Copyright 2013 Robert Clark
Though in the first test flight of the new version of the Falcon 9, the F9 v1.1, they did not stably "land" the first stage, SpaceX is optimistic they can solve the problem to get a reusable first stage:
SpaceX Hit Huge Reusable Rocket Milestone with Falcon 9 Test Flight (Video).
By Mike Wall, Senior Writer  October 17, 2013 02:01pm ET
SpaceX Hit Huge Reusable Rocket Milestone with Falcon 9 Test Flight (Video).
By Mike Wall, Senior Writer  October 17, 2013 02:01pm ET
Combining information from the Falcon 9 v1.1's maiden flight and the ongoing Grasshopper tests should help bring a rapidly reusable rocket closer to reality, SpaceX officials said.
"SpaceX recovered portions of the [Falcon 9 v1.1's first] stage and now, along with the Grasshopper tests, we believe we have all the pieces to achieve a full recovery of the boost stage," they wrote in the Oct. 14 update.
http://www.space.com/23230spacexfalcon9reusablerocketmilestone.html
SpaceX also plans to transition the halfscale Grasshopper VTVL test vehicle to a full scale Falcon 9 first stage:
Final flight of Grasshopper v1.0 sets new record.
By Brian Dodson
October 14, 2013
http://www.gizmag.com/grasshopperretiresaltituderecord/29384/
This article says this "Grasshopper 2", as it were, would have all 9 engines of the regular F9 first stage. However, discussions on other forums have said it would only have 3 engines. That would make sense since on stage return, you are using at most 3 engines, and moreover this way, you would not be risking an expensive 9 copies of the Merlins during these Grasshopper test flights.
Still, in point of fact there would be an advantage of using all 9 engines on this first stage Grasshopper, and with a full propellant load. In November, 2012 Elon Musk gave a lecture in London at the Royal Aeronautical Society.
SpaceX also plans to transition the halfscale Grasshopper VTVL test vehicle to a full scale Falcon 9 first stage:
Final flight of Grasshopper v1.0 sets new record.
By Brian Dodson
October 14, 2013
http://www.gizmag.com/grasshopperretiresaltituderecord/29384/
This article says this "Grasshopper 2", as it were, would have all 9 engines of the regular F9 first stage. However, discussions on other forums have said it would only have 3 engines. That would make sense since on stage return, you are using at most 3 engines, and moreover this way, you would not be risking an expensive 9 copies of the Merlins during these Grasshopper test flights.
Still, in point of fact there would be an advantage of using all 9 engines on this first stage Grasshopper, and with a full propellant load. In November, 2012 Elon Musk gave a lecture in London at the Royal Aeronautical Society.
About 30 minutes in, he gave the propellant fraction of the new Falcon 9 v1.1 first stage as around 96%, or perhaps 95.5%. The 96% propellant fraction number gives a 25 to 1 mass ratio. But at an Isp of 311 s for the Merlin 1D, the rocket equation gives a deltav of 311*9.81ln(25) = 9,800 m/s. Since the deltav to orbit is only about 9,100 m/s, this would allow a significant amount of payload. Then using the 9 engines and the full propellant load on the F9 first stage would allow in fact not just a VTVL test vehicle, but in fact a fully reusable and fully orbital vehicle. Amusingly, about 36 minutes into Elon's lecture someone asks a question about what he sees as the next big breakthrough in rockets after full reusability. Elon thinks for awhile and can't come up with an answer. He finally jokes maybe warp drive. Ironically, he already has the next big advance: a reusable SSTO.
In the blog post "The Coming SSTO's: Falcon 9 v1.1 first stage as SSTO" I had already discussed the F9 v1.1 first stage being used as a SSTO. But there I actually used the side boosters of the Falcon Heavy, which are based on the F9 v1.1 first stage, since they were supposed to have such a high mass ratio, at 30 to 1. However, this information in Elon's lecture on the first stage of the F9 v1.1 suggests it itself would have a surprisingly high mass ratio.
We'll enter this data into Dr. John Schilling's launch performance calculator to estimate the payload it could carry. On the SpaceX page on the Falcon 9 v1.1 the vacuum thrust is given as 6,672 kN. The Merlin 1D has a vacuum Isp of 311 s. We need to know the propellant mass of the F9 v1.1 first stage.
I found the propellant loads for the Falcon 9 v1.1 first stage and second stage in this environmental impact report:
Draft Environmental Impact Statement: SpaceX Texas Launch Site.
http://www.faa.gov/about/office_org/headquarters_offices/ast/environmental/nepa_docs/review/documents_progress/spacex_texas_launch_site_environmental_impact_statement/media/SpaceX_Texas_Launch_Site_Draft_EIS_V2.pdf
They're given on page 66, by the PDF file page numbering:
First and Second Stages
