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Formula sheet

Spacecraft Systems and Mission Engineering Formula Sheet

Spacecraft systems formulas for orbit period, delta-v, propellant, eclipse energy, battery sizing, radiator area, link budget, ADCS momentum, limits, and validation.

Branch
Aerospace Engineering
Content
Formula sheet
Updated
Revision
v1.1.0 ยท reviewed

This formula sheet collects first-pass relationships used in spacecraft systems and mission engineering. It covers orbit timing, velocity, delta-v, propellant mass, power and eclipse energy, battery sizing, radiator area, communications links, attitude-control momentum, data volume, reliability and validation margins.

Use these equations for mission screening, budget review and engineering interpretation. They do not replace flight dynamics tools, approved orbit analysis, qualified propulsion data, thermal-vacuum models, link analysis, radiation analysis, flight software verification or mission operations rules.

How to Use This Formula Sheet

Use this sheet as a mission-budget control surface. Start with the mission phase and mode: launch and early orbit, commissioning, nominal operations, payload pass, eclipse, downlink, maneuver, safe mode, degraded mode or disposal. A value that is acceptable in one mode can be mission-ending in another.

Then connect the budgets on one timeline. Orbit period drives eclipse time, contact windows, thermal environment and data return. Delta-v drives propellant and operations reserve. Power drives battery depth of discharge, heater duty and thermal rejection. Link margin drives downlink time and data storage. ADCS momentum drives pointing duration and desaturation needs.

Use the formulas for first-pass consistency and margin review. Use the validation package before claiming readiness, because spacecraft system risk often comes from cross-budget interactions rather than from one isolated subsystem calculation.

Basis and Validity Limits

The formulas in this sheet are screening equations. Circular-orbit speed and period use a simplified two-body assumption. Delta-v and propellant calculations do not include all maneuver execution losses, residuals, pressurant constraints, attitude constraints or operations reserve unless those are explicitly added. Low-thrust maneuvers require separate duty-cycle, pointing, power and timeline checks.

Power, battery and thermal formulas depend on mission mode, end-of-life degradation, temperature, duty cycle, heater logic, eclipse season, attitude, surface properties and view factors. A nominal power margin does not validate safe mode, battery recovery, cold-soak survival or thermal-vacuum correlation.

Communications and ADCS formulas are also conditional. Link budgets depend on ground station, weather, pointing, polarization, coding, contact schedule, data handling and regulatory constraints. Reaction-wheel momentum budgets depend on disturbance torque, wheel limits, desaturation opportunity, sensor validity, actuator authority and safe-mode transition logic. Reliability screens require common-cause, software, radiation and switching failures before they can support a mission claim.

Notation

SymbolMeaningTypical unit
rorbital radius from central-body centerm
R_EEarth mean radius for first-pass LEO workm
haltitude above reference radiusm
\mugravitational parameter\text{m}^3/\text{s}^2
T_oorbital periods
vcircular orbit speedm/s
\Delta vrequired mission velocity incrementm/s
I_{sp}specific impulses
m_0,m_finitial and final spacecraft masskg
Pelectrical powerW
EenergyWh or J
DODbattery depth of dischargedimensionless
\etaefficiencydimensionless
Aarea\text{m}^2
\epsilonsurface emissivitydimensionless
\sigmaStefan-Boltzmann constant\text{W}/\text{m}^2\text{K}^4
Hreaction-wheel angular momentum\text{N m s}
\tautorqueN m
Bcommunication bandwidthHz

Circular Orbit Speed and Period

Orbital radius:

r=R_E+h

Circular orbit speed:

\displaystyle v=\sqrt{\frac{\mu}{r}}

Orbital period:

\displaystyle T_o=2\pi\sqrt{\frac{r^3}{\mu}}

Ground tracks, drag, perturbations and station keeping require more detail, but these equations are useful for checking timing and order of magnitude.

