It is indeed very complicated, so the standards and procedures run to many thousands of pages. There is lots of math, and lots of software simulation, but those need inputs from long-running experiments with real devices that test all the hardware components until they break. This is a job for teams of specialist engineers who spend their entire careers making these kinds of computations. I have not done these particular calculations myself, but I have worked with people who did, and I have used similar methods to estimate expected space mission duration before on-orbit failure.
Personally, I don't find it horrible. I find it fun! Yes, it needs careful, painstaking, obsessive focus on vast numbers of tiny mathematical details, but those are good things, not bad.
- a) Is this calculation generally outsourced to a third-party specialist company
In the days when all the launches were made by big defense contractors like Boeing and Lockheed, they had teams of full-time specialists in house, because their houses were huge and rockets cost a bloody fortune, so they had plenty of money to pay such people. In the new era of many more launches at much lower cost, I don't know what the new launch providers do, so I am just wildly guessing here, but I would not be surprised to hear that, at least initially, they had outsourced this particular task to the very same teams of those big name companies, or set out to recruit some of those people directly.
Whoever does the work needs a large number of very expensive software tools, many of them built by the users over decades of high-precision work, and considered very valuable trade secrets. There are a few government provided tools, such as NASA's Debris Assessment Software, which
provides assessments that can verify compliance of a spacecraft, upper
stage, and/or payload with NASA's requirements for limiting debris
generation, spacecraft vulnerability, postmission lifetime, and entry
safety. Successful verification of a design in DAS demonstrates
compliance with NASA debris mitigation requirements. Historically, DAS
analysis has proven acceptable in meeting the compliance requirements
of many other agencies in the U.S. and around the world. It does not
address the inherent design reliability facets of NASA requirements
but addresses all Earth-related orbital debris requirements that make
up the bulk of the requirements in the NASA Technical Standard
8719.14C.
but to obtain it you must request permission, and accept the terms of a software usage agreement.
- b) If yes, what is the typical fee
I don't see a good way to get such information without getting a quote from such a firm, but you might find some interesting things using the search term "commercial space launch insurance". Launch costs have been decreasing, but the spacecraft themselves are still often very expensive, and the failure of a cheap rocket makes the whole mission a total loss, so spacecraft insurance policies are also very expensive. Some of the premium the mission owner has to pay to the insurance company to obtain that policy is used by the insurer to have its own team of engineers review and criticize the design team's reliability calculations, to make an independent assessment of risk that in part determines the premium, which also touches on your (a) and (d). Take a look at GAO-17-366, a US government report criticizing how the FAA was estimating space launch insurance costs in 2017. For historical interest, there are also FAA reports from 1998 and 2002 with space launch insurance statistics and some useful definitions.
- c) Does it have to be recalculated for every launch
Yes, definitely. Because each payload is a different mass, and is going into a different orbit, the same rocket will experience different loads, and may need to follow a different trajectory. Even if you are launching exactly the same thing to exactly the same place, you have to adjust for different wind speed and direction, and even different times of day, because that affects the likelihood of all sorts of conditions, such as the chance that a person will be outdoors without a roof to protect them from falling debris, or what kind of roof material they are under, and the ratio of that building's window glass area to floor area, and so on. Also, rockets are very rarely identical to each other, partly because the designers are always tinkering, and partly because sometimes you have to change suppliers of certain components, which may change their estimated reliability. This is where questions like "what makes something space-qualified" come in. Even if somehow you manage to launch an identical payload into an identical orbit using literally the same rocket, you would still have to change many of its probabilities of malfunction, because you now have a rocket that you know has launched successfully before: that makes some failures less likely, and other failures more likely, in rather complicated ways.
