Ridley Scott does it again. That’s lit­er­ally the only thing I can think of right now after I’ve fin­ished watch­ing the movie, the Mar­t­ian. I thought Alien was an amaz­ing movie but the Mar­t­ian is def­i­nitely bet­ter. I was sus­pi­cious that it was one of those dump, soul­less CGI movies and didn’t bother to catch it at the the­ater. Bad de­ci­sion. Scott is one of those rare di­rec­tors who can take any­thing and make it into a one of a kind movie you’ll re­mem­ber for ever. Oh yeah, five years from now I’ll re­mem­ber the Mar­t­ian al­right, not Grav­ity. Just so I won’t for­get any­thing let me write down the story.

Matt Damon gets left be­hind on Mars after a storm de­scends on the Aries 3 mis­sion site and threat­ens to tip over the MAV (Mars As­cent Ve­hi­cle). The rest of the six mem­ber crew jet­ti­sons out leav­ing Matt for dead. Matt is still alive but the com­mu­ni­ca­tions an­tenna has bro­ken off and pierced his suit, wound­ing him and de­stroy­ing the bio­m­e­try sen­sors. But it also plugs the tear in his suit, sav­ing his life. Matt must learn to sur­vive till the crew of Aries 4 gets there four years later. He grows food in­side the hab by bring­ing in Mars soil and using de­hy­drated human exc­reta as ma­nure. Matt is a botanist and he knows that you need 40 liters of water to farm a cubic meter of soil, he finds some pota­toes they’d set aside for thanks­giv­ing and uses those as seed to grow more pota­toes. To get the needed water he uses irid­ium to cat­alyze the de­com­po­si­tion of hy­drazine (which is NH₂NH₂) into ni­tro­gen and hy­dro­gen gas. Then he burns the hy­dro­gen gas using the oxy­gen in the hab at­mos­phere to get water. He soon has food to last him for more than 900 Mars days (of 25 hours each called a sol in the movie) in­stead of the 400 Mars day worth of food he had with him in the hab. Around this time he reequips one of the rover from parts scav­enged from the other and starts ex­plor­ing. To save en­ergy he shuts of the heater and in­stead uses a nu­clear ther­mo­elec­tric gen­er­a­tor to heat up the rover. By this time Nasa re­al­izes that Matt is alive from satel­lite pic­tures of the hab taken from Mars orbit. Matt finds the old pathfinder lan­der and brings it back to use it to com­mu­ni­cate with Earth which he does using ASCII and hexa­dec­i­mals. Un­for­tu­nately the hab ex­plodes de­stroy­ing the plants and leav­ing him just the rover for shel­ter. But Nasa fig­ures a way to bring him home ear­lier than the Aries 4 mis­sion by rerout­ing Her­mes around Earth and back to Mars so that Matt can ren­dezvous with Her­mes using the MAV from the Aries 4 mis­sion.

The vi­sual ef­fect is breath­tak­ing and the story is com­pelling. The story line takes place slightly in the fu­ture where I sup­pose the econ­omy is fa­vor­able enough that they got con­gress to fund eight Mars mis­sions. Well, I guess Don­ald Trump did be­come the pres­i­dent after all. The sci­ence is so ad­vanced that they have a near per­fect oxy­gen and water re­claimer, two things Matt doesn’t want for. A no­table ab­sen­tee was a 3D printer. Nasa is using a 3D printer right now on the In­ter­na­tional Space Sta­tion¹ but we don’t see one in the movie. A 3D printer would have come in re­ally use­ful. For ex­am­ple Nasa could just send Matt the code in on the radio, in the rover, and Matt could 3D print parts to re­place the Hab air­lock. 3D print­ers could even print food stuff from highly con­densed or­ganic mat­ter which would be more eco­nom­i­cal to bring along than de­hy­drated food stuff. Ul­ti­mately, I think if we ever got to Mars sci­ence would have pro­gressed enough that we would have ge­net­i­cally al­tered plants that could grow on Mars with less water. It is sur­pris­ing that the Hab doesn’t have plants and seeds other than those pota­toes Matt found set aside for thanks­giv­ing. That re­ally isn’t like Nasa. Good sci­ence would mean there’d be lots of seed, plant and tiny in­sects and an­i­mals they’d bring along to do ex­per­i­ments. After all, a botanist like Matt would be ultra in­ter­ested in grow­ing plants on Mars. (Why have him oth­er­wise?) Even oth­er­wise Nasa might be just smart enough to pack some for even­tu­al­i­ties like this.