The first stage of the Falcon 9 is approximately 12 ft by 150 ft, and includes nine Merlin 1D engines. The Merlin engine produces 146,000 lbs of thrust and contains a pumpfed gas generator cycle, turbine exhaust roll control, and hydraulic thrustvector control. The first stage consists of aluminum LOX and RP1 tanks that hold approximately 62,000 gallons (gal) of LOX and 38,000 gal of RP1. The second stage is approximately 12 ft by 41 ft, not including the fairing and payload, and uses one Merlin vacuum engine. The fairing (the top portion of the vehicle where the payload is encapsulated) would be 17 ft by 35 ft, and a smaller version may also be used. The second stage consists of approximately 15,000 gal of LOX and 9,000 gal of RP1 in tanks with a common bulk head.
The conversion factor between gallons and liters is 1 gallon to 3.7854 liters. So the amount of LOX in liters is 62,000*3.7854 = 234,700 liters, 234.70 m³. And the amount in liters of RP1 is 38,000*3.7854 = 143,800 liters, 143.8 m³. The density of LOX is 1140 kg/m³ and the density of RP1 is 820 kg/m³. So the total mass of propellant is 1140*234.7 + 820*143.8 = 385,500 kg. This is in the range of what has been estimated for instance on Ed Kyle's site of ca. 389 metric tons (mT):
Space Launch Report: SpaceX Falcon 9 v1.1 Data Sheet.
http://spacelaunchreport.com/falcon9v11.html#components
However, there is a significant difference from the estimates and the actual propellant load of the second stage. Using the same conversion factors and densities, the total propellant for the second stage is 92,670 kg, whereas Kyle gives it as approx. 64 mT.
Take, optimistically, the propellant fraction of the F9 v1.1 first stage as 96%, corresponding to a mass ratio of 25 to 1. Then the dry mass is 16,040 kg, which I'll round to 16,000 kg. In the calculator input the vacuum values for the Isp and thrust, as it takes into account the reduction at sea level. Select "No" for the "Restartable Upper Stage" option, otherwise the payload will be reduced. Select Cape Canaveral as the launch site at a launch inclination of 28.5 degrees to match the latitude of the launch site. Then the calculator gives an estimated payload of 5,147 kg:
I found the propellant loads for the Falcon 9 v1.1 first stage and second stage in this environmental impact report:
Draft Environmental Impact Statement: SpaceX Texas Launch Site.
http://www.faa.gov/about/office_org/headquarters_offices/ast/environmental/nepa_docs/review/documents_progress/spacex_texas_launch_site_environmental_impact_statement/media/SpaceX_Texas_Launch_Site_Draft_EIS_V2.pdf
They're given on page 66, by the PDF file page numbering:
First and Second Stages
The first stage of the Falcon 9 is approximately 12 ft by 150 ft, and includes nine Merlin 1D engines. The Merlin engine produces 146,000 lbs of thrust and contains a pumpfed gas generator cycle, turbine exhaust roll control, and hydraulic thrustvector control. The first stage consists of aluminum LOX and RP1 tanks that hold approximately 62,000 gallons (gal) of LOX and 38,000 gal of RP1. The second stage is approximately 12 ft by 41 ft, not including the fairing and payload, and uses one Merlin vacuum engine. The fairing (the top portion of the vehicle where the payload is encapsulated) would be 17 ft by 35 ft, and a smaller version may also be used. The second stage consists of approximately 15,000 gal of LOX and 9,000 gal of RP1 in tanks with a common bulk head.
The conversion factor between gallons and liters is 1 gallon to 3.7854 liters. So the amount of LOX in liters is 62,000*3.7854 = 234,700 liters, 234.70 m³. And the amount in liters of RP1 is 38,000*3.7854 = 143,800 liters, 143.8 m³. The density of LOX is 1140 kg/m³ and the density of RP1 is 820 kg/m³. So the total mass of propellant is 1140*234.7 + 820*143.8 = 385,500 kg. This is in the range of what has been estimated for instance on Ed Kyle's site of ca. 389 metric tons (mT):
Space Launch Report: SpaceX Falcon 9 v1.1 Data Sheet.
http://spacelaunchreport.com/falcon9v11.html#components
However, there is a significant difference from the estimates and the actual propellant load of the second stage. Using the same conversion factors and densities, the total propellant for the second stage is 92,670 kg, whereas Kyle gives it as approx. 64 mT.
Take, optimistically, the propellant fraction of the F9 v1.1 first stage as 96%, corresponding to a mass ratio of 25 to 1. Then the dry mass is 16,040 kg, which I'll round to 16,000 kg. In the calculator input the vacuum values for the Isp and thrust, as it takes into account the reduction at sea level. Select "No" for the "Restartable Upper Stage" option, otherwise the payload will be reduced. Select Cape Canaveral as the launch site at a launch inclination of 28.5 degrees to match the latitude of the launch site. Then the calculator gives an estimated payload of 5,147 kg:
Launch Vehicle:  UserDefined Launch Vehicle 