Delta-V and Propellant Mass

Tsiolkovsky rocket equation:

\displaystyle \Delta v=g_0I_{sp}\ln\left(\frac{m_0}{m_f}\right)

Mass ratio:

\displaystyle \frac{m_0}{m_f}=e^{\Delta v/(g_0I_{sp})}

Propellant mass:

m_p=m_0-m_f=m_0\left(1-e^{-\Delta v/(g_0I_{sp})}\right)

Include residuals, usable propellant, pressurant, tank constraints, duty cycle, pointing constraints and maneuver execution losses in a real budget.

Thrust, Burn Time and Impulse

Total impulse:

I_t=m_pg_0I_{sp}

Burn time for constant thrust:

\displaystyle t_b=\frac{I_t}{F}

Acceleration from thrust:

\displaystyle a=\frac{F}{m}

Low-thrust electric propulsion may have excellent propellant efficiency but long maneuver time. Mission operations must verify power, pointing, thermal state and duty cycle during the burn.

Eclipse Energy and Battery Sizing

Eclipse energy:

E_{ecl}=P_{ecl}t_{ecl}

Contingency-adjusted eclipse energy:

E_{req}=E_{ecl}(1+c)

Beginning-of-life battery energy required:

\displaystyle E_{BOL,req}=\frac{E_{req}}{f_{EOL}DOD_{max}\eta_{dis}}

Actual end-of-life depth of discharge for a selected battery:

\displaystyle DOD_{actual}=\frac{E_{req}}{E_{BOL,selected}f_{EOL}\eta_{dis}}

Battery sizing should be mode-based. Heater duty cycle, safe mode, degradation, temperature and current limit often dominate the real decision.

Solar Array Recharge Power

Energy that must be restored during daylight:

\displaystyle E_{recharge}=\frac{E_{req}}{\eta_{charge}}

Average daylight recharge power:

\displaystyle P_{recharge}=\frac{E_{recharge}}{t_{sun}}

Required solar-array beginning-of-life power:

\displaystyle P_{array,BOL}=\frac{P_{day}+P_{recharge}}{f_{aging}f_{pointing}\eta_{conv}}

where P_{day} is the mode-dependent daylight bus load.

Radiator Area in Vacuum

Radiative heat rejection to a cold sink:

Q=\epsilon\sigma A\left(T^4-T_{sink}^4\right)

Required radiator area:

\displaystyle A=\frac{Q}{\epsilon\sigma\left(T^4-T_{sink}^4\right)}

In a first cold-space screen, T_{sink} may be small compared with spacecraft radiator temperature. Real thermal design must include solar input, albedo, infrared radiation, view factors, conduction, contact resistance, heater logic and attitude.

Wavelength:

\displaystyle \lambda=\frac{c}{f}

Free-space path loss:

\displaystyle L_{fs,dB}=20\log_{10}\left(\frac{4\pi R}{\lambda}\right)

Received power:

P_{rx,dBm}=EIRP_{dBm}+G_{rx,dBi}-L_{fs,dB}-L_{misc,dB}

Thermal noise floor:

N_{dBm}=-174+10\log_{10}(B)+NF

Carrier-to-noise margin:

C/N=P_{rx,dBm}-N_{dBm}

Modulation, coding, pointing, polarization, weather, ground-station availability and interference set the real communications margin.

Data generated:

D=R_dt_{obs}

Required downlink time:

\displaystyle t_{dl}=\frac{D}{R_{dl}\eta_{protocol}}

Storage margin:

\displaystyle M_D=\frac{D_{storage}-D_{peak}}{D_{storage}}

Data budgets should include compression, packet overhead, retransmission, contact scheduling, onboard processing and ground-segment availability.

