Orbital rockets are now easy, page 2: solid-rockets for cube-sats.

Copyright 2017 Robert Clark

Introduction.

  In the blog post "Orbital rockets are now easy", I argued that with altitude compensation, liquid-fueled orbital rockets become within the capabilities of most university undergraduate labs.

 Here I'll argue solid-fuel rockets can also be built by amateurs to reach orbital space.

 I was impressed by this university teams launch to 144,000 feet of a suborbital solid-fuel rocket:

USC Rocket Propulsion Laboratory Breaks Record.
Amy Blumenthal | March 16, 2017
Student-run RPL launches rocket of own design to 144,000 feet.

LAUNCH OF FATHOM II ON MARCH 4, 2017. AT 144,00 FT FATHOM II WAS THE HIGHEST ALTITUDE FOR A ROCKET ENTIRELY DESIGNED AND MANUFACTURED BY STUDENTS
https://viterbischool.usc.edu/news/2017/03/usc-rocket-propulsion-laboratory-breaks-record/

  However, using essentially "off-the-shelf" components, you can get a 3-stage solid rocket of comparable size to actually reach orbit. Moreover, I was surprised to see after running a launch simulation program that you don't even need altitude compensation to accomplish it.

 By essentially off-the shelf, you could use solid-rocket motors sold to amateurs but with an addition of a lightweight carbon-composite casing that amateurs such as the USC team already have been making themselves for their own rockets.

 You would have Cesaroni Technology, or other solid motor manufacturer, make their motors with only a thin aluminum case, not meant to hold the full combustion chamber pressures. The amateurs would then make the carbon composite case rated for the full combustion chamber pressure.

 Cesaroni also makes commercial sounding rockets with carbon composite casings but this is likely to be more expensive than if the amateurs make it themselves.

Specifications.
 This motor by Cesaroni Technology, or similar motors by other manufacturers, could form the basis of the orbital rocket:

http://www.pro38.com/products/pro150/motor/MotorData.php?prodid=40960O8000-P

  This motor though only has a mass ratio of about 2.5 to 1, adequate for amateurs doing high power rocketry test flights, but not for an orbital rocket. However, you can save about half of the weight of the aluminum casing used by replacing it by carbon composite.

 Cesaroni was able to do this for the suborbital rocket SpaceLoft they constructed for UP Aerospace, Inc.:

CTI rocket motor successfully powers the launch carrying the ashes of astronaut and James Doohan - April 30, 2007.
On April 28th, a Spaceloft™XL rocket successfully completed a round-trip space flight launched from Spaceport America. This rocket was developed by UP Aerospace Inc. of Hartford, Conn. The rocket carried a wide variety of experiments and payloads, which included the cremated remains of Star Trek's "Scotty", James Doohan and NASA astronaut and pioneer Gordon Cooper. In addition, the cremated remains of more than 200 people from all walks of life were onboard. Also flown into space on the SL-2 Mission were dozens of student experiments from elementary schools to high schools to universities - from across America and worldwide - as well as innovative commercial payloads.

The flight was a successful demonstration of the rocket motor developed and built by Cesaroni Technology Incoporated (CTI). CTI started the design process in September of 2005. CTI specializes in low cost propulsion systems for military and space applications and used its experience to develop an affordable, reliable propulsion system for the rocket. The motor has a carbon fiber composite case and a monolithic solid propellant grain that is bonded to the casing.

Watch the post-launch coverage as carried by local television station KRQE here (9 Mb)
Watch the launch as carried by the BBC here (1 Mb)
Watch pre-launch coverage as carried by CTV Toronto here (9 Mb)
Watch pre-launch coverage as carried by CBC Toronto here (9 Mb)

Technical data for the UPA-264-C rocket motor

http://www.cesaroni.net/news.php

 So we'll assume the Cesaroni Pro150 can get a 0.8 propellant fraction by using carbon composite casing. We'll round off the propellant mass to 20 kg, and take the dry mass as 5 kg. We'll take this as the third stage of our rocket.

 For the lower stages, the size of stages commonly are in the range of 3 to 5 times larger than the succeeding stage. We'll take the second stage as 4 times larger than the third stage at a 80 kg propellant weight and 20 kg dry weight. For the first we'll take it as larger by an additional factor of 4 to a 320 kg propellant weight and 80 kg dry weight.

