. Units:

vertext = -v0/2a

and

vertexh = a(vertext)2 + v0(vertext) + h0

where

• vertext is the time (after rocket burnout) at maximum height• vertexh is the maximum height

v0 = 135,000 ft = 41,148 mh0 = 2,600 mph = 1,162 mpsSpace = 62 mi = 100 km = 100,000 m

So,

Time of Maximum Altitude = Rocket Burnout Time + vertext = 111.98 min

Maximum Altitude = vertexh = 110,027 m

::

Conclusion As a result of the spacecraft breaking the 100,000 m barrier, the space tourists aboard this particular suborbital spaceflight would have all proudly earned their Astronaut Wings.


A Virgin Galactic SpaceShipTwo on a flight test of the rocket system. Even at this lower altitude, the blackness of space can be seen overhead.


-v02a

http://www.tlcnm.net

00 www.STEMfortheClassroom.comwww.stemfortheclassoom.org

www.STEMfortheClassroom.com 00

The R.E.L. Skylon spaceliner taking off like any ordinary airliner would from a spaceport. Except this particular flying machine can go all the way into space!



Image: S.T.E.M. For the Classroom


Vocabulary• Latitude: The number of degrees north (or south) from the equator.• Orbital Altitude: The height above Mean Sea Level of an orbiting spacecraft.• Orbital Inclination: The angle that an orbit makes as it crosses the equator.

Narrative Across the pond is an innovative rocket company that toils away in relative obscurity designing a revolutionary new space launch vehicle system while being out-shined by comparable U.S. companies such as SpaceX and Virgin Galactic. This launch vehicle design is unique in that it looks and acts like an airplane from takeoff to landing; it just so happens that this particular airplane can remarkably fly all the way into Low Earth Orbit (LEO)! And the best part? It’s reusable, unlike those other expendable launch vehicles. Nice! This futuristic rocket company is called Reaction Engines, Ltd. out of the United Kingdom (www.reactionengines.co.uk), and they want to build the Skylon spaceplane that would fly from virtually any airport on the planet, and incredibly, would not require an expensive launch tower to perform a liftoff! This spaceplane would operate just like any ordinary airliner, except if you want to operate this bird, you’d better call it a spaceliner! The REL Skylon spacepl -er- spaceliner is designed to have ...........................................

a payload bay just like what the US Space Shuttle had, albeit a bit smaller. Can we derive the equation of the payload capability of the REL Skylon? Sure we can! The students in the High School Algebra 2 class at The Learning Community Charter School (www.tlcnm.net) in Albuquerque, NM certainly did!Fortunately for us, REL has made public their extensive data on the Skylon, including the lift capability. Using this information, a graph can be created. The horizontal axis represents the orbital altitude, and the vertical axis represents the payload weight. For the purposes of this exercise, we will be launching from – err – taking off from Spaceport America near Las Cruces, New Mexico, which is located at 33 degrees N Latitude.



http://www.reactionengines.co.uk


The orbital inclination will always be the same as the latitude of the launch site (unless, of course, you expend more propellant to go into a different orbital inclination), so the orbital inclination is 33 degrees. The blue line in the graph represents such an orbit. The red line represents an orbital inclination to the International Space Station (ISS) , and the yellowish-orange (orangish-yellow?) line represents a polar orbit.

::

AnalysisAs we can see from the graph, it not only takes more propellant to go higher but to also change the inclination. Therefore (and again for the purposes of this exercise), we will only be looking at the blue line, i.e., the “at latitude” line. For example, the graph tells us that the lift capacity of the REL Skylon at latitude to a 650 km orbital altitude is about 11,000 kg (11 mt). The endpoints of the graph show us that the REL Skylon can lift 14,250 kg to 250 km, and 9,790 kg to 800 km. Putting this together, we get two points!

(250, 14290) and (800, 9790)

We can now write the general linear equation in slope-intercept (y=mx+b) form for the payload launch capability, which is operating from a launch site of 33 degree Latitude going into a 33 degree inclined orbit:

PayloadAlt = mAlt + Payload0

where

• PayloadAlt = The weight going into space

• m = constant of variation• Alt = Orbital Altitude• Payload0 = Initial payload weight

Using the techniques of finding the slope between two points and the vertical-axis intercept, we get:

• m = -8.18• Payload0 = 16,335 kg

Therefore, our REL Skylon/Spaceport America Linear Equation becomes

PayloadAlt = -8.18Alt + 16335

Example What is the lift capacity of the REL Skylon operating from Spaceport America to an orbital altitude of 400 km? The input to the REL Skylon/Spaceport America Linear Equation is the orbital altitude, which is 400.