Reaction-Wheel Momentum Budget

Momentum accumulation from constant disturbance torque:

\Delta H=\tau_dt

End momentum:

H_{end}=H_0+\Delta H

Momentum margin:

\displaystyle M_H=\frac{H_{max}-H_{end}}{H_{max}}

Time to saturation:

\displaystyle t_{sat}=\frac{H_{max}-H_0}{\tau_d}

Momentum budgets should be reviewed by mission mode. A short pointing mode may pass while a long imaging or downlink pass saturates a wheel.

Reliability Screening

Constant failure-rate reliability:

R(t)=e^{-\lambda t}

Series reliability:

R_{series}=\prod_iR_i

Parallel one-of-two reliability for independent channels:

R_{1oo2}=1-(1-R_1)(1-R_2)

Reliability screening is not enough by itself. Common-cause failures, software faults, radiation effects, switching logic, safe-mode behavior and operations recovery often decide the mission outcome.

Worked Example 1: LEO Orbit Period and Speed

Estimate circular orbit speed and period for a spacecraft at:

h=550\ \text{km}

Use:

R_E=6378\ \text{km}
\mu=3.986\times10^{14}\ \text{m}^3/\text{s}^2

Orbital radius:

r=(6378+550)\ \text{km}=6928\ \text{km}=6.928\times10^6\ \text{m}

Speed:

\displaystyle v=\sqrt{\frac{3.986\times10^{14}}{6.928\times10^6}}=7586\ \text{m/s}

Period:

\displaystyle T_o=2\pi\sqrt{\frac{(6.928\times10^6)^3}{3.986\times10^{14}}}=5736\ \text{s}

Convert:

\displaystyle T_o=\frac{5736}{60}=95.6\ \text{min}

Engineering comment: this is a circular two-body estimate. Mission planning should add drag, Earth oblateness, access windows, local time of node, eclipse season and ground-station geometry.

Worked Example 2: Delta-V Propellant Mass

A 180\ \text{kg} spacecraft needs:

\Delta v=120\ \text{m/s}

Chemical thruster specific impulse is:

I_{sp}=220\ \text{s}

Propellant mass:

m_p=m_0\left(1-e^{-\Delta v/(g_0I_{sp})}\right)
m_p=180\left(1-e^{-120/(9.80665)(220)}\right)
m_p=9.7\ \text{kg}

Engineering comment: the propulsion budget should not select exactly 9.7\ \text{kg}. Add unusable propellant, maneuver error, temperature effects, residuals, valve leakage allowance, pressurization constraints and operations reserve.

Worked Example 3: Eclipse Battery Sizing

Eclipse load is:

P_{ecl}=42\ \text{W}

Eclipse duration:

t_{ecl}=36\ \text{min}=0.60\ \text{h}

Contingency:

c=20\%

Battery assumptions:

f_{EOL}=0.80
DOD_{max}=0.30
\eta_{dis}=0.92

Eclipse energy:

E_{ecl}=42(0.60)=25.2\ \text{Wh}

Contingency-adjusted energy:

E_{req}=1.20(25.2)=30.24\ \text{Wh}

Required beginning-of-life battery energy:

\displaystyle E_{BOL,req}=\frac{30.24}{0.80(0.30)(0.92)}=137\ \text{Wh}

Engineering comment: this is only the eclipse energy check. Battery current limit, heater peaks, safe mode, low-temperature capacity and degradation model must be validated before launch.

Worked Example 4: Radiator Area

A component must reject:

Q=28\ \text{W}

Target radiator temperature:

T=300\ \text{K}

Emissivity:

\epsilon=0.82

Use a cold-space first screen with:

T_{sink}\approx0\ \text{K}

Radiator area:

\displaystyle A=\frac{28}{0.82(5.67\times10^{-8})(300^4)}
A=0.074\ \text{m}^2

Engineering comment: this is optimistic if the radiator sees the Sun, Earth infrared, albedo or warm spacecraft surfaces. The final radiator design needs view factors, coatings, contact resistance, heater logic and thermal-vacuum correlation.