 Now for the specific impulses for the stages. This commercial solid motor has a similar sea level Isp as the Cesaroni solid rocket motor Pro150:

Star 37.
Thiokol solid rocket engine. Total impulse 161,512 kgf-sec. Motor propellant mass fraction 0.899. First flight 1963. Solid propellant rocket stage. Burner II was a launch vehicle upper stage developed by Boeing for the Air Force Space Systems Division. It was the first solid-fuel upper stage with full control and guidance capability developed for general space applications.
AKA: Burner 2;TE-M-364-1. Status: First flight 1963. Number: 180 . Thrust: 43.50 kN (9,779 lbf). Gross mass: 621 kg (1,369 lb). Unfuelled mass: 63 kg (138 lb). Specific impulse: 260 s. Specific impulse sea level: 220 s. Burn time: 42 s. Height: 0.84 m (2.75 ft). Diameter: 0.66 m (2.16 ft).
Thrust (sl): 33.600 kN (7,554 lbf). Thrust (sl): 3,428 kgf.
http://www.astronautix.com/s/star37.html

 So we'll estimate the vacuum Isp of the Cesaroni Pro150 to be in the Star 37's range of 260 s. However, rocket stages can get higher vacuum Isp's by using longer nozzles. A 285 s Isp is not uncommon for solid rockets motors with vacuum optimized nozzles, such as the Star 48.

 So we'll take the Isp for the second and third stage as 285 s.

 For the thrust, we'll take the thrust of the third stage as the same as the Pro150 of 8 N, and assume the thrust for the second and first stage scale according to their size so to 32 N and 128 N, respectively.

 Now use Dr. John Schilling's launch performance calculator to estimate the payload.

 The input page looks like this:

 Note there some quirks of this program you need to be aware of if you use it. First, always use the vacuum values for the Isp's and thrust numbers, since the program already takes into account the diminution at sea level. Second, always set the "Restartable Upper Stage" option to "No", rather than the default "Yes", otherwise the payload will be reduced. Third, always set the launch inclination to match the launch site latitude, otherwise the payload will be reduced. This is related to the fact that changing the orbital plane involves a delta-v cost. So for the Cape Canaveral launch site, the launch inclination should be set to 28.5 degrees.

Now, here's the result:



 So a payload to LEO of 7 kg. And this with standard nozzles, no altitude compensation required.

Structure.

  To save costs, I'm envisioning making the components as much "off-the-shelf" as possible. But among its standard products Cesaroni offers the Pro150 as the largest motor. So to get the larger second and first stages, we would have to combine multiple copies of this motor.

  I could cluster them in parallel, but for the first stage that would be 16 of them, and you would have the problem of simultaneous ignition with that many motors.

  So what I'm envisioning is take 4 copies of the Pro150 stacked vertically one on top of the other for the second stage, then cluster 4 of these second stage motors in parallel for the first stage.

 The question is though about the vertical stacking is how much the thrust scales in this case. If for the solid motors the propellant burned from the bottom upwards, then the thrust would be the same as for a single motor, you would just get 4 times longer burn time.

  But that's not how large solid motors work. Actually, they have a hollow region in the center so the propellant burns from the inside surface, proceeding outwards. In this case, you have a greater amount of propellant being burned per second because of the larger vertical surface area with the stacked segments. In fact, the thrust scales linearly with the number of segments.

 By the way, the reason why I don't just also stack the first stage vertically, is because of the thinness of the rocket that would result. The Pro150 is about 3 feet long and 1/2 foot wide. If you stacked vertically 16 for the first stage, 4 for the second, and 1 for the third, that would be a rocket 63 feet high but only 1/2 foot wide, for a ratio of length to width of over 120 to 1.

 This ratio of length to width is called the "fineness ratio". Rocket engineers don't like for it to be higher than about 20 to 1 because of the severe bending loads that would result. The upgraded version of the Falcon 9 has been noted for its long, skinny profile, and has a fineness ratio of about 20 to 1. The Scout solid rocket had a fineness ratio of about 24 to 1. Solid rockets can support a higher fineness ratio because their thicker walls can withstand higher loads. Still, 120 to 1 would very likely be too high.

  So to avoid this I decided to form the first stage by clustering in parallel four copies of the second stage. Note here these four clustered motors arranged around the second stage will provide the full thrust for the first stage while the central second stage motor will not fire until the four clustered motors are jettisoned.

