MAE 4262 ROCKETS AND MISSION ANALYSIS Rocket Equation

MAE 4262: ROCKETS AND MISSION ANALYSIS Rocket Equation and Losses Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk 1

ROCKET EQUATION: IMPORTANT TRENDS 2

TYPICAL DV MISSION REQUIREMENTS 3 http: //www. strout. net/info/science/delta-v/intro. html

DV CAPABILITY FOR VARIOUS ROCKETS REF: Space Propulsion Analysis and Design, by Humble, Henry and Larson 4

TYPICAL PERFORMANCE PARAMETERS (T and Isp)

GRAVITY • Remember that gravity on Earth (~ 9. 81 m/s 2) may be calculated fundamentally – Average radius of the Earth ~ 6, 378 km or 3, 963 miles – Mass of the Earth ~ 5. 9742 x 1024 kg • Some typical values for Earth: – High power amateur model rocket ~ 100, 000 ft, 30. 5 km, 19 miles • g/ge = 99% – Shuttle in LEO (altitude of 300 km, 186 miles) • g/ge = 91% – Satellite in GEO (altitude of 42, 000 km, 26, 000 miles) • g/ge = 1. 7% • Note that the radius of the moon is about 1, 737 km and mass is 7. 36 x 1022 kg – So g on the surface of the moon is about 1. 62 m/s 2 6

COMPARISON OF GRAVITY, DENSITY AND PRESSURE VERSUS ALTITUDE Gravity Pressure Density 7

COMPARISON OF GRAVITY, DENSITY AND PRESSURE VERSUS ALTITUDE Gravity Density Pressure Shuttle LEO 8

WHAT IS ACTUAL SCALE OF ORBITS? NOT EVEN CLOSE TO SCALE

WHAT IS ACTUAL SCALE OF ORBITS? GEO EARTH LEO, 300 km

WHAT IS ACTUAL SCALE OF ORBITS? GEO LEO EARTH

TYPICAL DRAG VARIATION FOR ROCKETS 12

Variation of lift and drag coefficient with Mach number of V-2 rocket missile based on body cross-sectional area with jet off 13

DRAG: SUPERSONIC MISSILE EXAMPLE 14

COMMENTS: LAUNCH FROM SURFACE OF EARTH • To get to orbit (or to escape), direction of travel must be parallel to Earth’s surface (not perpendicular) • We launch vertically off the surface of the Earth, WHY? – Gravity • When rocket is vertical, gravity is acting against T and V – Drag • V 2 dependence: Drag ↑ as rocket accelerates – Large effect in lower atmosphere – Acceleration of vehicle is almost constant even though mass is changing • Density dependence: r ↓ very rapidly in atmosphere (r/r. S. L. ~ 1% at 100, 000 ft) • All rocket pass through condition of maximum dynamic pressure (MAX Q) – Many rockets stay vertical through this part – Get through atmosphere as quickly as possible – BUT before rocket really starts to speed up 15

COMMENTS: LAUNCHERS • Need certain velocities to get to space (and stay in space), escape, insertion, transition velocities, etc. → give DV requirements • Don’t want to carry fuel (heavy fuel is working against you) – Burn fuel early in flight → high accelerations, V 2 ↑ – Atmosphere is counter argument: drag, dynamic pressure • Why not launch horizontal? – Less gravity loss – Drag loss is high, more time in atmosphere – Lots of structural stress – Launch might look different on moon • Vertical launch segment: – Get out of dense atmosphere quickly, but still at relatively low speed – Don’t spend too much time here (vertical segment contributes nothing to eventual vertical orbital velocity) – Highest gravity losses, but sustain them to get lower density then really increase DV 16

Variation in air density (r), velocity (V), altitude (h), and dynamic pressure (q) during a Space Shuttle launch 17