Use:

f=8.2\ \text{GHz}
R=1000\ \text{km}
EIRP=33\ \text{dBm}
G_{rx}=42\ \text{dBi}
L_{misc}=3\ \text{dB}
B=1.0\ \text{MHz}
NF=2\ \text{dB}

Wavelength:

\displaystyle \lambda=\frac{3.0\times10^8}{8.2\times10^9}=0.0366\ \text{m}

Free-space path loss:

\displaystyle L_{fs}=20\log_{10}\left(\frac{4\pi(1.0\times10^6)}{0.0366}\right)=170.7\ \text{dB}

Received power:

P_{rx}=33+42-170.7-3=-98.7\ \text{dBm}

Noise:

N=-174+10\log_{10}(1.0\times10^6)+2=-112\ \text{dBm}

Carrier-to-noise:

C/N=-98.7-(-112)=13.3\ \text{dB}

Engineering comment: this is a received-power screen. The real link margin depends on required E_b/N_0, coding, modulation, pointing loss, polarization, weather, implementation loss and ground-station scheduling.

Worked Example 6: Reaction-Wheel Momentum Margin

An imaging pass lasts:

t=2400\ \text{s}

Estimated disturbance torque:

\tau_d=18\ \mu\text{N m}=18\times10^{-6}\ \text{N m}

Wheel momentum at pass start:

H_0=0.020\ \text{N m s}

Operational wheel limit:

H_{max}=0.090\ \text{N m s}

Momentum accumulation:

\Delta H=\tau_dt=(18\times10^{-6})(2400)=0.0432\ \text{N m s}

End momentum:

H_{end}=0.020+0.0432=0.0632\ \text{N m s}

Momentum margin:

\displaystyle M_H=\frac{0.090-0.0632}{0.090}=0.298=29.8\%

Engineering comment: the pass has positive momentum margin. Validate the disturbance model, wheel-speed limits, desaturation opportunity, sensor validity and safe-mode transition before relying on this margin operationally.

Validation Evidence Package

Before accepting a spacecraft budget calculation, assemble evidence that connects the formula result to a mission mode, configuration, timeline and verification record. Confirm:

  1. Mission phase, mode and configuration are explicit.
  2. Units and time bases are consistent.
  3. Mass, power, thermal, data, pointing and propellant budgets use the same mission timeline.
  4. Degradation and end-of-life assumptions are visible.
  5. Safe mode is checked separately from nominal mode.
  6. Communication budgets include ground segment and scheduling constraints.
  7. Attitude-control margins include actuator saturation and momentum management.
  8. Thermal calculations include environmental boundary assumptions.
  9. Reliability arguments include common-cause and switching failures.
  10. Verification evidence is tied to build, configuration, environment and pass/fail criteria.

Also include the mission timeline revision, mass properties, power-mode table, thermal boundary case, communication contact assumptions, ADCS mode, flight software configuration, environmental assumptions, end-of-life factors, uncertainty or margin policy, safe-mode criteria and retest trigger. A spacecraft calculation should state what evidence would invalidate the margin after design, integration, environmental test or operations changes.

Common Formula Mistakes

Common mistakes include:

  • closing a power budget for average load while missing eclipse or heater peaks;
  • sizing propellant without unusable mass, execution loss or operations reserve;
  • checking a link budget without contact duration or pointing loss;
  • treating a reaction wheel as unlimited torque storage;
  • validating subsystem tests while omitting integrated mission scenarios;
  • using beginning-of-life capacity or power as if it were end-of-life capability;
  • adding redundancy without validating switching logic and common-cause failures;
  • reporting margins without the mode, mission phase or configuration that produced them.

Additional mistakes include checking each subsystem with a different timeline, using beginning-of-life solar or battery capability in an end-of-life claim, accepting a downlink budget without data volume and contact duration, and treating safe mode as a scaled-down nominal mode instead of its own operating case.

Spacecraft systems work is budget discipline. A formula is useful only when it is tied to a mode, margin, failure case and verification record.

REF

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