Payload400 = -8.18(400) + 16335 = -3272 + 16335

= 13,063 kg

Conclusion The REL Skylon spaceliner could lift around 13 mt to a 400 km, 33 degree LEO from Spaceport America, which can be verified by the graph. For comparison, the SpaceX Falcon 9 has a published lift capacity of about 13 mt to a 28.5 degree LEO. The major difference, of course, is this one would be reusable, which would help drive the operational costs down. So, why are we still using throwaway rockets in the 21st Century again?


An R.E.L. Skylon spaceplane ready for takeoff for a journey all the way into space!


BIGELOWSPLASH 1


A Bigelow space station comprised of one (1) BA-330 habitat module, two (2) Sundancer habitat modules, and one (1) Propulsion Bus/Docking Node (PB/DN). This particular configuration can house 12 crew and has a total pressurized volume of 690 m3. Two (2) Boeing CST-100 Crew Modules are used to ferry the crew back to earth.



Vocabulary● BA-2100: A Bigelow module. Volume: 2,100

m3. Weight: 100 mt. Crew: 16, Cost: $500M.● BA-330: A Bigelow module. Volume: 330 m3.

Weight: 25 mt. Crew: 6. Cost: $125M.● Falcon Heavy: An expendable launch vehicle

from SpaceX. Payload: 53 mt. Cost: $150M.● Propulsion Bus/Docking Node (PB/DN): A unit

used to reboost the space station due to orbital decay coupled with a module that allows Bigelow modules to be attached together. Weight: 17 mt. Cost: $75M.

● Space Launch System Block I-A (SLS I-A): An expendable launch vehicle from NASA. Payload: 105 mt. Cost: $750M.

Narrative To live and work and be productive in space, you have to have a place to call home. While the Earth is certainly a great place to go home to (free air! free water! free - well, you get the point), it would become very expensive indeed if every time you knocked off your shift in space you took a ride back to Earth, and then got up the next day to fly back into space again! The obvious answer is to place your home, your city if you will, in space. Shopping around for what's available to use to build our city, we happily find Bigelow Aerospace, makers ........

of the famous BA-330 and BA-2100 space station habitat modules (www.bigelowaerospace.com), where the names of the modules denote the pressurized volume of each unit. These inflatable modules go into space, where they, well, you know, inflate. Astronauts then move into what is essentially a balloon in space (aren't all space modules really just that?). Ah, but what a balloon! Crew capsules can dock at either end, and power is derived from solar panels, while excess heat is dumped into the biting cold of space using radiators. It even has windows! Home sweet home, indeed.

Analysis To launch these excellent habitat modules into space, we obviously need a launch vehicle. Once again, shopping around for what's available to use to launch our city we find two Expendable Launch Vehicles (ELV). These rockets haven't been built yet, but so long as funding continues they will be one day. Since the SLS I-A ELV can carry 105 mT into Low Earth Orbit (LEO), and one BA-2100 weighs 100 mT, it can carry only one unit at a time. We will call this the “BA-2100 Stack.” The Falcon Heavy ELV can lift 53 mT to LEO, and each BA-330 weighs 25 mT, so it can carry 2 units at a time (assuming, of course, that it could fit in a payload shroud). We will call this the “BA-330 Stack.” The PB/DNs each weigh 17 mT, so 3 units will fly on the Falcon Heavy ELV to LEO. This will be called the “PB/DN Stack.”




http://www.bigelowaerospace.com

BA-2100 Stack● Cost: (1) SLS IA + (1) BA-2100 = $750M +

$500M = $1,250M● Weight: (1) BA-2100 = (1) 100 mt = 100 mt● Volume: (1) 2,100 m3 = 2,100 m3

● Crew: (1) 16 = 16 AstronautsBA-330 Stack

● (1) Falcon Heavy + (2) BA-2100 = $150M + (2) $125M = $400M

● Weight: (2) BA-330 = (2) 25 mt = 50 mt● Volume: (2) 330 m3 = 660 m3

● Crew: (2) 6 = 12 AstronautsPB/DN Stack

● (1) Falcon Heavy + (3) PB/DN = $150M + (3) $75M = $375M

● Weight: (3) 17 mt = 51 mt

So let's build us a city in space, shall we? Hey, if the high school students at The Learning Community Charter School (www.tlcnm.net) in Albuquerque NM can have fun with this, then so can we.