An­other cu­ri­ous thing is why they’d allow the crew to wait till even a day be­fore a major storm. At this point Nasa wouldn’t know a lot about storms on Mars and they’d be cau­tions since their pre­dic­tion could be to­tally off. They’d even seem to know just how much force was nec­es­sary to knock the MAV off its feet. It all seems very con­ve­nient or very in­con­ve­nient de­pend­ing on whether you’re Ri­d­ley Scott or Matt Damon. All in all the sci­ence seems pretty solid even if its a bit shoddy now and then, like how did Vogel make a bomb in just half an hour. It would have taken Rip­ley’s sci­ence de­part­ment at least a day to come up with that plot point. And blow­ing up an air­lock to let the air out? Bad move. How did Com­man­der Lewis cal­cu­late the thrust that would gen­er­ate? The tear in the air­lock is not cylin­dri­cal. The noz­zle shape and size can in­flu­ence thrust dra­mat­i­cally². Ever blown up a bal­loon and let if loose? Be­sides that you’d have to know the vol­ume of air ejected and the pres­sure. I men­tion this be­cause Jo­hans­son even makes a pre­dic­tion re­gard­ing the final dis­tance and rel­a­tive ve­loc­ity at one point. I don’t know about you but I’d love to have that gal on my space­ship.

Matt de­com­poses hy­drazine by using an irid­ium cat­a­lyst to get hy­dro­gen and ni­tro­gen. This has some basis in re­al­ity. Mono­pro­pel­lant thrusters used for at­ti­tude con­trol uses hy­drazine as fuel. The de­com­po­si­tion of hy­drazine is ac­tu­ally cat­alyzed by gran­u­lar alu­mina coated with irid­ium³. In fact that what Matt uses in the movie. I been won­der­ing how Matt got so much Irid­ium since irid­ium is very rare and ex­pen­sive. But the prob­lem is that the de­com­pos­tion of hy­drazine is very very exother­mic (mean­ing it re­leases en­ergy). The re­ac­tion takes place at about a $1000^{0}$ C. Be­sides, hy­drazine de­com­poses to give hy­dro­gen, oxy­gen and am­mo­nia. A fact thats not men­tioned in the movie. The evolved hy­dro­gen would spon­ta­neously react with the oxy­gen in the hab at­mos­phere but we see Matt strug­gling to light the hy­dro­gen on fire. The evolved am­mo­nia would mean­while sat­u­rate the hab at­mos­phere, poi­son­ing Matt. I’m not sure the hab life sup­port sys­tem is de­signed to elim­i­nate am­mo­nia.

The fun­ni­est part of the movie is ac­tu­ally not even con­nected to the movie. Its Pro­ject El­rond. The Coun­cil of El­rond in the Lord of the Ring de­cides to en­trust Frodo with the task of de­stroy­ing the One Ring by toss­ing it in to Mount Doom. Guess which actor was at the Coun­cil of El­rond. Sean Bean.

“You don’t just walk in to Mor­dor.”

The only sci­ence that just might con­fuse the reader is the lot of or­bital ma­neu­ver­ing and jig­gling that goes on. As far as I know the gen­eral pub­lic still thinks rocket are fired straight up into orbit. The nerds will for­give me if I err. But you see that’s not the case. I’ll take a mo­ment to ed­u­cate the gen­eral pub­lic. The faint of heart should go no fur­ther. But it’s not that hard. It’s just rocket sci­ence after all.

Let’s de­fine what an orbit is first. An orbit is the path of a freely falling body. Toss that pen off your desk and it fol­lows a par­a­bolic path to the ground. Fire a pro­jec­tile and it fol­lows a par­a­bolic path to the enemy. A parabola is not quite a cir­cle nor it is it a straight line. Parabo­las are ac­tu­ally a class of curves of the sec­ond order. The first order being straight lines. Be­lieve it or not cir­cles are re­lated to the parabola or more ex­actly to its cousin, the el­lipse. When you start throw­ing a body faster and faster par­al­lel to the ground, after a cer­tain point the parabola traces out the cur­va­ture of the Earth and it goes on for­ever around the Earth. Not re­ally. Fric­tion from the at­mos­phere would slow it down and it would hit the earth a few miles away. But if you get high enough, say a hun­dred kilo­me­ters, you’re high enough so that there’s not much air left and an orbit is pos­si­ble. This time in­stead of a parabola the path traced is an el­lipse – es­sen­tially two parabo­las joined back to back. Each orbit has a char­ac­ter­is­tic total en­ergy. If the orbit is not a per­fect cir­cle (which it rarely is) po­ten­tial en­ergy is ex­changed for ki­netic en­ergy and vice versa and the orbit bobs up and down dur­ing each orbit lead­ing to an el­lipse.

