Tuesday, August 1, 2017

Altitude compensation attachments for standard rocket engines, and applications, Page 5: metal foil expandable nozzles.

Copyright 2017 Robert Clark

 In prior posts I gave some possibilities for achieving altitude compensation, [1],[2], [3], [4].The importance of this is they increase the payload both for single stage and multistage rockets.

 Another possibility is illuminated by this:

 Only it would use pressurize gas rather than popcorn to expand out the nozzle.

   Bob Clark


1.)Altitude compensation attachments for standard rocket engines, and applications.

2.)Altitude compensation attachments for standard rocket engines, and applications, Page 2: impulse pressurization methods.

3.)Altitude compensation attachments for standard rocket engines, and applications, Page 3: stretchable metal nozzles.

4.)Altitude compensation attachments for standard rocket engines, and applications, Page 4: the double aerospike.

Sunday, May 7, 2017

Test flights of the Falcon Heavy for missions to the moons of Earth and Mars, Page 1.

Copyright 2017 Robert Clark

  The SpaceX Red Dragon lander mission to Mars on the Falcon Heavy has been pushed back to 2020, perhaps to return a Mars surface sample. SpaceX though plans for two Falcon Heavy test flights for the latter part of this year, 2017.

  Elon has discussed testing recovery of the stages on these first test flights which will reduce payload. He has also discussed putting a "fun" payload on one of them, like his cheese wheel on the first Dragon test flight.

  I suggest instead missions be undertaken of great scientific and practical importance, missions to the moons of Earth and Mars. 

Flight to a Permanently Shadowed Crater on the Moon.
 Abundant evidence suggest ice water deposits in the permanently shadowed craters on the Moon. This has been proposed to be used to produce orbital propellant depots. This would radically reduce the mass that would need to be launched to orbit for a Mars mission since most of this mass is just propellant. 

 There have also been some tantalizing indications from the LCROSS mission of valuable metals in the shadowed craters. Then the first space mining missions may be to the Moon rather than the asteroids.
 I'll estimate the delta-v to land on the Moon using this diagram:

 The delta-v to GTO (geosynchronous transfer orbit) is 2.5 km/sec. Then after that according to the delta-v diagram we need an additional 3.2 km/sec to land on the Moon. We wish to use the cargo version of the Dragon to land on the Moon. This weighs about 5.5 metric tons (mT) fueled with its own propellant, while the FH can get 26.7 mT to GTO. 

 So the idea would be to get extra delta-v by using the smaller mass of the Dragon capsule. To estimate this we'll need the specs for the upper stage of the Falcon Heavy, same as for the Falcon 9's upper stage, 348 s Isp for the Merlin 1D FT, approx. 107.5 mT propellant load, and approx. 4 mT dry mass. Then the delta-V this upper stage achieves with the 26.7 mT payload is 348*9.81*Ln(1 + 107.5/(4 + 26.7)) = 5,136 m/s.

 So by reducing the payload mass from 26.7 mT to 5.5 mT we want this upper stage to achieve a delta-v of:

5.1(to reach GTO) + 3.2(to land on the Moon) = 8.3 km/sec .

 And calculating the delta-v of the stage with the reduced payload we get:

348*9.81*Ln(1 + 107.5/(4 + 5.5)) = 8,572 m/s.

 This is above the needed 8.3 km/s, though close. Actually we'll get somewhat better than this because the lower stages having to loft a lighter payload will be able to provide more delta-v than before.

 Also, we actually will use the Draco thrusters on the Dragon to do the actual landing since the FH upper stage would put the capsule to high up if it were to land vertically, and it's thrust is so high achieving the stable landing is made difficult.

 That raises another difficulty because of the low thrust of the Draco thrusters. There are 18 Dracos of the cargo Dragon each of thrust level 400 N, for a total of 7,200 N. This can lift 7,200/9.81 =  734 kg in Earth's gravity. In the lunar gravity at 1/6th g, the Dracos could lift, 4,404 kg. But the fueled mass is 5,500 kg.