Example The folks at Bigelow Aerospace have made it easy for us: they've already designed a very nice space station! It's called the "Hercules Resupply Depot," but we'll just call it "Home." Thanks, Bigelow! As you can see from the poster, we'll need three (3) BA-2100s, six (6) BA-330s, and three (3) PB/DNs to complete the design. We can therefore calculate how many "stacks" we'll need:

(3) BA-2100 = (3) BA-2100 Stacks● (3) $1,250M = $3,750M● (3) 100 mt = 300 mt● (3) 2,100 m3 = 6,300 m3

● (3) 16 Crew = 48 Crew(6) BA-330 = (3) BA-330 Stacks

● (3) $400M = $1,200M● (6) 25 mt = 150 mt● (6) 330 m3 = 1,980 m3

● (6) 6 Crew = 36 Crew(3) PB/DN = (1) PB/DN Stack

● (1) $375M● (3) 17 mt = 51 mt

Adding everything up we get:

● Total Cost: $5,325M● Total Weight: 501 mt● Total Volume: 8,280 m3

● Total Crew: 84 Astronauts

Notice that our volume is close to the 8,300 cubic meters advertised in the Hercules image.

Conclusion Currently, the International Space Station (I.S.S.) is orbiting the Earth, but is due to retire in the near future. There really is no comparison, however, in their home and ours:

COST WEIGHT VOLUME CREWHERCULES: $5,325M 501 mt 8,280 m3 84I.S.S.: $150,000M 450 mt 837 m3 6

While the two space station weights are nearly equal, the total volume is nearly ten (10) times the International Space Station with 14 times as many crew, all for 1/30th the cost. It's keeping up with the Jones’ in reverse! Maybe when the I.S.S. retires in the near future the international community can move on up to a bigger and better place that’s a wee bit more affordable, si?



SPACEPORTSPLASH 1


SPACEPORTSPLASH 1

The layout of Spaceport America near Las Cruces, NM. It is the very first facility ever designed and built from the ground up as a spaceport. Virgin Galactic, among many other tenets, will be moving in soon to begin spaceflight operations.

Image: Spaceport America


Vocabulary● Altitude: The distance a spacecraft is above a

given point on the ground● Distance From Spaceport: The ground

distance from the edge of the runway to the spacecraft

● Glide Distance: The distance the Landing Laser measures to the spacecraft

● Glide Slope (θ): The angle a spacecraft makes to the horizontal

● Landing Laser: The laser used to determine the distance to a spacecraft

Narrative Any spacecraft returning from space is always out of propellant. This is because all of the propellant is used up during the trip into space; consequently, there is none available for the trip back. Wait. What? So how does it land? All machines that have wings can glide, that is, fly with the engine turned off. Some glide better than others, but still, they all glide. The key to gliding in an unpowered spacecraft is speed. The faster the spacecraft flies through the atmosphere, the more efficient the wings. Thus a more efficient glide. Altitude is the other important component, in that altitude allows the spacecraft to build up speed if needed. This is why spacecraft ..............

Image: Reaction Engines Ltd.

come in with their nose down; they are maintaining their airspeed. As they cross over the edge of the runway, the nose is pulled up and the spacecraft flattens out it’s glide as air is packed underneath the wings. It’s then just a simple matter of letting the spacecraft sink to a gentle touchdown. Once on the ground the nose is kept in the air “wheelie” fashion, so that speed can be bled off without using brakes because they can get very hot very quickly. After the nose comes down on its own, the brakes can then be (sparingly) applied. Eventually, the spacecraft rolls to a full stop. Back home once again!




Analysis So what was the Altitude and the Distance from the Spaceport of the spacecraft that just landed? Good question! Why, just the other day some high school students at The Learning Community Charter School (www.tlcnm.net) were asking the same question. Great minds surely do think alike. We need two pieces of information to get started: the Glide Distance (GD) and the Glide Slope (θ). We will use the Landing Laser to determine this distance and the angle. It will take a “snapshot” of this information whenever we need. Once we have Glide Distance and Glide Slope, which is to say, once we have a side and an angle, we can form a right triangle.

Therefore, we can use trigonometric identities to solve for the other two sides. We can also see that the Glide Distance becomes the hypotenuse of the right triangle. Moreover, since sine is defined as the opposite side divided by the hypotenuse, and cosine is defined as the adjacent side divided by the hypotenuse, we can write the two trigonometric equations using this information.

● DIST = GD cosθ● ALT = GD sinθ

So now we can calculate the altitude and the distance to the Spaceport for our spacecraft. Note: we will be using degrees instead of radians to keep things a little simpler.

Example The Landing Laser at this moment in time reads a Glide Slope of 35 degrees with the spacecraft at 15 miles distant. What is the altitude and ground distance from Spaceport America (www.spaceportamerica.com) of the spacecraft? We must first convert 15 miles to meters, which comes to about 24,140 m. Therefore,

● ALT = (24140)sin(35) = 13,846 m● DIST = (24140)cos(35) = 19,774 m

Conclusion So when the spacecraft is about 15 miles away, it is at an altitude of 13,846 m (8.6 mi) above the ground, and is 19,774 m (12.3 mi) away. Is there a way that we can check our results? Sure there is! Use the Pythagorean Theorem.

