Rocket Delta-v, Staging and Orbital Refueling Calculator

Stage masses and propulsion

NASA Glenn's Ideal Rocket Equation gives $\Delta v_i=g_0 I_{sp,i}\ln(m_{0,i}/m_{f,i})$, with standard gravity $g_0=9.80665\,\mathrm{m/s^2}$. Each initial and final mass includes the entire carried upper stack. The first-stage dry mass and retained propellant are removed only after its burn. Retained propellant remains in the burnout mass. Effective exhaust speed also gives $\dot m=T/(g_0I_{sp})$, burn duration $(p-r)/\dot m$, and ignition thrust-to-weight $T/(m_0g_{local})$.

Screening estimates for ascent

The selected loss estimate is $g_{local}(t_1\sin\gamma_1+t_2\sin\gamma_2)+D_{mean}t_{atm}/m_{mean}$, where $D_{mean}=\bar q C_D A$ and $m_{mean}=m_0-\dot m_1t_{atm}/2$. Mean angles and dynamic pressure are user assumptions. These lumped approximations follow the force components discussed by NASA Glenn and MIT Lecture 14, pages 6–7. They are not a trajectory integration.

Constrained optimal staging

At fixed payload and combined stage wet mass, redistribute 5–95% of stage wet mass to the booster. Preserve each stage's dry fraction, retained fraction, and Isp. Calculate both stage mass ratios at each allocation; compare an interior numerical maximum with both boundaries. Engine sizing, tank dimensions, burn-time changes, and losses are outside this ideal optimization. A separate dry-mass sweep holds receiver propellant capacity, inventory, reserve, payload, and Isp fixed to show the effect of an uncertain receiver dry mass. A negative gain can occur if the current allocation is outside the permitted interval.

Delivery campaign

Derived here from mass conservation: the inventory just after delivery $k$ is $M_k=\min(C,aM_{k-1}+\eta m_t)$, with $a=\exp(-b\Delta t)$. The first delivery occurs at $\Delta t$, not at zero. Departure inventory is $M_N\exp(-b h)$. Search counts 0–200 for the first successful departure. The final transfer is reduced to the target stock required before the wait, subject to capacity. Cargo left aboard the final tanker is reported separately. Accepted mass is summed on every visit. A separate sequential inventory ledger accumulates evaporation during each storage interval and the final wait: $B=\sum_{k=1}^{N}M_{k-1}(1-e^{-b\Delta t})+M_N(1-e^{-b h})$. This ledger does not read the received-mass total or departure-stock result. Initial inventory plus offered cargo is then compared with departure stock, independently summed boil-off, and transfer losses. The count does not include a receiver launch, a depot launch, or failed flights.

All default vehicle masses, Isp, thrust, reserve, delivery cargo, and cadence values are illustrative engineering assumptions. The 100–180 tonne receiver dry-mass interval and 80–160 tonne cargo interval are user-editable scenario ranges, not statistical uncertainty bands or published Starship performance ranges. No dated mission claim is needed for the calculation.

Two serial stages burn sequentially at constant effective thrust and Isp, without a separation coast, crossfeed, parallel boosters, fairing discard, or throttle. Dry masses exclude payload and all propellant. The orbital receiver is the upper stage, with the same dry mass, capacity, retained propellant, payload, and Isp; change the inputs when the mission configuration changes. Enter tanker cargo already available at rendezvous; ascent and return propellant are outside this quantity.

Boil-off is a continuous fraction of current inventory per day, not a fixed fraction of full-tank capacity. For example, 0.001/day corresponds to a daily loss of $1-e^{-0.001}\approx0.09995\%$. NASA NTRS 20110013691, attached page 1, describes a particular passively insulated depot design with modeled losses below 0.05% per day. This supports a thermal-design sensitivity study, not the chosen default or its application to Starship. A fixed heat leak can instead give nearly constant mass loss, especially at low fill; that regime needs a thermal model.

Transfers are instantaneous at completion, and only actually offered mass incurs transfer inefficiency. The receiver has sufficient ullage within its entered usable capacity. All flights succeed. The equivalent propellant inventory does not track oxidizer/fuel mixture changes.

Use for preliminary mass and delta-v budgets, education, and sensitivity studies. Positive Isp, thrust, dry masses, gravity and delivery interval are required. Loaded propellant must exceed retained propellant; starting and target inventory must fit receiver capacity, and the target must cover retained propellant. Drag exposure must fit inside the first burn.

Ascent outputs are scalar estimates. They do not prove orbital insertion, lunar landing, launch feasibility, or a safe vehicle design. Gravity varies with altitude; real drag depends on the trajectory and changing Mach number. Upper-stage T/W below one can be acceptable in orbit. The allocation optimizer covers only the stated mass interval and fixed mass fractions; engine and structural feasibility require separate analysis.

The campaign is capped at 200 deliveries. “>200 or unreachable” means the target is not met inside this horizon; reported inventory and losses then describe that horizon, not a successful mission. A target above capacity after the final hold may be impossible regardless of flight count. Values with insufficient departure inventory for the reserve give zero usable departure delta-v.

Reference note: MIT Lecture 14 page 9 prints 0.368−0.063=0.295; the subtraction is approximately 0.305. Its subsequent two-stage payload fraction therefore is not used as a reference. The verified case instead uses the independently stated, rounded single-burn mass fraction on page 8.