 There are a couple of things we can do to lighten the Dragon. We could remove both the parachute and thermal protection systems since the capsule won't be returning to Earth in this mission.  These weigh about 5% each of the landed mass, so about 10% all together. So this shaves 420 kg off the landed mass.

 Another possibility would be to replace the Dracos with Superdracos, which have many times greater thrust. But I'm not sure how well these would fit in the same housing for the Dracos.

 We could also remove most of the pressure vessel for landing on the airless Moon. From images of the Dragon's pressure vessel, this could be a significant mass:

 For the rover, we might use a copy of the Mars Pathfinder mission. NASA often makes two or more copies of its spacecraft for testing purposes. Then we could use one of these copies. This weighed only 264 kg for the lander plus 10.5 kg for the Sojourner rover.

 Other possibilities for a lightweight rover might be those being developed independently by entrants to the Google Lunar X-prize:

Flight to Phobos, the mysterious moon of Mars.
 A great scientific mystery also is the make-up of the Mars moon Phobos. Flyby missions showed it to have surprisingly low density. Serious scientific speculation included that it may actually be hollow. Current theories are though that it may be analogous to a "rubble-pile" type asteroid. This is not known for sure however. A lander mission may help to resolve the issue.

 Note also that key to Elon's plan for manned flights to Mars is getting the fuel for the return trip from Mars. Taking the fuel from the Martian moons instead would have advantages such as low gravity for getting the fuel to an orbiting propellant depot. Then these first flights to the Martian moons could serve as scout missions for water ice deposits.

  The Falcon Heavy test flights this year will be outside the optimal launch window in 2018. This means they will require higher delta-v to reach Mars, and higher delta-v to slow down on reaching the destination. This limits the mass that can be transported to and landed on Mars, in addition to the expense of the extra in-space stages required.

 Then I will suggest here a method that has long been proposed for arriving at Mars but never attempted, aerocapture. This slows down a craft arriving at Mars by plunging deep within the atmosphere so that minimal propellant burn is required. Note, that if these tests missions using aerocapture succeed then this will suggest it will work to solve the problem of landing large masses on Mars such as a crew habitat, a key enabling technology for manned flights to Mars.

 For the delta-v required to depart from Earth I'll use the orbital calculation program:

Trajectory Planner.

 This provides the delta-v's required for the Hohmann tranfer orbits between the various planets. The program provides pork-chop plots that allow you to estimate departure and arrival delta-v's dependent on departure time.

 The program though uses Modified Julian Date format, which can be converted to standard date format here:

 For a Dec. 23, 2017 departure, which is given in Modified Julian Date format of 58110 in the "Trajectory Planner", the delta-v Hohmann transfer delta-v is 6.155 km/s. We then need to calculate the delta-v needed on leaving Earth orbit. On the Orbiter-Forum discussion forum for the Orbiter space simulation program this formula was provided by member Dgatsoulis:

 \Delta V = \sqrt{V_{\infty}^2 + V_{esc}^2} - V_{orb}

where V_{\infty} is the hyperbolic excess velocity (departure deltaV from trajectory planner).

V_{esc} is the local escape velocity, aka the escape velocity for the parking orbit altitude.

V_{esc} = \sqrt{\frac{2GM_{planet}}{R_{planet}+alt}}

where G is the gravitational constant, M_{planet} is the planet's mass, R_{planet} is the planet's radius and alt is the altitude of the parking orbit.

V_{orb} is the parking orbit velocity.

V_{orb} = \frac{V_{esc}}{\sqrt{2}} 

 Same applies for arrival. If you want to simply calculate the periapsis velocity and not the orbit insertion/injection dV, then don't use the V_{orb} term.


 So the  delta-v on leaving Earth orbit is:
 This 1.11 km/sec more than the usual delta-v to make a Trans Mars Injection during the optimal departure windows of 3.8 km/s.