(8.6)2 + (12.3)2 = 73.96 + 151.29 = 225.25

The result is close enough to 15 squared. We therefore are on track to a safe landing!

θ

GDALT

DIST

A Virgin Galactic SpaceShipTwo coming in for an unpowered glide landing. Note the nose down attitude of the spacecraft as it aims for the edge of the runway. Note also that this particular spacecraft configuration has no engine!



http://www.spaceportamerica.com


Delta V

Astronauts Kowalsky and Dr. Stone on their way to the International Space Station from a scene in the hit movie “Gravity” starring George Clooney and Sandra Bullock.

Image: Warner Bros.

and the


of the Situation



In the recent film release, “Gravity,” astronauts are left stranded at the Hubble Space Telescope (HST) after a pretty horrific accident. Since one of the astronauts was wearing a Manned Maneuvering Unit (MMU), they used the nifty apparatus to get to the International Space Station (ISS). We like this movie, even though much has been written in many space blogs have been written about how this feat is virtually impossible, simply because of the different orbital heights and orbital inclinations. The verdict: impossible! So the question is whether there is a way to find out if the accusation is true? I mean it’s not like there’s a Pre-Calculus class out there in America that is actually running a S.T.E.M. project that calculates Delta V, is there? Well, guess what? There really is. As it turns out, The Learning Community Charter School (www.tlcnm.net), a High School in Albuquerque, NM, has a Pre-Calculus class that is running a S.T.E.M. for the Classroom Delta V project. They even have an app for it! Moreover, they presented their findings to their class on October 17, 2013. Now the question is can we use their app to determine the required Delta V that our hapless astronauts need? Sure we can! Let’s plug in some numbers and find out how. Note: For this exercise, we will ignore the orbital inclination and destination positioning. Uh…don’t ask. The equations needed are a bit sticky, but they work nicely, especially in a spreadsheet. Here’s some technical terms to match what the students had to work with: the periapsis is the lowest point of an elliptical orbit, while the apoapsis is the highest point.Since the astronauts start at the higher orbital altitude, and try to thwart their momentum instead of applying an opposite thrust, we’ll only concentrate on the Apoapsis Delta V. First things first: if you want to change your orbital altitude, you must know the orbital altitudes that you want to change from and to. Looking up the particulars of HST and the ISS, we get that the HST Orbital altitude is 354 miles and the ISS Orbital Altitude is 205 miles (your mileage may vary, since their orbits are decaying). Of course, we have to always first convert everything to S.I. units for the app. Therefore, since ........................

one mile equals 1.609 km, The HST orbital altitude is at 570 km and the ISS orbital altitude is 330 km. We’re now set to tackle the question. To the Batmobile app!

The inputs are: Periapsis: 330 and Apoapsis: 570 (we can ignore the On-Station Time). The output is: Apoapsis Delta V is 67 meters per second (mps). Not much of a change needed, huh? But can the MMU handle it? The MMU is rated at 24.4 mps. Since 24.4 is no where near 70, we therefore can safely conclude that the MMU does not have the capability of reaching the ISS from the Hubble. Too bad this kind of stuff isn’t taught at a High School somewhere, huh? Oh wait. There is a High School out there somewhere that is trying their best to provide students with relevant math topics after all. This entire lesson was taught at TLCCS, and will continue to be taught, along with other innovative and fun S.T.E.M. stuff geared for Pre-Algebra, Algebra 1 and Algebra 2 classes. Besides, wouldn’t this lesson be better than wasting time on Standardized Tests? Isn’t this a better way of getting students to a deeper level of understanding in mathematics and critical thinking? By the way, the movie has great human drama and has got some great special effects. However, what’s projected on the screen simply ain’t happening. Our students proved as much given all the above.


Vocabulary• Apoapsis Delta V Burn: The rocket firing at the highest point of a Transfer Orbit.• Periapsis Delta V Burn: The rocket firing at the lowest point of a Transfer Orbit.• Transfer Time: The time between apoapsis and periapsis Delta V rocket firings.

Narrative Getting from one orbital altitude to another not only takes delta v, it also takes time. It isn’t as simple as aiming yourself and firing your rocket. Orbital mechanics is a ballet in space, where the lower your orbit the faster you go, and vise versa. Any change in orbital velocity and you wind up in an elliptically-shaped orbit, instead of the fairly stable circular orbit.

