Chapter 4 — Key Takeaways (Getting to Orbit)

A one-page reference. Reread this before an exam, or before you budget any launch.

The one idea

Orbit is sideways, not up. An orbit is a horizontal speed, not an altitude. A satellite falls continuously toward Earth while the curved surface drops away beneath it at the same rate — perpetual free fall that keeps missing the ground. Height is cheap; the $\sim 7.8\ \text{km/s}$ of sideways speed is the whole cost.

The key equations (with symbols and units)

Equation Use it to find Symbols
$v_{\text{orbit}} = \sqrt{\mu/r}$ circular orbital speed (km/s) $\mu = 3.986\times10^5\ \text{km}^3/\text{s}^2$; $r$ = orbital radius (km)
$\text{drop} = d^2/(2R)$ how far Earth curves away over horizontal distance $d$ $R$ = Earth radius; matches free-fall drop $\tfrac12 g t^2$
$\Delta v_{\text{grav}} = \int g\sin\gamma\,dt$ gravity loss (m/s) $g \approx 9.81\ \text{m/s}^2$; $\gamma$ = flight-path angle above horizontal
$\Delta v_{\text{drag}} = \int (D/m)\,dt$, $D = \tfrac12\rho v^2 C_d A$ drag loss (m/s) $\rho$ = air density; $C_d$ = drag coeff.; $A$ = frontal area
$\Delta v_{\text{launch}} \approx v_{\text{orbit}} + \Delta v_{\text{grav}} + \Delta v_{\text{drag}} - v_{\text{rot}}$ total launch delta-v (km/s) $v_{\text{rot}} = 0.465\cos\phi$ km/s (eastward)

The launch delta-v budget to LEO (memorize the stack)

Line Approx. Note
Orbital speed (LEO) ~7.8 km/s $\sqrt{\mu/r}$; lower orbits are faster
Gravity loss ~1.5 km/s range 1.2–1.6; the big thief
Drag loss ~0.1 km/s small; $\rho$ and $v$ peak at opposite times
Launch delta-v (no credit) ~9.4 km/s what Chapters 1 & 3 quoted
Earth-rotation credit −(up to 0.46) km/s eastward, equatorial; penalty if westward

Definitions at a glance

Term One-line meaning
Gravity turn Lift off vertical, kick over a few degrees, then hold thrust along velocity and let gravity bend the path to horizontal.
Pitch program The scheduled pitch-angle-vs-time (or altitude) that steers the vehicle from vertical to horizontal.
Gravity loss Delta-v lost because part of the thrust fights gravity during the climb: $\int g\sin\gamma\,dt$.
Drag loss Delta-v lost to aerodynamic drag during atmospheric ascent: $\int (D/m)\,dt$.
Ascent trajectory The full path + pitch/throttle program from lift-off to insertion; the optimum minimizes total losses within structural limits.
Suborbital Reaches space (~100 km) but not orbital speed; a ballistic arc that falls back.
Orbital flight Has the ~7.8 km/s sideways speed to keep falling around Earth without descending.

Decision aid — "which idea applies?"

You want… Use
the sideways speed for a circular orbit $v = \sqrt{\mu/r}$
the extra delta-v beyond orbital speed add gravity loss (~1.5) + drag loss (~0.1)
the gravity-loss rate right now $g\sin\gamma$ (max straight up, zero horizontal)
the launch-site benefit subtract $0.465\cos\phi$ km/s (eastward)
whether one stage can do it compare $e^{\Delta v/v_e}$ with the $\sim 12.5$ ceiling (Ch. 3)

Common pitfalls

Pitfall Reality
"Space starts at 100 km, so just climb 100 km." Reaching space is a ~1.4 km/s hop; orbit is ~7.8 km/s sideways — ~30× the energy.
"Go straight up; it's the shortest path." Straight up maximizes gravity loss ($\sin\gamma = 1$) and leaves zero orbital velocity.
"Rise slowly to save fuel." The hover trap: thrust-to-weight ≈ 1 burns delta-v at rate $g$ for zero speed. Climb briskly (T/W ~1.2–1.5).
"Astronauts are beyond gravity." Gravity at 400 km is ~90% of surface; they float because they're in free fall, orbiting.
"Drag is the big loss near Mach 20." Drag is ~0.1 km/s; when the rocket is fast the air is nearly gone. Gravity is the big loss.

Numbers worth memorizing

  • Orbital speed (LEO) $\approx 7.8\ \text{km/s}$; launch delta-v to LEO $\approx 9.4\ \text{km/s}$.
  • Gravity loss $\sim 1.5\ \text{km/s}$; drag loss $\sim 0.1\ \text{km/s}$.
  • Earth-rotation credit up to $\sim 0.46\ \text{km/s}$ (eastward, equatorial).
  • Reaching orbit is $\sim 30\times$ the energy of a 100 km suborbital hop.
  • The launch mass ratio (~19–23 at $v_e = 3$ km/s) exceeds the single-stage ceiling → staging is mandatory.

Mission / astrotools additions this chapter

  • MDR: added the launch delta-v line (~9.4 km/s to LEO) as the top of the delta-v budget; noted it is the launch vehicle's job, separate from the spacecraft's own maneuvers.
  • rocket.py: ascent_delta_v(v_orbit, gravity_loss, drag_loss, rotation_bonus).