 We need to calculate how much mass the FH upper stage could get to this higher delta-v of 4.89 km/s. By the FH specs it can get 16.8 mT to Trans Mars Injection.This FH upper stage with the 16.8 mT payload mass can do 348*9.81Ln(1 + 107.5/(4 + 16.8)) =  6,211 m/s delta-v. So with a smaller mass we want to achieve 6,211 + 1,110 = 7,321 m/s delta-v. This can be done with a 10 mT payload:

338*9.81Ln(1 + 107.5/(4 + 10)) = 7,377 m/s.

 Now we have to calculate how much is the speed on arrival at Mars.  The Trajectory Planner gives the "arrival" speed as 4.275 km/s. However, again this is not the speed the spacecraft would have on entering Mars's atmosphere. This is instead the speed at which it arrives at Mars's position in its orbit around the Sun, i.e., the Hohmann orbit delta-v needed to be supplied to match Mars' solar orbital speed.

 To get the entry speed into Mars' atmosphere, use the Dgatsoulis formula above without the Vorb term. Using 5.0 km/s as the escape velocity for Mars we get:

 If all we wanted was to slow down to enter Mars orbit then we would subtract off from this by aerocapture to bring the speed down to Mars' orbital velocity of 3.56 km/s. However we also want to be put it on a trans Phobos insertion from Mars. By the delta-v chart above we need an additional .9 km/s, so to 4.46 km/s. So by the aerocapture we only need to slow it down by about 2.11 km/s.

 This should be well within the capabilities of aerocapture. However, the payload mass may be as high as 10 mT. The question is could the dragon's approx. 10 square meter base provide sufficient air drag to slow down that high mass, and would its heat shield be thick enough?

 In follow up posts I'll present some preliminary calculations that suggest that plunging deep into Mars atmosphere, skimming the tree-tops so to speak, should allow large masses such as this to be slowed at such high entry speeds.

 With the payload of the FH as high a 10 mT, the rovers and equipment that could be transported could be 4.5 mT above the 5.5 mT fueled weight of the cargo Dragon. But according to the delta-v chart we still need 0.5 km/s delta-v to land on Phobos. The cargo Dragon has a delta-v capability of about 600 m/s with its Draco thrusters for the Dragon capsule alone. So this should be sufficient, but it would not be if the extra cargo was several metric tons. So we could keep the cargo low as for a Mars Pathfinder sized rover or we could add additional propellant tanks to increase the landing capability.

 The possible cargo carried by the Dragon being as high as 4.5 mT suggests though we should try to make use of that cargo space. One possibility would be the processing equipment to produce ISRU (in situ resource utilization) propellant. Perhaps a rocket to do a sample return. Possibly orbiting imaging spacecraft for Phobos or Mars. Others?

    Bob Clark

Note: thanks to members of Keithth G and DGatsoulis for helpful discussions on this topic and member Piper, for writing the Trajectory Planner program.

Tuesday, April 25, 2017

About the launch abort system for the New Shepard capsule.

Copyright 2017 Robert Clark

 Blue Origin has revealed the format of its suborbital tourism capsule for the New Shepard suborbital launcher:

Take a Peek Inside Blue Origin’s New Shepard Crew Capsule.
Published: 29 Mar , 2017
by Nancy Atkinson

   The cylinder in the middle is the launch abort motor. It is only supposed to fire in case of an emergency to pull the capsule away from the rocket launcher.

 Normally, it would not even fire. Still its presence inside the passenger cabin is rather disconcerting. Moreover, it is a solid rocket motor. For solid motors, the combustion chamber is the entire rocket, so if a failure, i.e., a breech does occur it can happen anywhere along the motors length.

 A Blue Origins video animation from 2015 shows the solid rocket escape motor with handholds at about the 2:25 point:

 Be careful to mind your head while floating though!

   The reason Blue Origin decided to put the abort motor inside the cabin likely was for reasons of positioning of the center of gravity(CG) with respect to the center of pressure(CP). A well known rocket stability rule of thumb is the center of pressure should be below the center of gravity

The trunk and fins helped that for the SpaceX launch abort test by bringing the CP rearward:

 But compare this to the Blue Origin abort test:

  Notice that the capsule is gyrating while the rocket motor is firing. This would be very unpleasant for the passengers since they would be subjected to high g's while being thrown right and left, albeit while strapped in.