Transferring from one orbit to another is accomplished by changing a circular orbit into an elliptical orbit The Transfer Time is the time it takes for the spacecraft to follow the path of the yellow curve. To get from one orbit to another, we must push ourselves off, and then brake when we get there. We

coast in between these two maneuvers. We can calculate how much time we will coast from one point in space to another.

AnalysisThe equation to calculate the transfer time from one orbital altitude to another is given by one of the Hohmann Transfer Orbit Equations:

where• Ts is Transfer Time in seconds• r1 is the smaller orbital radius• r2 is the larger orbital radius• µ (mu) is Standard Gravity = 9.81 m/s2

• π (pi) is the constant 3.14159265358…

The number π is involved in the equation because we are actually going around in circles.

Example Let’s suppose you want to go from the Hubble Space Telescope (HST) to the International Space Station (ISS). How long would it take to get there? The orbital radius is simply the sum of the orbital altitude plus the radius of the Earth. So for the HST (r1), we get 6,708.14 km, and for the ISS (r2) we get 6,948.14 km. Plugging everything in the equation, we get 2,673 seconds of transfer time. If we divide that by sixty (seconds per minute) we get 44.55 minutes.

Conclusion It takes about half an orbit to transfer between two orbital altitudes that are not that far apart.


Image: Warner Bros.

Image:Wikipedia


Apollo 17 astronaut Dr. Harrison H. Schmitt gets his picture taken by his Commander Gene Cernan on the surface of the Moon next to the Lunar Roving Vehicle (LRV). The LRV was the very first vehicle designed and built to operate only on the lunar surface.

Image: NASA


Vocabulary● Crew Module (CM): The part of the spacecraft

where the astronauts live and work● Crew Module Weight: The total weight of the

CM including provisions and the crew● Crew Size: The number of astronauts aboard

a spacecraft● Mission Duration: The total time necessary to

accomplish a space mission

Narrative A long time ago, in a galaxy very near our own, there once existed a wondrous vision of the future, where spacecraft were bountiful, and imaginations could soar. Given our astronautics concepts for S.T.E.M. Education projects the Boeing (www.boeing.com) Space Tug Study (circa 1971) exemplifies this mighty fine vision. Notably, it was a complete design that not only included a Crew Module (CM), but also an Engine Module (EM), the latter to be fully discussed in a future issue in this series. The CM and EM were designed to attach together, and very much similar to the Apollo spacecraft that heralded NASA's endeavors in the so-called space race of that era. Hence, a near twin to the Apollo design. Indeed, the CM and EM craft included a design for a lander kit that could be attached for the sole purpose of landing on the moon. (This scenario ......

will also be discussed later in this series.) Overall, the Boeing Space Tug design was inspirational, as well as functional. Moreover, as an innovative spacecraft it was well ahead of its design and era. Remarkable! That being said, the space tug was never built. However, we can still make something good out of its advanced design. Namely, our S.T.E.M. classroom projects geared to this study posits our students working on real space missions using a real spaceship model and real numbers. In short, there are no virtual implications in our applied concepts and mathematics. Note: Boeing, as of this publication, is a top contender for NASA's Commercial Crew Development (CCDev) program, where they, and companies such as the SpaceX Dragon, compete for NASA's coveted Crew Capsule contract.

Analysis Rocket scientists obsess over spacecraft weight. They also obsess over other things like power consumption. Still, the weight factor seems to be the biggest obsession of all. Spacecraft CM weight thus depends on many factors, such as the number of astronauts needed or the duration of the mission. Some spacecraft component weights will also vary depending on these and other factors, such as Environmental Control and Life Support systems, while others will weigh the same no matter what, such as the CM structure itself. The Boeing study provided a graphic of the CM, ..............

Image: The Boeing Co.

The Boeing CST-100 Crew Capsule



http://www.boeing.com

where we can see the actual numbers that the Boeing engineers came up with. Again to mention the point, it must be stressed how our S.T.E.M. classwork projects use real numbers on a real spaceship design. As we can see from the following graphic, a crew of fifteen will allow for a 2-day mission, and the CM will weigh 9,386 pounds, whereas a crew of three yields a 50-day mission and weighs in at 9,755 pounds (all weights include the crew). True, these figures look suspiciously like points on a Cartesian graph, which means that they are linear equations. Let's see if we can derive these equations, just like the high school students at The Learning Community Charter School (www.tlcnm.net) did! We'll use the awesome Slope-Intercept (y=mx+b) form. First, we'll make the Crew Size the independent variable. Therefore we get the points (3, 50) & (15, 2) and (3, 9755) & (15, 9386). We can thus calculate the slope of the two linear equations.

● Slope1 = = -4

● Slope2 = = -30.75

Plugging in one of the points, get the intercept.