The other im­por­tant vari­able that char­ac­ter­izes an orbit is the or­bital in­cli­na­tion. Imag­ine a plane pass­ing through the equa­tor. The angle the orbit makes with this plane is called the in­cli­na­tion. A $0^0$ in­cli­na­tion means the or­bit­ing space­craft is al­ways above the equa­tor. A $90^{0}$ in­cli­na­tion on the other hand means that the orbit passes di­rectly above the poles.

Let’s de­scribe the rocket en­gine. A rocket en­gine works on the basis of New­ton’s third law, that every ac­tion has an equal and op­po­site re­ac­tion. The sim­plest math­e­mat­i­cal model of a rocket as­sumes that we throw of lit­tle peb­bles in the other di­rec­tion from which we want to travel; at reg­u­lar in­ter­vals and ob­vi­ously very fast. Each lit­tle peb­bles has a mass and there­fore gives a lit­tle kick to the rocket in the di­rec­tion we want to travel. This “kick” is equal and op­po­site to the force we threw the peb­ble with. But the rocket doesn’t fly as fast as the peb­ble be­cause it is much more mas­sive. This is a more ef­fec­tive way than the rocket grow­ing two hands and push­ing it­self off. Of course the rocket could just grow wings while at it al­though the at­mos­phere is too thin for wings where the rocket wants to go. (Aside: per­haps Icarus and his son did grow wax wings but the at­mos­phere was too thin that high up and also there was ice grow­ing on the wings be­cause it was so cold up there.) In a rocket en­gine the role of the peb­ble is played by the gas formed by the com­bus­tion of rocket fuel. Each gas mol­e­cule acts as a lit­tle peb­ble and the kick given by all the gas mol­e­cules is called the thrust. The vari­ables that char­ac­ter­ize the rocket en­gine are the ef­fec­tive ex­haust ve­loc­ity of the gas, $v_{e}$ and the mass flow rate which is just the rate at which the fuel is used up. Both these vari­ables de­ter­mine the spe­cific im­pulse of the rocket en­gine or the $I_{sp}$. The $I_{sp}$ is a mea­sure of the over­all ef­fi­ciency of an en­gine.

The re­sult of the above strat­egy for lift off gives us a neat lit­tle equa­tion called the Tsi­olkovsky Rocket Equa­tion (TRE). This equa­tion is exact for the model given above. In fact you need to as­sume only that to get this equa­tion.

\[ \Delta v = v_{e}\cdot ln\left ( \frac{m_0}{m_i} \right ) \]

Please don’t have a panic at­tack! Ig­nore every­thing else and just focus on the $\Delta v$ (say delta vee). The $\Delta v$ is just the change in ve­loc­ity. So if you’re just start­ing out with a given amount of fuel and the pro­vided rocket you can achieve a speed of $\Delta v$. If you’re at a cer­tain speed v then you can achieve a speed of v + $\Delta v$. Rocket sci­en­tist al­ways talk in terms of $\Delta v$ and not dis­tance. To get off the earth and into orbit re­quires a $\Delta v$ of v₁. Es­cap­ing from Earths grav­i­ta­tional pull re­quires an­other $\Delta v$, call it v₂. Slow­ing down on reach­ing Mars re­quires a $\Delta v$ of v₃. Add that all to­gether and you get a $\Delta v$ of v₁ + v₂ + v₃. So the rocket sci­en­tist then pre­sum­ably goes out into his back­yard and choses a rocket that can de­liver a $\Delta v$ of just that. The good thing with $\Delta v$ is that it’s rel­a­tively in­de­pen­dent of the rocket’s de­sign. In­stead just tell the TRE the power of the en­gine and how much crew, food and fuel you’re haul­ing and it chucks out the $\Delta v$. At least if does so in the­ory. You ac­tu­ally have to be Nasa to be crazy enough to do that sort of stuff in real life. In­ter­est­ingly, you don’t need a rocket to use the TRE. Fart­ing in space is the­o­ret­i­cally gov­erned by this equa­tion, al­though we can’t vouch for the after ef­fects.