 Then for these reasons I suggest giving the New Shepard a trunk with fins as has the SpaceX Dragon capsule.

 This could be done by instead of having the ring structure at the top of the New Shepard stay attached to the New Shepard, let it act as the trunk for the capsule:

 Then you would move the solid rocket abort motor down into this structure, so it is no longer inside the passenger compartment.

 However, this ring structure does have a function as far as the landing of the New Shepard rocket; it holds the fins and the speed brakes used during the landing:

 So how could we maintain those functions if that ring structure is instead attached to the capsule? Two possible approaches you could duplicate it so the New Shepard has its own as does the capsule. 

 Or another possibility would be to have the ring structure only detach along with the capsule only during an abort scenario. For the normal launch, with no abort, the ring structure would stay attached to the New Shepard rocket, carrying also inside the abort motor, while the capsule detaches for the normal flight to suborbital space.

 But if there is a need for an abort, the solid rocket abort motor would fire carrying the ring structure and the capsule away from the New Shepard. In this scenario where there would need to be an abort presumably there would be a failure of the New Shepard anyway and you would not expect to recover it.

  Bob Clark

Wednesday, March 15, 2017

Satellite dishes and satellite phones for radio astronomy and passive radar detection.

Copyright 2017 Robert Clark

Asteroid Detection.
 In the blog post "Combined amateur telescopes for asteroid detection", I suggested using multiple small amateur telescopes in concert to act as a giant astronomical instrument to make dim observations in the optical range. Could we do the same with multiple satellite dishes or satellite phones to make dim radio observations? 

 There is a technique called "passive radar" that uses reflected radio waves from aircraft that originate from surrounding radio transmissions such as from television and radio stations:

Passive Radar.
3. Typical illuminators
Passive radar systems have been developed that exploit the following sources of illumination:
Analog television signals
FM radio signals
Cellular phone base stations
Digital audio broadcasting
Digital video broadcasting
Terrestrial High-definition television transmitters in North America
GPS satellites (GPS reflectometry).
Satellite signals have generally been found to be inadequate for passive radar use: either because the powers are too low, or because the orbits of the satellites are such that illumination is too infrequent. The possible exception to this is the exploitation of satellite-based radar and satellite radio systems. In 2011, researchers Barott and Butka from Embry-Riddle Aeronautical University announced results claiming success using XM Radio to detect aircraft with a low-cost ground station.

 The difficulty in using satellite transmissions for the detections previously is that they just use a single ground station for the reception of the reflected signals. Instead of this, suppose we used millions of satellite dishes or radios or satellite phones to make the detections?

 As with the case of multiple amateur telescopes, you couldn't form a coherent signal from this method. But like in the optical case you could make correlations from which you could make a probabilistic estimate of the likelihood of an actual detection.

 There is an additional difficulty however. We are envisioning using satellites at geosynchronous orbit, about 35,000 kilometers out in space. We would detect asteroids closer than this distance by their blocking the satellite signals from being detected by satellite dishes or phones.

 However, the asteroids would tend to direct the reflected signals back out to space rather than towards the Earth, except for the case where the asteroid is along a line from the satellite towards the limb of the Earth, and with the dishes/phones along the limb. But this would be relatively few asteroids and dishes/phones so precisely placed in the right position.

 So in actuality for this method to work we would be looking for holes, deletions, in the signal. Such deletions in the satellite signal would be small for each dish or phone. But by correlating the signals of millions of them we can determine statistically that it represents a real detection.

 This would only be for detecting asteroids rather close in, since they would be inside the distance of geosynchronous orbit. This would still be useful since from multiple observations we could determine their orbits. And such asteroids that came so close in would have a higher probability of presenting an impact hazed on a future orbital pass.

 But could we also detect asteroids further out? Some proportion of the signal from the GEO satellites likely escapes past the sides of the Earth to proceed to the other side. And this proportion of the signal likely is increased by the signals bouncing off the ionosphere. Then these signals could proceed further outwards to be reflected back to Earth by more distance asteroids.