● b1 = 2 - (-4)(15) = 62● b2 = 9386 - (-30.75)(15) = 9,847.25

Therefore, the two linear equations are (C = Crew)

● Mission Duration = -4C + 62● WeightCM = -30.75C + 9847.25

Note: The Mission Duration is measured in days, and the CM Weight is measured in pounds. In this case, there was no real need to convert to the metric system, so we decided to keep this exercise simple.

Example Suppose you want to take a crew of 10 astronauts on a space mission. What would be the mission duration of the mission? Most importantly, what is the CM weight?

Using the linear equations that we just derived, we plug in C = 10 into the equations.

● Mission Duration = -4(10) + 62 = 22 days● WeightCM = -30.75(10) + 9847.25 = 9,540 lbs

Conclusion For this particular design, a crew of 10 would have allowed the spacecraft to stay aloft for about 3 weeks, and would have weighed under 10,000 pounds. Nice! The Boeing Space Tug Study spacecraft, if it had been built, would have been a most formidable and versatile spacecraft, well ahead of its time.


9386 - 975515 - 3

2 - 5015 - 3



Vocabulary● Dry Weight (M1): The weight of the spacecraft

fully loaded excluding propellant● Engine Module (EM): The part of a spacecraft

that holds the propellant tanks and the rocket engine.

● Exhaust Velocity: The velocity of the escaping gas exiting a rocket.

● Gross Weight (M0): The weight of the spacecraft fully loaded including propellant

● Inert Weight: Weight of the EM● RL10 Engine: The rocket engine used in the

Engine Module (EM)● Standard Gravity (g0): The acceleration due to

free fall, equal to 9.80665 m/s2

● Specific Impulse (ISP): The force with respect to the amount of propellant used per unit of time. This is why the units are in seconds.

Narrative When last we visited the Boeing Space Tug Study, (www.boeing.com) we learned about the Crew Module (CM) and how to calculate the weight based on the number of crew aboard. For example, a crew of ten gets a 3 week mission weighing in at around 10,000 lbs (4,536 kg). However, the CM isn’t going anywhere unless there is a way to make it go. Enter the Engine Module (EM). This machine has everything you need to go on a .........

space mission: propellant tanks, reaction control system, batteries, and of course, the reliable RL10 rocket engine. This engine is useful in that it can be restarted several times during a space mission, burning the highly efficient Liquid Hydrogen (LH2) fuel. Very nice! The CM would have attached to the EM, and the two would have become one, as they say.

It really would have been quite an awesome machine, because the EM would have carried any type of payload, whether it was the CM or satellites. It was designed to fit in the payload bay of the US Space Shuttle, where it would be let loose in space to perform its space tug duties, then fit back inside another shuttle for the ride back home for refurbishing and reuse. What a great idea! So can we use this Boeing information to make a real-world space mission of our own? You bet your last rocket!

Analysis Fortunately, Boeing left behind some great graphics of their work. Not only is there a great image of what the EM would have looked at on the inside, we also have some astronautics numbers! Thanks Boeing!

Image: Mark Wade

Image: NASA

One of the many excellent uses for a Space Tug: boosting satellites to

Geosynchronous Earth Orbit!





The most useful equation in all of rocketry is appropriately called the Rocket Equation, also known as the “Tsiolkovsky Equation” named after its discoverer.

ΔV = VEXHln( )

It tells rocketeers how much of a change in velocity (ΔV) is available given the payload weight, the propellant weight, and the exhaust velocity (VEXH) of the gasses being expelled by the rocket. The exhaust velocity of a rocket is the engine’s ISP multiplied by the standard gravity of Earth. Using the graphic from the Boeing study, we see that the RL10 at the time had an ISP of 460 s. We can thus calculate the engine’s VEXH.

VEXH = ISP • g0 = (460)(9.80065) = 4,511 meters per second (mps)

We see on the graphic that the inert weight of the EM is 5,610 lbs. This mean that if we include the CM with 10 crew on a 22 day mission that weighs 9,540 lbs, the dry weight of the EM becomes 15,150 lbs. Finally, the graphic shows that the propellant weighed 39,800 lbs. We now have everything we need to solve the rocket equation.

● M1 = WeightINERT + WeightCM = 15,150 lbs● M0 = M1 + WeightPROPELLANT = 54,950 lbs

Note: we will leave the units in pounds instead of converting to the metric system since the units cancel in the equation.

Putting everything together, we get

ΔV = (4511)ln( ) = 5,812 mps

Example The students at The Learning Community Charter School (www.tlcnm.net) had designed a mission to go from a Bigelow space station in Low Earth Orbit (LEO) to a wayward satellite at about 11,000 km altitude to conduct repairs. Will they make it? Looking up the ΔV requirements to go from LEO to the satellite, we see that we need a 2,868 mps change in velocity. It therefore takes twice the ΔV requirements for the round-trip, or 5,736 mps.