 It would be possible though to get a single vertically stacked motor using multiple segments if the segments were shorter, resulting in a smaller rocket. For instance, there is a market for cubesats at only 1 kg mass to orbit. If you made the solid motor segments only about 1/2 foot long by cutting the Pro150 into 6 segments, you could take one of these smaller segments as the third stage, 4 segments for the second stage, and 16 segments for the third stage.

Cost.

 The Cesaroni Pro150 retails for about $3,000 and in general the Cesaroni solids cost in the range of $100 per kg of the motor mass:

Cesaroni O8000 White Thunder Rocket Motor.   
$3,099.95
Product Information
Specification
Brandname:  Pro150 40960O8000-P                  Manufacturer:  Cesaroni Technology
Man. Designation:  40960O8000-P                    CAR Designation:  40960 O8000-P
Test Date:  4/10/2008                                   
Single-Use/Reload/Hybrid:  Reloadable             Motor Dimensions mm:  161.00 x 957.00 mm (6.34 x 37.68 in)
Loaded Weight:  32672.00 g (1143.52 oz)         Total Impulse:  40960.00 Ns (9216.00 lb/s)
Propellant Weight:  18610.00 g (651.35 oz)       Maximum Thrust:  8605.10 N (1936.15 lb)
Burnout Weight:  13478.00 g (471.73 oz)          Avg Thrust:  8034.50 N (1807.76 lb)
Delays Tested:  Plugged                                      ISP:  224.40 s
Samples per second:  1000                                  Burntime:  5.12 s
https://www.csrocketry.com/rocket-motors/cesaroni/motors/pro-150/4g-40kns-reloads/cesaroni-o8000-white-thunder-rocket-motor.html

 So take the cost of the third stage, derived from the Cesaroni Pro1050, as $3,000, and the second stage 4 times larger as $12,000, and the first stage larger by an additional factor of 4 as $48,000. So $63,000 for a smallsat launcher with a 7 kg payload to orbit.

Applications.
 Several universities have created their cubesats and smallsats to be launched piggyback on large rockets such as the Falcon 9. However, the solid rocket launcher formed from essentially off the shelf components could be built by any interested university itself thus creating their own launcher and satellite.

 Despite the small size of such satellites, and their low construction cost, the launch cost is still not cheap when sent piggyback. SpaceX for their latest incarnation of the Falcon 9 is charging about $60 million for a 20,000 kg payload to LEO, about $3,000 per kilo. But the price is much higher than that for small payloads that have to ride piggyback on launchers. For instance Spaceflight Industries charges about $100,000 per kilo to book such flights. But a 1 kg cubesat launch would only cost in range of $9,000 for one of these dedicated solid-rocket launchers.

 The remaining entrants to the Google Lunar X-Prize will have to pay expensive launch costs for their spacecraft to the Moon. But with the university teams using their own solid rocket launchers, the launch costs would be so cheap the teams could afford to make many attempts to win the lucrative $30 million prize to soft-land on the Moon, and many more teams could have remained in the race.

 Also, both DARPA and the Army funded programs to develop such small dedicated launchers (liquid fueled), with their ALASA and SWORDS programs. But both their programs failed. They wanted to get about 25kg to 50kg to orbit for a launch cost of $1,000,000, about $20,000 to $40,000 per kilo. But the small size solid rockets will be able to beat this price, moreover will be more operationally responsive by using solid rockets. In a follow up post I'll discuss the payload can be more than doubled by using altitude compensation reducing the per kilo cost even further.


National security implications.
 Recently, there has been some discussion on creating Ultra Low Cost Access to Space (ULCATS). See for instance this study:

FAST SPACE: LEVERAGING ULTRA LOW-COST SPACE ACCESS FOR 21ST CENTURY CHALLENGES.
http://www.airuniversity.af.mil/Portals/10/Research/documents/Space/Fast%20Space_Public_2017.pdf

 Most of the discussion has been about how this would improve U.S. capabilities. But surprisingly little has been about what are the national security implications of any university world-wide and many knowledgeable amateur groups world-wide launching their own rockets to orbit.

 To prepare for this, which will be here like tomorrow, this discussion must begin now.


    Bob Clark

Note: thanks to former aerospace engineer and math professor GW Johnson for helpful discussion on this topic on the NewMars.com forum:

Amateur solid-fueled rockets to *orbital* space?
http://newmars.com/forums/viewtopic.php?id=7763







NASA Technology Transfer for suborbital and air-launched orbital launchers.

