SpaceX - And why 10 minutes matter

[legroeder2k]

Hi together,

I was part of the live stream yesterday where you asked for some whiteboard results why a 10 minute launch delay would be bad - so I tried to bring it down to some mathematical arguments.
First of all, please excuse any spelling mistakes and also take a note that I'm nowhere near a rocket scientist but rather someone who likes KSP and is interessted in orbital mechanics and some of the values are simplified / guesswork (I tried to mark guesswork with  ).

So to answer the question why a 10 minute launch delay is bad we need to take a look at some facts at first. To do any change in the orbit a space craft needs what's called delta-v, the potential to change in velocity. Given the mission profile flown with CrewDragon demo-1 we just need to care about the capsules delta-v budget - so let's start by looking into what we have.

By looking into https://www.faa.gov/about/office_org/headquarters_offices/ast/environmental/nepa_docs/review/launch/media/fonsi_dragon_pad_abort.pdf we can find, that Crew Dragon capsule carries ~1388kg (3060 pounds) of NTO/MMH fuel. Given it's dry mass of 9525kg from wikipedia we can use the ideal rocket equation to calculate the total available delta-v for it.

dv = v_e * ln ( m₀ / m₁ )
where
m₀ is the initial mass of 9525kg + 1388kg of fuel = 10913kg
m₁ is the final dry mass of 9525kg
v_e is the exhaust velocity in vacuum of NTO/MMH of 3347m/s
so dv = 3347 m/s * ln(10913 / 9525) = 455 m/s

From the Demo-1 launch the capsule is brought to a roughly 280km by 280km orbit by the second stage and is then finally separated from the Falcon-9 this is why I only look into delta-v budget of the Dragon capsule. The 280km temporary orbit is something I'm unsure of but I found resources in form of TLEs (https://sattrackcam.blogspot.com/2020/05/the-trajectory-of-upcoming-crew-dragon.html) which lead to a rough 280km orbit and looking into Space-X mission profile on https://www.spacex.com/launches/?utm_source=morning_brew it's seems that they are doing a different phasing which might cost some more energy as my calculated Hohmann transfer down below.

The ISS orbits earth in a 400 by 420km orbit, for simplifying I'll take a 400km circular orbit as those 20km just add around 12m/s for the transfer.

So, completly independent of being 10 minutes late or not CrewDragon needs to perform a transfer from its 280km orbit to the ISS orbit. I calculated the energy needed for this using the calcuations I found here: https://www.faa.gov/about/office_org/headquarters_offices/avs/offices/aam/cami/library/online_libraries/aerospace_medicine/tutorial/media/III.4.1.5_Maneuvering_in_Space.pdf



Removing the 68m/s of delta-v from our 455m/s and we are left with 387m/s for flying around and deorbiting.

Speaking of deorbiting, the capsule must come back at some point in time which will use up some fuel. Some users at stack exchange did some math for the space shuttle (https://space.stackexchange.com/questions/12011/how-could-a-90-m-s-delta-v-be-enough-to-commit-the-space-shuttle-to-landing) which it turns out that around 90m/s can be enough to get back, so lets remove that from our capsule as well and we are left with 297m/s for orbital shenanigans.

Up until this point it's all the same for a correctly timed launch or a 10 minute late launch. After 10 minutes the ISS with it's orbital velocity of around 7,673km/s will have travelled 4600km from it's optimal point but as the capsule is lower it will slowly catch up - so it's just a bit more of waiting for the astronauts to get to the ISS - no big deal as Wikipedia states the CrewDragon is laid out for a week in space or 210 days docked to the ISS.

But - while waiting for 10 minutes on the launch pad - not only does the ISS travel 4600km but the earth also rotates away from the ISS.

Given that the earth rotates by 360°/day it turns by 0,25° per minute (2,5° in 10 minutes) so the orbit will be misaligned by 2,5° waiting for 10 minutes.
More specifically for a launch to the same inclination the right ascension of the ascending node will differ by 2,5° and if launched pointing towards the ISS their inclination will differ by those 2,5°.
The RAAN (right ascension of the ascending node) is the angle in the equator between the ascending node (where the orbit crosses the equator from south to north) and the point of vernal equinox.