- d) Who verifies the calculation
When applying for a license to operate a space launch vehicle, one of the things the applicant is required to provide is all the information required by the FAA to make their own assessment. This is described in the U.S. Code of Federal Regulations, Title 14, Chapter III, Subchapter C, Part 440, Appendix A, "Information Requirements for Obtaining a Maximum Probable Loss Determination for Licensed or Permitted Activities". This includes, for example,
III. Flight Operations
E. Trajectory data as follows: Nominal and 3-sigma lateral trajectory
data in x, y, z and x (dot), y (dot), z (dot) coordinates in
one-second intervals, data to be pad-centered with x being along the
initial launch azimuth and continuing through impact for suborbital
flights, and continuing through orbital insertion or the end of
powered flight for orbital flights.
F. Tumble-turn data for guided vehicles only, as follows: For vehicles
with gimbaled nozzles, tumble turn data with zeta angles and velocity
magnitudes stated. A separate table is required for each combination
of fail times (every two to four seconds), and significant nozzle
angles (two or more small angles, generally between one and five
degrees).
for launch and
II. Flight Operations
E. Nominal and 3-sigma dispersed trajectories in one-second intervals,
from reentry initiation through landing or impact. (Coordinate system
will be specified on a case-by-case basis)
F. Three-sigma landing or impact dispersion area in downrange (±) and
crossrange (±) measured from the nominal and contingency landing or impact
target. The applicant is responsible for including all significant
landing or impact dispersion constituents in the computations of
landing or impact dispersion areas. The dispersion constituents should
include, but not be limited to: Variation in orbital position and
velocity at the reentry initiation time; variation in re-entry
initiation time offsets, either early or late; variation in the
bodies' ballistic coefficient; position and velocity variation due to
winds; and variations in re-entry retro-maneuvers.
G. Malfunction turn data (tumble, trim) for guided (controllable)
vehicles. The malfunction turn data shall include the total angle
turned by the velocity vector versus turn duration time at one second
intervals; the magnitude of the velocity vector versus turn duration
time at one second intervals; and an indication on the data where the
re-entry body will impact the Earth, or breakup due to aerodynamic
loads. A malfunction turn data set is required for each malfunction
time. Malfunction turn start times shall not exceed four-second
intervals along the trajectory.
for re-entry. That's just a summary of the full technical requirements, which are represented in part by the many detailed equations, tables, and graphs in Part 420, defining exactly which computations are required to be performed by a particular method. The FAA in turn may consult with NASA and the Department of Defense, who each have their own long and detailed list of standard requirements. Any or all of these agencies may choose to employ engineering specialty consultant contractors to assist in the application review. NASA-STD-8719.25, "Range Flight Safety Requirements", is a good place to start reading; representative contents are
4.5 Debris Risk Assessment
4.5.2 An assessment of risk to the public and workforce due to debris shall account for each of the following as a function of flight-time
or loss-of-control-time:
a. All potential debris, generated intentionally or not, that could
cause a casualty, including debris that could affect someone on the
ground or on a waterborne vessel, or cause an aircraft mishap. Note:
Casualty models used in range safety risk assessments typically
evaluate certain impact parameters, such as kinetic energy, and
incorporate thresholds on those parameters that define when a debris
impact has the potential to cause a casualty or down an aircraft.
These thresholds may change as our knowledge of human
vulnerability/aircraft vulnerability evolves. Sources of the latest
casualty and aircraft impact thresholds developed for use by the range
safety community include RCC 321, and Air Force Space Command Manual
(AFSPCMAN) 91-710, Air Force Space Command Range Safety User
Requirements Manual.
b. All populated areas in the overflight area that could be impacted
by the debris.
c. The probability of the debris impacting each populated area, which
takes into account the probability of vehicle failure.
d. The effective casualty area of the impacting debris, which accounts
for the cross-sectional area of the debris, average size of a person,
and the effects of any explosive debris.
e. The population density of each populated area. Note: The assessment
should consider any risk mitigation factors associated with each
population, such as sheltering and time of day of the flight.
f. Debris variability, including size, shape, aerodynamic properties,
weight, and potential to survive to impact.
g. The sources of debris variability, including breakup conditions.
h. The uncertainties in the state vector at the instant of jettison or
destruct and any correlations used.
i. Any velocity imparted to the debris fragments during jettison,
destruct, or breakup.
j. The influence of atmospheric variability, including winds.