Conclusion For this particular spacecraft configuration, the ΔV capacity of 5,812 mps exceeds the required ΔV of 5,736 mps. We therefore conclude that this mission can be safely flown with this configuration and recommend that crew selection begin immediately. Boeing truly had a remarkable space vehicle on the drawing board back in the day, no?

m0m1

5495015150



Enter the Boeing Space Tug Study (circa 1971). (www.boeing.com) The study included a design for a Crew Module (CM) and an Engine Module (EM) that were connected together. The spacecraft held one last surprise: a lander kit designed to be attached to the spacecraft to form a reusable lunar lander. The lander kit included the landing legs, a landing radar, extra batteries, etc., and weighed 890 lbs. Boeing truly outdid themselves back then. Despite how awesome the Boeing space tug was, it was never funded, which is to say, it was never built. But what if it had? Would there have been a way to pay for not only the development and engineering cost, but the day-to-day operating costs as well? We estimate a lunar investment of no more than $40B (USD) total.

Vocabulary● Landing Kit: Includes the lunar landing legs,

infrastructure, landing radar, etc.● Lunar Investment: The amount of money

needed to fully fund a mission to the Moon● Lunar Material: A certain substance that can

only be found on the Moon● Powered Ascent Initiation (PAI): The lunar

liftoff ΔV requirements equal to 1,890 mps● Powered Descent Initiation (PDI): The lunar

landing ΔV requirements equal to 2,181 mps● Return On Investment (ROI): The percent

increase of a profitable investment

Narrative Going into space is an expensive proposition; there really is no way around this important fact. The old adage “No bucks, no Buck Rogers” certainly encapsulates this sentiment. This (probably) is why only nations so far have attempted this feat. But there is a sea change underway... Private industry has been getting into the act, albeit still with a little help from the government. So far, however, the Return On Investment has been woefully inadequate. There has to be a way to not only pay for everything that is needed for a trip to, say, the Moon, but to also make a reasonable ROI (we define a “reasonable” ROI as greater than 10%).

Image: Beijing Institute of Space System Engineering

The Chang’e-3 lunar lander and Yutu lunar rover on the surface of the Moon.

Imag

e: M

ark

Wad

e




Analysis There exists on the Moon a commodity that is relatively easy to mine and has sold for $442,500 per carat (a high quality diamond goes for $5,000/carat). The Boeing space tug with a lunar lander kit could have easily brought back this Lunar Material. But how much of it? To the Rocket Equation! Note: for a detailed treatment of this equation, please see our previous article.

Example A Payload Tray weighs 1,500 lbs, and the science payload we will deposit on the lunar surface is 8,250 lbs. We will replace the science equipment with the same amount of this lunar product. We will continue to use the CM from the previous articles.

m1 = Inert + Kit + CM + Tray + Payload= 5610+890+9540+1500+8250 = 25,790 lbs

m0 = m1 + Propellant= 25790 + 39800 = 65,590 lbs

The mass ratio therefore becomes 2.54:1.

Since the rocket engine nozzle has to be retracted to make room for the landing, the ISP loses 3 seconds. Thus the VEXH is 4,482 mps, and the delta vee is ............

ΔV = VEXHln(ratio) = (4482)ln(2.54) = 4,183 mps

Looking up the ΔV budget requirements for a lunar landing and liftoff reveals 4,071 mps. Therefore, this spacecraft configuration is sound. This means that we can bring back 8,250 lbs (3,742 kg) of Lunar Material. However, we will not use $442,500/carat as our average selling price. Instead, we will use $2,500 per carat. There are 5,000 carats per kilogram, so our average selling price is $7,500,000/kg. Our Gross income is $46,776,768,768. Subtracting the lunar investment from this amount equals $6,776,768,768.

Conclusion As every high school math student knows, just like the students at The Learning Community Charter School (www.tlcnm.net) knows, the percent increase is defined as new minus old divided by old times 100.

ROI = %Increase = (100) = 16.94%

So we get to walk on the Moon, we get to reuse the lander, we get our money back, and with a tidy profit! Now, what can this very mysterious Lunar Material possibly be? The answer is: why, lunar material of course; better known as Moon rocks!

Reference: http://en.wikipedia.org/wiki/Moon_rock

The Apollo 17 Lunar Module “Challenger” on the surface of the Moon. This would be the last landing on another celestial body of a spacecraft with a human crew (so far)...