The T in TRE stands for Tsi­olkovsky, Con­stan­tine Tsi­olkovsky. He was a Russ­ian school teacher who de­rived this equa­tion for the first time⁴. He also sug­gested that a rocket en­gine might work in the vac­uum of space and con­jec­tured that a space suit might be re­quired. The guy also fig­ured out that get­ting a rocket to orbit and doing it in one piece is nearly im­pos­si­ble. The eas­ier way is to jet­ti­son stage after stage till you’re in orbit. You see, most of the mass of the rocket is in the fuse­lage and the fuel. Each kilo­gram of fuel prac­ti­cally has to carry the mass of fuel tank re­quire to carry it­self and the pre­vi­ous kilo­gram of fuel and all the rest of the kilo­gram that came be­fore. A bet­ter way is to just cut off that extra mass after a cer­tain pe­riod of time. The an­swer is stag­ing. Most Amer­i­can rock­ets have three stages. Some like the Ar­i­ane 5 has just two. But So­viet rock­ets had lots of stages. Dur­ing the space race Nasa de­cided to build more pow­er­ful rocket en­gines and cut down the num­ber of stages. The So­vi­ets on the other hand had smaller en­gines but had more of them packed in smaller stages. An al­ter­na­tive to stag­ing is called clus­ter­ing⁵ which was also de­scribed by Tsi­olkovsky. If you’ve seen pic­tures of old So­viet rock­ets like the R-7 and the Mol­niia then you’ve seen clus­ter­ing. The four rocket en­gines sur­round­ing the rocket core pivot out­wards and de­taches after their fuel has been de­pleted. This was nec­es­sary so that rocket en­gine would not have to be ig­nited in space.

Now Her­mes, the space­ship that car­ries Matt Damon and the crew of Aries 3 to Mars is built in orbit by launch­ing smaller parts called mod­ules in rock­ets and as­sem­bling it in orbit like we built the In­ter­na­tional Space Sta­tion. Launch­ing a rocket means that it needs to go into an orbit. But keep in mind that in an orbit a space­craft goes around in a el­lip­ti­cal path. In a per­fect world we would launch the rocket right at the hori­zon to max­i­mize ef­fi­ciency. But due to fric­tion with the at­mos­phere its more eco­nom­i­cal and fuel ef­fi­cient to launch the rocket straight up for the first few min­utes and then slowly tilt it so that ul­ti­mately the rocket fol­lows a curve up into orbit. The rea­son is that in the lower at­mos­phere the air is very dense. So if you go side­ways you’ll spend more time in the lower at­mos­phere and lose a lot of ve­loc­ity. In­stead fol­low­ing a curved path that is op­ti­mized by rocket sci­en­tists gets you the best of both worlds. The morale is to spend as lit­tle fuel as pos­si­ble stuck in the at­mos­phere. Lit­tle is rel­a­tive. A space­craft spends most of its fuel just to get out of the at­mos­phere and in to orbit. A rocket never needs to spend as much $\Delta v$ as dur­ing launch.

The sec­ond pro­ce­dure after launch is called the apogee burn. When you have got­ten out of the at­mos­phere, you’re coast­ing along with en­gines shut in a par­a­bolic path that will ul­ti­mately have you crash­ing down into the sea some­where. But along this par­a­bolic path there’s a point where the al­ti­tude of the space­craft above the sur­face of the earth is max­i­mum called the apogee of your orbit. Fir­ing your en­gine along the di­rec­tion of travel at this point (or doing a pro­grade burn as it is called) causes the parabola shape of your orbit to trans­form into an el­lipse just above the at­mos­phere. With this burn you’re of­fi­cially in orbit and if you’re a Nasa as­tro­naut you’ll get a badge that in­forms you of this⁶. At about an al­ti­tude of a 160km or so above the Earth’s sur­face an orbit would last a day or two be­fore de­cay­ing. That’s why a rocket in LEO (Low Earth Orbit) needs to fire its thrusters now and then to main­tain orbit. But an orbit above 600km or so is fairly safe from at­mos­pheric drag forces and can spend a cou­ple of years safely there.

The third pro­ce­dure is dock­ing. Get­ting a satel­lite into orbit is easy. But get­ting two space­craft into orbit and mak­ing them con­nect to each other? Not so easy. In fact it re­quired two tries for Nasa to get it right. The first dock­ing in space was done dur­ing the Gem­ini mis­sions when Gem­ini 6A and Gem­ini 7 con­nected in space. The main point here is to launch when the tar­get (which gets launched first) is right above your head. Get­ting into a close enough orbit that is a tiny bit lower than the tar­get gets you closer to the tar­get space craft as a lower orbit means a higher ve­loc­ity. A higher orbit on the other hand takes you away from the tar­get since you’re now mov­ing slower com­pared to the tar­get. Doing pointed burns gets the space­craft closer and closer till you’re mov­ing to­wards the tar­get. The rel­a­tive ve­loc­ity is mea­sured along the axis that passes through the tar­get and your space­craft. Space crafts are very frag­ile and at this point in­stead of the main en­gine the space craft uses its at­ti­tude jets which are gas jets pow­ered by mono­pro­pel­lants like hy­drazine. They are al­ways fired pointed away from the tar­get to pre­vent dam­age to the solar sails of the tar­get. Then the two space­crafts are docked to­gether.