Launch Site:  Cape Canaveral / KSC 
Destination Orbit:  185 x 185 km, 28 deg 
Estimated Payload:  5147 kg 
95% Confidence Interval:  1242  9908 kg 
This is surprisingly high for a stage using engines without an especially high Isp. However an SSTO reaches its best performance when using altitude compensation. Let us suppose we use altitude compensation so that the engines on the first stage have the same vacuum Isp as the Merlin Vacuum at 340 s. 

Note that because of the higher Isp, the thrust is also increased. On that SpaceX page on the Falcon 9 v1.1, the thrust of the single Merlin Vacuum on the upper stage is given as 801 kN. So 9 would have a thrust of 7209 kN, which I'll round to 7,210 kN. Select "Optimal" in the calculator for the "Trajectory". Then the calculator gives the result:


This is remarkable as being near the payload cited by SpaceX for the full two stage Falcon 9 v1.1 of 13,150 kg. But for a fair comparison we should see also how high the payload would get for the two stage F9 when altitude compensation is also given to the first stage. The calculation here is made difficult by the fact that we don't know the propellant fraction of the upper stage, so we can't calculate the dry mass from the known propellant mass of 92,670 kg. 

For the upper stage much smaller than the first stage, the mass ratio would not be as great. It is known that as you scale up a rocket the mass ratio improves. The reverse is also true, when you scale down a stage the mass ratio becomes worse. The acceleration at burn out for just an empty upper stage, and payload would also be rather high. Then I'll take the mass ratio for the upper stage at only 10 to 1, giving a 9,200 kg upper stage dry mass. Let's calculate first what the calculator gives as the payload for the present case using the standard Merlin 1D at 311 s Isp. The calculator gives: 



Rather close to the actual value of 13,150 kg. Now we'll calculate it for the case where the first stage has been given altitude compensation to get a 340 s Isp. We'll change the Isp input to 340 s and also increase the thrust to 7,210 kN as before. Then the calculator gives: 



This is a significant increase but not nearly as dramatic as the increase for the SSTO case. For the SSTO case the payload more than doubled. But for the TSTO case it increased by less than 25%. This could mean the SSTO could approach that of the TSTO on a cost per kilo basis. Elon Musk has said the Falcon 9 first stage takes up about threequarters of the cost of the Falcon 9: 

Musk lays out plans for reusability of the Falcon 9 rocket October 3, 2013 by YvesA. Grondin 

Performance hit for reusable rockets:http://www.nasaspaceflight.com/2013/10/muskplansreusabilityfalcon9rocket/ 