Image: NASA

$6776768768$40000000000



REFLECTION

Why Learn Mathematics Using a S.T.E.M. Approach?by Joe ManessTLCCS Mathematics Teacher

I am one lucky guy. The students at The Learning Community Charter School (TLCCS) in Albuquerque, NM are as fearless as they are diligent and mindful given their astronautics and aerospace projects. There really is no other way to describe them. They take on each S.T.E.M. assignment with vigor and curiosity, then tackle each math problem with attitude and tenaciousness. So how did I get so lucky? Was it because I work at a charter instead of one of the non-charter high schools? I don't think that is it. The smaller classroom sizes do matter, regardless what our State's Secretary of Education claims. My classes are the proof. Any high school can get these results if they had the courage (read, "funding") to reduce the class sizes. Or was it because of the backing of the most awesome principal to ever captain the helm of a school? Viola Martinez certainly does take education seriously. But that isn't it either. While the full-throated support I got from her for these projects certainly helped (a lot), it wasn't really necessary. If my principal had just ignored me, I would have still moved forward with the projects anyway. That being said, was it just a better caliber of students? Well, that's not it either. My classes are just as diverse as the rest of the population, with just as diverse capabilities and interests as everyone else. Perhaps I am a lucky fellow because of the Common Core standards. Please. The Common Core standards are certainly applied to every lesson taught. Then again, that's only because S.T.E.M. education covers all the standards. Besides, these projects do not "teach to the test", so that fact alone rules it out. So, again, I ask the question: How did I get so lucky? Here's my answer. . .

::

One cold morning I walked into my class to begin a math lesson involving aerospace technology. My students were looking at me like I was from some other planet, maybe even from some other dimension of time. Slowly, the students began to comprehend ideas such as the quadratic equation and the parabola and the maximum or minimum of a curve. ......

They learned how high Virgin Galactic's SpaceShipTwo would go, and to visualize the spaceflight as a graph. I began to wonder who these people were, and from what far away place they came from. Or was I the one out of place? I then went on to another class and did the same thing, only with these students we used natural logarithmic equations to determine the change in velocity of an orbiting spacecraft using information from a study completed decades earlier. It was only then that I realized that I was indeed on the wrong planet (which accounted for the earlier stares from the so-called students), and that they were some sort of alien species from some other multiverse. My suspicions were confirmed when I asked the creatures to write about their S.T.E.M. education experiences, and that the top essay would be published on this very page of this very magazine. Alas, it was not to be. Instead, it was an impossible task to pick a winner, because every entity in these two classrooms had written eloquently and passionately about parabolic spaceflight and delta V calculations. I actually wept after reading their essays, for I understood that I would now have to fill this page with something else, and that I would never see my beloved home planet again. . .

::

To be honest, I have no earthly idea how I got so lucky. I do admit to being a space and math nerd, and I have delved into space history to piece together these projects that you find within these pages. I have the backing of my principal, even if I don't have the backing of my State’s educational system. I even have the backing of my second-in-command, Dr. Rich Holtzin, who always gets to wear his editor's hat on our articles. Sheesh! Yet none of it, even collectively, is it. Ultimately, it's really about the students. Thus it's not about the school, the principal, or even the Common Core. It has always been about the students. They have been allowed the freedom to make mistakes and grow, which in turn, fostered a learning community. By Jove, I think that's it!


A Reusable Spaceliner in Low Earth Orbit Reaction Engines Ltd., an aerospace company out of England, has created a truly remarkable flying machine in the Skylon spaceplane. The Horizontal Takeoff/Horizontal Landing design has been the dream of rocket scientist for decades. Earlier designs had been sound engineering-wise, but had paid the price in horrendous weight because of the three different types of engines required. The first engine was a normal jet engine burning jet fuel used for supersonic flight. The second kind was a RAMjet engine burning jet fuel used for hypersonic speeds. The last engine was a rocket engine burning Liquid Hydrogen (LH2) for the portion of the flight outside the atmosphere. So, it used three different engines and two different fuels. R.E.L. instead uses one type of engine throughout the three flight regimes. This hybrid engine is called SABRE (Synergetic Air-Breathing Rocket Engine). When in jet and RAMjet modes, the oxidizer is taken from the atmosphere; Liquid Oxygen (LO2) is only used during the rocket phase. Thus, the engine burns only LH2 all of the time, so instead of three engines and two fuels, we get one engine using one type of fuel. Nice! One of the topics in this magazine deals with the payload capability of the Skylon spaceliner given the location of the launch site and the orbital inclination, then comparing the results to the SpaceX Falcon 9 Expendable Launch Vehicle.


a place where students and their teachers take S.T.E.M. education waaaaay too seriously...

All images: The Learning Community Charter School

S.T.E.M. For the Classroom Magazine 2014 - 2015

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S.T.E.M. For the Classroom Magazine 2014 - 2015