To cope with those misalignments CrewDragon needs to perform a so called plane change which covers either the inclination change and the change of the RAAN depending on where you do the burn.

Given a simple plane change where only the direction changes the equation is rather simple the forumla looks like this:

dV = 2 * dV_initial * sin( theta / 2 )
where
dV_initial is the current orbital speed and
theta is the plane change angle

As this burn is normally performed prior to the transfer burn let's calculate it for the 280km orbit:

dV = 2 * 7,742 km/s * sin (2,5°/2) = 0,337km / s = 337 m/s

And our 297m/s left over are busted - and those were only there because we calculated for a completly empty capsule.

The problem with plane changes is that it's not just speeding up but you are going really fast in one direction (7,742 km/s) and you have to stop and accelerate in a different direction again.
(This is also covered by the PDF i linked in above)

Given this ideal and simplified conditions delaying the launch by up to 8,8 minutes works and after that the capsule does not have enough fuel to reach to ISS by any means.
Having this really costly plane change manuever is also the reason why they need to wait up until Saturday now because they can only launch when the ISS passes over the launch site directly.

Hope this shed some light and I also hope I did not made any big errors. (If you do the calcuations for a lower 200 by 200km orbit the transfer takes 110m/s and the plane change 339m/s also busting the budget so the initial orbit should not make a big difference)

[dr.diesel]

Welcome to the forum!

I look forward to the day where the launch window, and the weather, is only a minor consideration.

[legroeder2k]

Thank you!
So do I - makes all those mind boggling calcuations much easier.

[Rerouter]

I believe the actual deltaV required is slightly less, they have some wiggle room in the path the rocket takes to its first orbit, and during this it is much cheaper to correct for that inclination change as your starting from a lower initial velocity, e.g. the first quarter of the burn you have a slightly different angle to decrease the plane change amount then use the last quater to come more in plane with where you want to end up. kind of like an S-curve.

[llkiwi2006]

The most efficient way to launch into a different plane is by doing a dogleg turn in the early stages of the flight. The Falcon 9 has a lot of excess performance and can definitely perform such a maneuver, it is more of a case of SpaceX deciding it is not worthwhile writing and certifying software to do it. For comparison, ULA's Atlas V has this capacity so their (at least uncrewed) launches to the ISS (e.g. OA-7) has a 30-minute launch window. Also crewed launches have a lot more constraints on their trajectory to allow a safe abort scenario, so a dogleg might not be possible.

[Rerouter]

I will admit I kind of overlooked that, that the path a human flight takes needs a high chance of fast recovery from anywhere along its path, so they are probably still a little limited in how they skew things.

[legroeder2k]

And i want to thank you for the input - there were quite some ideas I didn't had because "straight" up and decoupling maneuvers from each other is much easier in KSP than doing fancy combined things - I'm sure I will learn something 

[Domagoj T]

Another consideration is the fact that Falcon 9 uses superchilled liquid oxygen (which is not common, most rockets us "regular" liquid oxygen). That is why LOX is loaded at the last minute. As soon as it is loaded it starts to warm up and the performance starts to drop. With regular liquid oxygen, it is sitting at boiling point (-183°C) and will not warm up at all, there is only some boil-off, but that is easily fixed by topping off the tanks. With F9, if LOX warms up it has to be unloaded and new cold oxygen loaded.

[iMo]

Except the temperature of the LOX they fill the tanks in the last minutes because of the girls and guys who are working in the capsule since T-2.5hours. Would be not safe to have them there with the tanks full loaded, imho..
PS: it looks like NASA can have a SpaceX rocket (incl. the outfit of the astronauts and the supporting personnel) painted any colour that it wants so long as it is white or black

[Domagoj T]

Except the temperature of the LOX they fill the tanks in the last minutes because of the girls and guys who are working in the capsule since T-2.5hours. Would be not safe to have them there with the tanks full loaded, imho..