 The strength of the signal leaking past Earth would be reduced so the reflected signals would also be reduced. But in this case you are making actual positive detections rather than looking for holes in the signal so all in all the results could be just as effective as in the close in asteroid case.

Aircraft detection.
 A problem with detecting aircraft on intercontinental flights is that when they fly over the oceans they fly too far from the radar stations on land to be detected. Then perhaps the method of satellite signal detections by multiple dishes/phones can be used to track such aircraft as well. This may give a us a method to finally locate the missing airliner Malaysian Airlines Flight 370. The flight was lost three years ago but there may have been some satellite TV, radio or phone customers who saved programs or phone conversations at that time for which the recorded digital data can be reviewed to reconstruct a detection of the aircraft.

  Bob Clark

Tuesday, March 14, 2017

A smaller, faster version of the SpaceX Interplanetary Transport System to Mars, Page 2: triple cores for larger payloads.

 Copyright 2017 Robert Clark

 In the blog post "A smaller, faster version of the SpaceX Interplanetary Transport System to Mars", I suggested using just the upper stage of the ITS to get a booster for a Mars rocket, using an existing Ariane 5 core as an upper stage. This would be much cheaper and faster than the 7,000 metric ton, 42 engine booster that SpaceX was planning.

 Elon Musk says SpaceX plans to have the smaller upper stage built by 2020. So we could possibly have a Mars transport system by then since the Ariane 5 as an upper stage already exists. However, by using triple cores of the ITS upper stage we could also get a system of the larger size SpaceX is proposing.

 We'll input the data into Dr. John Schillings payload estimation program. In the calculator, select "No" for the "Restartable Upper Stage" option, rather than the default "Yes", otherwise the payload will be reduced. Select Cape Canaveral as the launch site, and input 28.5 degrees for the launch inclination to match the latitude of Cape Canaveral, otherwise the payload will be reduced.

 We'll also use the 382 s Isp of the vacuum version of the Raptor. Altitude compensation allows even engines used on first stage boosters to have the same vacuum Isp as upper stages engines.

 We'll use also crossfeed fueling. As I have argued before this is a well-known technique having been used for decades on jet airliners. To emulate crossfeed fueling with the Schilling calculator, enter in 2/3rds the actual propellant load in the field for the sideboosters, and enter in (1 + 2/3) times the actual propellant load in for the first stage propellant load.

 So in the side boosters propellant field enter in (2/3) * 2,500,000 kg = 1,667,000 kg. And in the first stage propellant field enter in (1 + 2/3) * 2,500,000 = 4,167,000 kg.

 For the thrust fields, enter in the vacuum thrust for 9 vacuum Raptors, since the calculator always takes as input the vacuum values, even for first stages and side boosters. The vacuum thrust for the 382 Isp vacuum Raptor is 3.5 meganewtons, 3,500 kN. So 9 would be 31,500 kN. Enter in also the vacuum Isp 382 s.

 For the second stage, we'll increase the vacuum thrust of the Vulcain engine on the Ariane 5 to 1,450 kN in accordance with an increased vacuum Isp of 465 s, since we can get this higher vacuum Isp by just using a nozzle extension. For the dry mass input 12,000 kg and propellant 158,000 kg. Inputting these specs in the calculator results in:

Mission Performance:
Launch Vehicle:  User-Defined Launch Vehicle
Launch Site:  Cape Canaveral / KSC
Destination Orbit:  185 x 185 km, 28 deg
Estimated Payload:  504575 kg
95% Confidence Interval:  426107 - 597674 kg

"Payload" refers to complete payload system weight, including any necessary payload attachment fittings or multiple payload adapters

This is an estimate based on the best publicly-available engineering and performance data, and shou
ld not be used for detailed mission planning. Operational constraints may reduce performance or preclude this mission.