4.5.3 A debris risk assessment for any property identified under paragraph 4.2.3.b shall account for:
a. All potential debris (intentionally or unintentionally generated)
that could cause property damage, which accounts for the specific
nature of the property.
b. The cross-sectional area of the debris and the potential for damage
due to any explosive debris.
c. Debris variability, including size, shape, aerodynamic properties,
weight, and potential to survive to impact.
d. The sources of debris variability, including breakup conditions.
e. The uncertainties in the state vector at the instant of jettison or
destruct and any correlations used.
f. Any velocity imparted to the debris fragments during jettison,
destruct, or breakup.
g. The influence of atmospheric variability, e.g. winds.
h. The probability of the debris impacting the property, which
accounts for the probability of vehicle failure and the location,
size, and shape of the property.
8719.25 is only 37 pages, but it starts with a page-long list of other requirements documents incorporated by reference. The ones they include from DOD are listed at https://public.ksc.nasa.gov/kscsma/requirements-documents-dod-range-safety-requirements/ One of those, also mentioned in the quotation above, is Space Systems Command Manual 91-701, Range Safety User Requirements Manual. It is published in 7 volumes, totaling 659 pages. It lays out in great detail not only what performance is required, but also what steps are required to test or simulate performance, how the results shall be documented, and management procedures which must be used by any company engaged in a contract to sell rockets and similar things to the U.S. Government. Volume 7 begins with a ten-page list of other referenced documents from standards-making bodies, from AIAA S-113A (2016), "Criteria for Explosive Systems and Devices on Space and Launch Vehicles", to UFC 3-340-02, "Structures to Resist the Effects of Accidental Explosions", and everything in between.
- e) Are there any publicly available calculations
I do not think there is any public source of data for actual operational launch systems, because those would necessarily include key design features of dual-use military technology falling under the international Missile Technology Control Regime, the U.S. ITAR, and foreign equivalents. Even within the U.S. commercial launch community, no company wants to share its sensitive design and performance data with its rivals. Detailed launch vehicle reliability models are required by the government to make its decisions on which license applications to approve, but that is considered proprietary information of each competitor separately, and the government is prohibited by its own contracting regulations from sharing that data with others. Anything patented is public, but that's exactly why these companies rarely patent anything, instead hoping to keep their trade secrets private for as long as possible.
However, there are public government documents which present example calculations for your reference. One of them is the FAA document you linked in your question: Advisory Circular 431.35-1, "Expected Casualty Calculations for Commercial Space Launch and Reentry Missions", 2000. Page 3 says
In order to explain the basic methodology, this advisory circular uses
simplified examples. The number of possible events and inputs are
selected to allow the reader to focus on the process, rather than on
the large number of events and situations that may actually need to be
considered, in performing a specific Ec analysis. Actual
analyses are generally far more extensive, yet all utilize the basics
described in this document. The methodology described here provides
acceptable approaches. However, the user is cautioned that the
applicant is responsible for demonstrating that the inputs to the
formula and assumptions made are appropriate for the situation being
addressed.
The rest of the document has worked examples with numbers plugged in. The probability distributions on page 20 are my favorite part. However, this is an old document, and the newer ones are more detailed and have better graphics. AC 431.35-1 was replaced in 2013 by the Flight Safety Analysis Handbook, which is nearly ten times as long, and thus has tons more fun math and graphs and tables. If you want to plug stuff into equations yourself, this is the document you want. It was joined in 2021 by AC 450.115-1A, "High Fidelity Flight Safety Analysis", which treats similar material in a rather different style.
- f) Is there any information on how non-US agencies might perform the calculation differently?
This answer is already way too long, but a place you might get started is Japan's Conditions and Methods for Calculating Expected Casualties (Launch Vehicles), which compares the US procedures with those of ESA (Europe) and CNES (France)