Actually, this sequence is backwards from the established NASA (and all other space agencies) protocols, specifically because SpaceX wanted to use subcooled oxygen. Prior to Falcon 9, procedure was to fuel the rocket and only then would the crew (and support personnel) be allowed to approach and board. This was a topic of a heated debate some two years ago when SpaceX proposed this sequence.

[iMo]

The today's sequence is something I would always go with - as it is the safest. In case of fire or explosion the supporting personnel around the capsule would be lost otherwise..

[legroeder2k]

I've found some more data about the DM-2 mission at https://everydayastronaut.com/dragon-2-dm-2-spacexs-first-crewed-mission/

One thing is the payload is announced there with 12055kg which I understand is the complete wet mass of the dragon capsule  - bringing it's available delta-v down to about 400m/s. (9525 kg dry + 1388 kg fuel + 1142kg capsule payload)

Another, way more interessting thing is the mission profile for Falcon 9 a bit down in the post. There Tim Dodd explains that for safety reasons the flight profile is really shallow to ensure a safe abort at any time.
Which makes my rough calculations worse because the transfer from a lower orbit to ISS needs more delta-v. Not talking about the plane change, as the orbital velocity is higher in lower orbits. Taking 180km and not 280km in the calcuations means for transfer they have to spent 127m/s and for the 2,5° inclination change i get 340m/s. Exceeding the 400m/s quite a bit.

And due to the fact that having a shallower trajectory also brings the issue of having to fight atmospheric drag for longer they really might not have enough fuel margin in the Falcon 9 to do much wiggling at all.

[nfmax]

The today's sequence is something I would always go with - as it is the safest. In case of fire or explosion the supporting personnel around the capsule would be lost otherwise..

Where the safety systems make use of nitrogen tetroxide & mono-methyl hydrazine, it must be pretty damn dangerous

[iMo]

Sure, but the probability the hydrazine+N2O4 bailout engines/tanks explode (or leak) is lower than with the main LOX+kerosene tanks, imho..
PS: In a different thread I wondered why they do not use solid propellant bailout engines, but somebody argued the hydrazine engines are better suited (ie. used for orbital maneuvers too).. 

[Domagoj T]

PS: In a different thread I wondered why they do not use solid propellant bailout engines, but somebody argued the hydrazine engines are better suited (ie. used for orbital maneuvers too).. 

SuperDracos were designed to be multi purpose. They provide launch abort capability, but were also intended to be used for propulsive landings. NASA decided they didn't want crewed vehicles to use propulsive landing, but SuperDracos were presumably already part of the design so they stayed. We may still see them in that role in the future.
Since they are hydrazine engines, the added benefit is that they use the same propellant and oxidizer as the much smaller Draco engines which are used for orbital maneuvering, so in normal operation when you don't use launch abort procedure, you can still use that propellant. If you had solid motors, that would be dead mass for the entirety of mission. Also, being hypergolic engines, they can be restarted numerous times, and offer variable thrust and burn length capability, neither of which can be done with solid motors.

In a nominal mission, I don't think SuperDracos are used for orbital maneuvering. Their specific impulse is worse than Dracos, so the small guys do that part.

[westfw]

19 hours seemed like a really long time to get from orbit to actual docking.I was curious - if they had been able to launch in the original launch window, would it have been a shorter trip?

[Domagoj T]

The time from launch to docking was planned to be the same for original and backup launch window.
Once it becomes routine, they will likely be able to shed tome time, but in this case they took it slow to test some of the systems on the Dragon.

As a side not, the record for fastest launch to docking is held by Progress MS-11 and clocks in at just three hours and 22 minutes.
That being said, usually the docking procedure is slow on purpose to minimize the usage of thrusters in the vicinity of the station (you don't want exhaust residue caking the station), and lower overall risk and damage in case of collision.