 This is comparable to the payload mass of the expendable version of SpaceX's ITS. This would save greatly on development costs when not having to develop the larger booster. The launch cost would also be greatly reduced since judging by the Falcon Heavy, using triple cores only increased the price 50% over that of the single core rocket.

    Bob Clark

Sunday, October 30, 2016

Altitude compensation attachments for standard rocket engines, and applications, Page 4: the double aerospike.

Copyright 2016 Robert Clark
(patent pending)

 On the blog page "Altitude compensation attachments for standard rocket engines, and applications", I suggested various methods to accomplish altitude compensation with already existing engines. One method was a sort-of "inverted aerospike". It consisted of a movable spike pointed inward, rather than pointing outward as with the standard aerospike:

 There are two disadvantages to this method. First the spike has to be movable so that adds mechanical complexity. Secondly, the size of the outer, fixed nozzle in order to achieve high Isp at high altitude has to be large. But this nozzle will be used all the way from the ground, so this will induce high drag at low altitude.

 The reason why this nozzle has to be large is because you are not really using the altitude compensating capacity of a shaped spike on exit from the nozzle. The only purpose of the movable spike is to vary the size of the exit plane of the nozzle, to provide a variable area ratio.

 But could we use a fixed nozzle and the usual outward-pointing aerospike? This would have the advantages that we could use the altitude compensating capacity of the usual aerospike, so we could use a shorter nozzle, and also have a fixed spike, reducing mechanical complexity. 

 The problem with this with a usual engine is you would need to change to a toroidal combustion chamber, an expensive change to an engine. So instead of this, we will also use an inward pointing spike so that the exit of the nozzle has a toroidal shape:

  This now has two advantages. We will be using this as an attachment to a usual ground-firing engine and nozzle. Since these already expand the exit gases to a certain extent, you would need a much shorter, slimmer and lighter outward-pointing spike to accomplish the rest of the expansion at high altitudes. The usual aerospike has to accomplish the full expansion from ca. 100 bar combustion chamber pressures to near vacuum pressures at high altitude, requiring a large and heavy spike.

  Another advantage is that nozzles for sea-level-firing engines actually overexpand the exit gases at sea level. This is because you want a longer nozzle to achieve at least moderate performance also at high altitude. But now, with the addition of the inward-pointing spike you can reduce the pressure at exit of the nozzle to that of sea level by reduction of the exit plane area. This will also improve the performance at sea level.

   Bob Clark

Tuesday, October 4, 2016

A smaller, faster version of the SpaceX Interplanetary Transport System to Mars, UPDATED, 10/15/2016.

Copyright 2016 Robert Clark

 In the blog post "A SpaceX Heavy Lift Methane Rocket, Page 2", I proposed some architectures for a Mars transport rocket. This was based on a quite large 1,800,000 lb. vacuum thrust of the methane-powered Raptor.

 However, recently Elon Musk discussed the current version of the SpaceX Mars transport model called the Interplanetary Transport System (ITS). Here they are going back to a smaller version of the Raptor, at ca. 660,000 lb. vacuum thrust. In this version however, their booster will be quite large at ca. 7,000 metric tons (mT) gross weight.

 Because of the reduced size of the Raptor this will require 42 engines on the booster. However, interestingly the size of the upper stage will be similar to the size of the booster I discussed in "A SpaceX Heavy Lift Methane Rocket, Page 2". So you could get a Mars launch booster by using this upper stage instead as a booster.

But because of the smaller engines in the SpaceX formulation they will use nine of the Raptors on the upper stage. I stated in the earlier blog post I wanted to use at most 5 of the larger Raptor engines to emulate the safety record of the 5 large engines on the Saturn V booster. However, SpaceX seems to have gotten the 9 engines on the Falcon 9 to work, and in any case you could just use the booster to send the cargo and habitats to space and use high safety rockets to launch the crew to meet up with the transport craft in space.

 The objection could be made however, that this is supposed to be just an upper stage, not a booster stage. However, at his IAC presentation of this Mars transport system he stated that the upper stage in both the spaceship and tanker form could be SSTO. Furthermore, the tanker he said could be used a fast intercontinental transport craft. This means necessarily they would have to be able to launch from the ground. So it is not too much of stretch to assume they could be used as a first stage.