Copyright 2015 Robert Clark

 I have become enamored of NASA's Morpheus lunar lander project. In the post "NASA Technology Transfer for manned BEO spaceflight", I discussed how it can be used to produce a manned lunar lander, or asteroidal lander, for a few 10's of millions of dollars, far less than the $10 billion estimated to be needed by NASA. And in "NASA Technology Transfer for Orbital Launchers", I discussed how its engines could be used for the small orbital launch system Firefly, resulting in a significant reduction in the launcher's development costs.

 I don't think NASA fully appreciates the usefulness of the Morpheus development. Here I'll show how the Morpheus itself can be used to produce suborbital launchers, and also the stages for orbital launchers. For instance the Morpheus can be used to provide the solution to DARPA's ALASA air launched, small orbital system.

 The Wikipedia page on the Morpheus gives its propellant load as 2.9 metric tons (mT) and dry mass as 1.1 mT. Its methane/LOX engine has an Isp of 321 s with a thrust of 24 kN, 2,450 kilogram-force (kgf).

 Note this means when fully fueled the single engine could not lift the vehicle in Earth's gravity. The single engine of course would be fine for its intended purpose as a lunar lander at 1/6th gravity. However, for a Earth launch system we'll use a half-size vehicle to be launchable with a single engine. Rounding off this gives it a propellant mass of 1.5 mT and dry mass of .5 mT. Compared to the full Morpheus this will have only two spherical propellant tanks instead of four, one each for the liquid methane and LOX.

 Since this will be reaching high velocity through Earth's atmosphere it will have to be streamlined. Then we'll place the two propellant tanks inline vertically. We'll also need an aeroshell. To save weight we could make the aeroshell composite. Another possibility would be to make the aeroshell inflatable. Since the aeroshell would not need to be load-bearing and with the possibility to make it inflatable we'll assume it adds only a small proportion to the weight. We could save additionally weight by making the tanks out of aluminum-lithium alloy, titanium, or composites. Alternatively, we could use a cylindrical tank to hold the propellants to eliminate the need for an aeroshell.

 Suborbital Case.

 This page gives the required delta-v for a suborbital flight as in the range of ca. 2,400 m/s:

Flight Mechanics of Manned Sub-Orbital Reusable Launch Vehicles with Recommendations for Launch and Recovery.
Mechanical and Aeronautical Engineering Department, University of California, Davis, CA 95616-5294
Marti Sarigul-Klijn Ph.D. and Nesrin Sarigul-Klijn*, Ph.D.
An approximate delta V to reach 100 km is 7,000 to 8,000 fps (2,100 to 2,400 m/s) for vertical takeoff, with slightly less delta V needed for air launch, and significantly more required for horizontal takeoff.
http://www.spacefuture.com/archive/flight_mechanics_of_manned_suborbital_reusable_launch_vehicles_with_recommendations_for_launch_and_recovery.shtml

  Now, at a 1.5 mT propellant load, .5 mT dry mass, .25 mT payload, and 321 s Isp, the vehicle can do a delta-v of 321*9.81ln(1 + 1.5/(.5 + .25)) = 3,460 m/s, sufficient for a suborbital flight.

 There are commercial opportunities for suborbital flight with NASA. Also using two to four copies or scaled up that many times this could also be used for a suborbital tourism vehicle.

DARPA Air-Launched Orbital Vehicle.

 DARPA is funding research into a small air-launched system called ALASA, As described in the blog post "Dave Masten's DARPA Spaceplane, page 2: an Air Launched System", high altitude supersonic air-launch at Mach 2 can cut 1,600 m/s from the delta-v needed for low Earth orbit. This would reduce the delta-v that needed to be supplied by the rocket from 9,100 m/s to 7,500 m/s.

 We'll use two copies of the half-size Morpheus firing in parallel and cross-feed fueling. Cross-feed fueling allows the upper stage to have its full level of fuel after staging, unlike the usual case with parallel staging. As in the earlier blog post, we'll again use the Star 17 solid stage as the final, orbital stage:

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

 Then we can get a payload of 55 kg to orbit by supersonic air-launch:

321*9.81ln(1 + 1.5/(.5 + 2.0 + .124 + .055)) + 321*9,81ln(1 + 1.5/(.5 + .124 + .055)) + 280*9.81ln(1 + .110/(.014 + .055)) = 7,690 m/s.


  Bob Clark

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