Dave Masten's DARPA Spaceplane, page 2: an Air Launched System.

Copyright 2014 Robert Clark

 In the blog post Dave Masten's DARPA Spaceplane, I discussed using SpaceX Falcon 1 or Falcon 9 stages to achieve DARPA's XS-1 reusable first-stage spaceplane. Another DARPA program ALASA seeks to send smaller payloads of 45 kg to orbit for $1 million using air-launch. 

 DARPA has already awarded a contract to Boeing to produce the ALASA system:

Boeing Targets 66 Percent Launch Cost Reduction with ALASA.
By Mike Gruss | Mar. 28, 2014

The ALASA rocket, measuring 7.3 meters long, would be attached to the underbelly of a Boeing-built F-15E fighter aircraft. DARPA says taking off from a standard airport runway would allow the Defense Department to launch from almost anywhere. Credit: Boeing artist's concept.

http://www.spacenews.com/article/military-space/40023boeing-targets-66-percent-launch-cost-reduction-with-alasa

 However, using the Falcon 1 upper stage may provide a fast, low cost means to produce such a system. Masten Space Systems could develop this as well since it would provide a much reduced cost proof-of-principle for their larger spaceplane that in itself would still be profitable.

 SpaceX has said the Falcon 9 first stage accounts for 3/4 of the cost and the upper stage, 1/4. If we assume a similar ratio for the $8 million Falcon 1, then we might estimate the cost of the upper stage as $2 million. However, unlike with the Falcon 9, the Falcon 1 upper stage uses a much smaller and simpler engine in the pressure-fed Kestrel and it is a much smaller stage in comparison to the first stage than is the case with the Falcon 9. Then I'll estimate its cost to be, say, $1 million. That would already be at the $1 million max cost DARPA wants per launch for the ALASA system.

Solid Rocket Motor Expendable Stage version.
 To get the launch cost below $1 million we would need reusability. If we got 10 launches from the Kestrel powered booster, that would be $100,000 per launch for just this lower stage. Then staying below the $1 million max cost would depend on the cost of the upper stage. There were some small solid rocket stages that were below $1 million in cost such as the Star 17 solid rocket motor:

Encyclopedia Astronautica.
Star 17
Solid propellant rocket stage. Loaded/empty mass 124/14 kg. Thrust 19.60 kN. Vacuum specific impulse 280 seconds.
Cost $ : 0.580 million.
Status: Out of production.
Gross mass: 124 kg (273 lb).
Unfuelled mass: 14 kg (30 lb).
Height: 0.98 m (3.21 ft).
Diameter: 0.44 m (1.44 ft).
Span: 0.44 m (1.44 ft).
Thrust: 19.60 kN (4,406 lbf).
Specific impulse: 280 s.
Specific impulse sea level: 220 s.
Burn time: 18 s.
Number: 25 .
http://www.astronautix.com/stages/star17.htm

 This is not currently in production but there are probably some remaining in storage or some of comparable size.

 According to Ed Kyle's page on the Falcon 1, the F1 upper stage had a 0.36 metric ton (mT) dry mass, 3.385 mT propellant mass and 327 sec Isp. However, the Kestrel engine used only had a 3,175 kgf (kilogram-force) thrust, i.e., less than the stage weight. So we'll cut down the propellant load to 2.5 mT.

 Now we'll use the fact that airlaunch actually can result in a significant reduction of the required delta-v that needs to be supplied by the rocket to reach orbit and therefore a significant increase in payload. This is described in some reports by Sarigul-Klijn et.al.:

Air Launching Earth-to-Orbit Vehicles: Delta V gains from Launch Conditions and Vehicle Aerodynamics.
Nesrin Sarigul-Klijn University of California, Davis, CA, UNITED STATES; Chris Noel University of California, Davis, CA, UNITED STATES; Marti Sarigul-Klijn University of California, Davis, CA, UNITED STATES
AIAA-2004-872
42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, Jan. 5-8, 2004
http://pdf.aiaa.org/preview/CDReadyMASM04_665/PV2004_872.pdf  [first page only]

 The conclusions are summarized in this online lecture:

A.4.2.1 Launch Method Analysis (Air Launch).
For a launch from a carrier aircraft, the aircraft speed will directly reduce the Δv required to attain LEO. However, the majority of the Δv benefit from an air launch results
from the angle of attack of the vehicle during the release of the rocket. An
ideal angle is somewhere of the order of 25° to 30°.
A study by Klijn et al. concluded that at an altitude of 15250m, a rocket launch with the
carrier vehicle having a zero launch velocity at an angle of attack of 0° to
the horizontal experienced a Δv benefit of approximately 600 m/s while a launch
at a velocity of 340m/s at the same altitude and angle of attack resulted in a
Δv benefit of approximately 900m/s. The zero launch velocity situations can
be used to represent the launch from a balloon as it has no horizontal velocity.
Furthermore, by increasing the angle of attack of the carrier vehicle to
30° and launching at 340m/s, a Δv gain of approximately 1100m/s
was obtained. Increasing the launch velocity to 681m/s and 1021m/s produced a
Δv gain of 1600m/s and 2000m/s respectively.
From this comparison, it can be seen that in terms of the Δv gain, an airlaunch is 
superior to a ground launch. As the size of the vehicle decreases, this superiority 
will have a larger effect due to the increased effective drag on the vehicle.

https://web.archive.org/web/20120229141110/https://engineering.purdue.edu/AAE/Academics/Courses/aae450/2008/spring/report_archive/reportuploads/appendix/propulsion/A.4.2.1%20Launch%20Method%20Analysis%20(Air%20Launch).doc

 A speed of 340 m/s is a little more than Mach 1, while subsonic transport aircraft typically cruise slightly below Mach 1. So the delta-V saving could still be in the range of 1,000 m/s with air launch even using a standard subsonic jet, a significant savings by the rocket equation.

 And this study found by using a supersonic carrier aircraft you could double the payload of the Falcon 1:

Conceptual Design of a Supersonic Air-launch System.
43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit
8 - 11 July 2007, Cincinnati, OH
http://www.ae.illinois.edu/m-selig/pubs/ClarkeEtal-2007-AIAA-2007-5841-AirLaunch.pdf

 The idea had been that airlaunch can't result in much of an improvement in payload since jet transports typically cruise only around 300 m/s, so, it was thought, you would only subtract this off the delta-v needed to reach low Earth orbit (LEO), which is about 9,100 m/s. However, there is also the altitude the aircraft can achieve and another key factor is the high altitude launch means you can use the higher Isp and higher thrust vacuum versions of the engines. The Isp advantage can be quite significant. For instance the Merlin 1C only had a vacuum Isp of  304 sec, but the Merlin Vacuum, being optimized only to operate at near vacuum conditions, had an Isp of 340 sec.

 The Boeing version of the ALASA system will use the F-15E fighter jet for the airlaunch. This has a Mach 2.5 maximum speed at altitude and can carry 10,400 kg payload. So we'll use this also for our system. Following the Sarigul-Klijn et.al. paper, the Mach 2.5+ max speed of the F-15E  is above the 681 m/s air launch speed needed to reduce the delta-v to orbit by 1,600 m/s. This will bring the delta-v that needed to be delivered by the rocket down to about 7,500 m/s.

 Using the reduced propellant load for the Falcon 1 upper stage of 2.5 mT then with a 45 kg, 0.045 mT, payload, an (F1 upper stage + Star 17) rocket could get a delta-v of:

327*9.81ln(1 + 2.5/(0.36 + 0.124 + 0.045)) + 280*9.81ln(1 + 0.110/(0.014 + 0.045)) = 8,500 m/s.

 This is high enough that a cheaper subsonic carrier, which according to Sarigul-Klijn can still subtract off about 1,000 m/s from the required delta-v, could be used instead of the Mach 2.5 F-15E. 

 Let's also estimate how much higher payload we could get using the reduction of delta-v to 7,500 m/s allowed by using the F-15E. Taking the payload to be 80 kg, 0.08 mT, we get:

327*9.81ln(1 + 2.5/(0.36 + 0.124 + 0.08)) + 280*9.81ln(1 + 0.110/(0.014 + 0.08)) = 7,560 m/s.

 So we could actually exceed the DARPA requirements to get 80 kg to LEO.

Two Falcon 1 Upper Stage Version.
 Instead of using an expendable solid rocket as the upper stage, we could use instead a second Falcon 1 upper stage. This will allow the possibility of getting a fully reusable system. We'll have both stages firing in parallel to be able to get a T/W greater than 1. We'll also use cross-feed fueling to maximize payload. For the upper stage that reaches orbit, we'll give it the full 3.385 mT propellant load since this stage doesn't have to have a T/W greater than 1. Then using the reduced 7,500 m/s required delta-v to orbit, we could transport 240 kg to LEO:

327*9.81ln(1 + 2.5/(0.36 + 3.745 + 0.240)) + 327*9.81ln(1 + 3.385/(0.36 + 0.240)) =7,530 m/s.

 We could also improve the mass ratio of these stages and increase the payload by switching to lightweight aluminum-lithium alloy for the propellant tanks. This could save as much as 25% off the tank weight. 

  Bob Clark