An advantage of making this smaller upper stage the actual booster is that Elon has said they will have a development craft within 4 years. So if we make now a smaller upper stage to go with it, we could have a valid Mars transport craft at that early date. If we made this new upper stage correspondingly 1/3rd size, then we would be able to get 1/3rd the crew size to Mars, so a crew of ca. 35 to Mars.

 However, interestingly we might be able to use already existing upper stages on existing rockets for the upper stage, for instance possibly the famous Centaur upper stage used on the Atlas V or the Ariane 5 core itself used here as an upper stage.

 We can estimate how much we could get to LEO using the ITS tanker as the booster and the Ariane 5 core as the upper stage. The required delta-v to LEO is 30,000 ft/sec about 9,100 m/s:

Modern Engineering for Design of Liquid-Propellant Rocket Engines, p.12

 Since we can get a high 465 s vacuum Isp for a hydrolox engine just by using a nozzle extension we'll assume that value for the Ariane 5 engine. Also we'll assume we can get a vacuum 382 s Isp for the ITS tanker by using altitude compensation.

 Then for the propellant and dry mass values for the ITS tanker and Ariane 5 core we could get 225 metric tons to LEO:

382*9.81ln(1 + 2500/(90 + 170 + 225)) + 465*9.81ln(1 + 158/(12 + 225)) = 9,140 m/s.

 This makes clear another key advantage of this architecture: whereas the original SpaceX ITS would require five flights of the ITS to refuel the upper stage spaceship, with this smaller version a single flight would be able to carry the spaceship to orbit as well as its fuel for its flight towards Mars.

 But what would be the crew size for this smaller upper stage? We can estimate it by making a comparison to the delta-v possible in accordance with the stats of the ITS spaceship:

382*9.81ln(1 + 1950/(150 + 450)) = 5,400 m/s.

 So we want the Ariane 5 case to be able to reach a delta-v of 5,400 m/s when fully refueled and firing in space headed towards Mars:

465*9.81ln(1 +158/(12 +55)) = 5,500 m/s.

 So this is a payload of about 55 metric tons. This is about 1/8th the mass for the ITS case, so we can estimate the crew size to be 1/8th also, so to a crew of 12.

  Elon in his IAC presentation says the ITS carrying its 100 member crew might be able to reach Mars in 80 days at a particular close Mars opposition. This is dependent on the departure delta-v however. In the blog post "Propellant depots for interplanetary flight". I noted that at a higher departure delta-v possible by using a smaller 6 metric ton habitat for only a crew of 3, the Ariane 5 used as the in space propulsion stage might be able to make it to Mars in only 35 days, when leaving at such a particularly close Mars opposition.

  Bob Clark

UPDATE, 10/15/2016:

 Dr. John Schilling's launch performance calculator is back up. This allows us to produce a more accurate payload estimate. The vacuum thrust for the 382 Isp Raptor is 3.5 meganewtons, 3,500 kN. So 9 would be 31,500 kN. We'll also increase the vacuum thrust of the Vulcain engine on the Ariane 5 to 1,450 kN in accordance with the increased vacuum Isp of 465 s. Inputting these and the other specs in the calculator results in:

Mission Performance:
Launch Vehicle:  User-Defined Launch Vehicle
Launch Site:  Cape Canaveral / KSC
Destination Orbit:  185 x 185 km, 28 deg
Estimated Payload:  177767 kg
95% Confidence Interval:  150063 - 210818 kg

"Payload" refers to complete payload system weight, including any necessary payload attachment fittings or multiple payload adapters

This is an estimate based on the best publicly-available engineering and performance data, and should not be used for detailed mission planning. Operational constraints may reduce performance or preclude this mission.

 So the calculator estimates 178 metric tons. This is less than the 225 metric ton estimate using just the rocket equation, but it still means a single flight could carry enough payload to fully refuel an Ariane 5 core upper stage for a flight to Mars.

  Bob Clark