After the Her­mes has been loaded and fully fu­eled the next phase is to es­cape from the Earths grav­i­ta­tional pull. At the des­ig­nated spot on the orbit fir­ing the en­gines will cause the orbit to rise until the space craft achieves es­cape ve­loc­ity which for the earth is 11 km/s. This is called a Hohmann trans­fer. A trans­fer orbit to Mars is fea­si­ble every 25 months⁷. Once the space craft drifts away from earth it is tech­ni­cally or­bit­ing the sun. A few minor course cor­rec­tion will send it to­wards the Mars in­ter­cept point. Once there Mars will catch the Her­mes in its grav­ity well. But here’s the catch. The Her­mes will not be in a full orbit but in an orbit know as a hy­per­bola. Just like the parabola sends the space­craft crash­ing down to earth and the el­lipse sets it in a sta­ble orbit the hy­per­bolic tra­jec­tory takes the Her­mes into Mars orbit and then again away from Mars. In essence it’s called a flyby. But the Her­mes needs to be in orbit to land on Mars. So we slow down by burn­ing against the di­rec­tion of travel (called a ret­ro­grade burn) and low­er­ing the en­ergy of the orbit so that an el­lip­ti­cal orbit is formed.

The plan to save Mark Whit­ney in the movie called Pro­ject El­rond re­quires the Her­mes not to enter Earth orbit. In­stead after a re­sup­ply run the space­ship flies by the earth and sling­shots around the Earth to travel to Mars again. Sling­shots around planet is a very fuel ef­fi­cient way to travel the solar sys­tem al­though rather slow. The basic prin­ci­ple is that al­low­ing the planet to cap­ture the space­craft in a hy­per­bolic orbit and then al­low­ing it to leave, ac­cel­er­ates the space­craft. In fact the space­craft steals an­gu­lar mo­men­tum from the planet. It’s an easy way to gain much needed $\Delta v$. Voy­ager 2 was ini­tally put in a Hohmann trans­fer orbit to Jupiter. A sling­shot around Jupiter sent it to Sat­urn. At Sat­urn an­other sling­shot sent it to Uranus. The gen­eral term is for this is a grav­ity as­sist. In fact, the­o­ret­i­cal cal­cu­la­tions ⁸ show that sling­shots around a grav­i­ta­tional sys­tem of three bod­ies con­fig­ured in a pre­cise way can ac­cel­er­ate a fourth body to near the speed of light.

Even after get­ting to Mars the Her­mes does not go into orbit. In­stead Matt Damon comes up in the MAV in parabola so that they meet at a cer­tain point where Com­man­der Lewis does an EVA to res­cue him. There is a rea­son why the rel­a­tive ve­loc­ity must be small. Dock­ing with a space craft is a per­fect ex­am­ple of an in­elas­tic col­li­sion. Dur­ing the col­li­sion, Lewis’s and Damon’s or­bits merge into a new one ac­cord­ing to the law of con­ser­va­tion of mo­men­tum. In fact after that both Lewis and Damon would drift away from the Her­mes if it weren’t for their tether. Since the Her­mes is so large the col­li­sion changes its mo­men­tum very lit­tle.

The final ques­tion is, was Matt Damon worth sav­ing? Was risk­ing a bil­lion dol­lar space­ship and the life of five peo­ple for a sin­gle man’s life worth it? I think it is. Hu­man­ity needs he­roes to in­spire us. To fight the fight we can­not fight. To go where we can­not go. If Matt was not saved then we prac­ti­cally al­lowed the man to die. And the best thing about hu­man­ity is that we never leave a man be­hind.

As a side note, Her­mes is the mes­sen­ger to the Gods and es­pe­cially close to Ares in Greek mythos. Ares is the God of war and later the Ro­mans adopted him as their God, Mars. So yeah, who bet­ter to res­cue a stranded Mar­t­ian than Mars best buddy.

Credit

Image Credit Nasa, 20th Cen­tury Fox and Don P. Mitchell.

Ref­er­ence