This would put it at about $40 million out of the $54 million for the full rocket. Then the cost per kilo for the SSTO would be $40,000,000/12,068 = $3,314 per kilo, while for the TSTO it would be $54,000,000/17,056 kg = $3,166 per kilo. The benefits of the SSTO would be even more dramatic in the reusable case. In the Nasaspaceflight.com article Elon says the loss in payload for the F9 for returning just the first stage to the launch site was about 30%. This is interesting because he said in another interview the loss in payload for returning both stages would be a loss of about 40%: Elon Musk on SpaceX’s Reusable Rocket Plans. By Rand Simberg February 7, 2012 6:00 PM Despite the dangers, Musk is clearly a fan of the rocketpowered approach. He told PM that SpaceX has come up with a solution to make both the lower and upper stages of the Falcon 9 reusable. (The Dragon capsule that will fly atop the rocket has already demonstrated that it can be recovered in the ocean after it splashlands with a parachute, though SpaceX is building verticallanding capability into that as well.)http://www.popularmechanics.com/science/space/rockets/elonmuskonspacexsreusablerocketplans6653023 These two quotes together could mean the payload loss from making the upper stage also reusable is 10%, assuming Elon was being consistent between the two quotes. Then a question arise: would the payload loss from the making the SSTO reusable also be just 10% of the payload? This doesn't seem likely, for if you changed the relative sizes of the first and upper stages while keeping the payload the same, then the extra added components for the upper stage such as heat shield, landing legs, and propellant reserve for landing should also change. It should not stay as the same 10% of the payload, regardless of the size of the stage. So we'll need to do use some other sources to see how much payload would likely be lost under the reusable SSTO case. Payload Lost for a Reusable SSTO.We need a heat shield, landing legs, and reserve propellant for the landing. This interesting discussion between noted spacehistorian Henry Spencer and a former manager for both the DCX and X33 programs, Mitchell Burnside Clapp, is about the relative benefits of horizontal versus vertical landing of RLV's:Horizontal vs. vertical landing (Henry Spencer; Mitchell Burnside Clapp). http://yarchive.net/space/launchers/horizontal_vs_vertical_landing.html Burnside Clapp conservatively estimates the propellant that needs to be kept on reserve for the landing amounts to about 30 seconds of engine firing. Spencer optimistically estimates it might be as low as 10 seconds. I'll estimate it as 20 seconds. Assume the engine used for the landing has similar sea level Isp as the Merlin at 282 s. But this is not for the full firing of all engines as would be needed for takeoff of a fully loaded rocket. We'll assume we only need enough thrust for the dry mass of the stage, as the needed reserve propellant is a small proportion of this. Taking the dry mass of the first stage as 16,000 kg, 157,000 N, the flow rate of such an engine would be (flow rate) = (thrust)/(exhaust velocity) = 157,000N/2370m/s = 57.5 kg/s. And the propellant for a 20 second burn would be 1,150 kg, 7% of dry mass. For the heat shield, it will be the PICAX material of SpaceX. The mass for this heat shield used for the Dragon has been estimated in the range of 226 kg. However, the video SpaceX has released of a reusable Falcon 9 shows a heat shield on the upper stage that extends partially down the side of the stage. Then I'll estimate the mass as double that of the Dragon at 550 kg. For the landing gear the example of the lighweight gear for the B58 suggests it can be as low as 1.5% of the landing weight: Landing gear weight (Gary Hudson; George Herbert; Henry Spencer). http://yarchive.net/space/launchers/landing_gear_weight.html With lightweight composites this might be reduced to 1% of the landed weight, 160 kg. The total of all three of these extra systems for reusability would then be 1,860 kg, about 12% of the 16,000 kg dry weight. This would need to be subtracted off from the delivered mass to LEO. Then the reusable F9 v1.1 first stage would have a payload to LEO of 10,200 kg. Comparsion of Costs of Reusable SSTO, Partially Reusable TSTO, and Fully Reusable TSTO.First, under the partially reusable case of just the first stage being reusable, this would subtract off 30% of the payload, so from 17,056 kg to 11,940 kg. Now assume the first stage is reusable 10 times and this cuts the cost of that stage by a factor of 10, so to $4 million per flight. Then the upper stage being expendable would be $14 million, i.e. $54 million  $40 million, and the total cost would be $18 million per flight, at a cost per kilo of $1,500 per kilo.Now compare to the reusable SSTO case. Again assume 10 uses at a cost of $4 million per flight. Use the reusability loss estimate above that lowers the payload to LEO to 10,200 kg. Then the cost per kilo would be only $390 per kilo(!)Perhaps a fairer comparison though would be to the fully reusable TSTO case. This would cut the payload by 40% so from 17,056 kg to 10,230 kg. Since we're using the full rocket 10 times, assume the cost is cut to $5.4 million per flight. This would be a cost per kilo of $527 per kilo. So the reusable SSTO would carry about the same payload but at a better cost per kilo. Admittedly though this conclusion is based on very rough estimates for the propellant reserve needed for landing and the mass needed for the heat shield for a long rocket stage compared to that of a capsule. Bob Clark Update, October 18, 2014: The calculations here were assuming the Falcon 9 v1.1 had payload to LEO of 13,150 kg. However, as discussed in the post "Golden Spike" Circumlunar Fights, Page 2 this payload is actually that of the partially reusable version. The actual payload of the expendable version is ca. 16,600 kg. Then assuming altitude compensation increases the payload of a TSTO by 25%, the Falcon 9 v1.1 with altitude compensation on the first stage would have a payload of ca. 20,000 kg. So in the last section with comparisons of the price per kilo of a reusable SSTO and TSTO, the fully reusable TSTO with 40% loss should have a payload of 12,000 kg. This would still mean the reusable SSTO would have a lower price per kilo than the fully reusable TSTO. UPDATE, October 25, 2014: SSTO's achieve their best usefulness with altitude compensation. Low cost methods of giving already existing engines altitude compensation are discussed here: Altitude compensation attachments for standard rocket engines, and applications. http://exoscientist.blogspot.com/2014/10/altitudecompensationattachmentsfor.html 
