Slow Launch

Every time I see a rocket vault off the pad, I wonder why the engineers didn't make the propellant tanks bigger. Adding propellant to any stage always increases the delta-V delivered, so long as you ignore gravity losses.

It takes a lot of thrust and energy expended just to get the rocket to hover stationary above the pad. The delta-V lost is called the gravity loss. As the rocket gets closer to orbital velocity, gravity losses drop. At low velocities, gravity losses are just time * acceleration by gravity.

Gravity losses, as it turns out, are a big deal. As you add propellant, you logarithmically increase the delta-V, but you linearly increase the gravity losses. At some point, gravity losses overtake the increased delta-V.

The tradeoff between the two is different for different kinds of rockets. In a solid-fuelled rocket, the entire stage is a big combustion chamber, that must contain the gas pressure used to accelerate the vehicle upward. Large pressure vessels are heavy, and so the fuel container is a large fraction of a solid rocket's mass. Pressure-fed liquid-fuelled rockets don't actually have the combustion chamber in the tanks, but the tanks must hold higher pressure than the combustion chamber, so the mass penalty is similar.

Pump-fed liquid-fuelled rockets hold their propellants at a small fraction of the combustion chamber pressure, and so their tanks are a small fraction of the weight of a similarly-sized solid rocket.

Let's take a look at the two extremes. First, a kerosene/LOX liquid-fuelled rocket, like the Saturn V or Falcon 9. I've made up a table to show the decreasing performance return of steadily larger and larger tanks. Here I've presumed a base rocket with an Isp of 290, a first-stage thrust of 750,000 kg, a Gross Lift-Off Weight (GLOW) of 500,000 kg, and a first stage burnout weight (this includes the upper stages) of 150,000 kg. This rocket has an initial acceleration of 1.5 G (but remember you lose 1 G to earth). Incremental tankage weighs just 2.5% of the incremental propellant stored. I'm assuming that the engine thrust stays fixed. Delta-V numbers are in meters/sec. Tower clearance times are a little high, as they assume no acceleration beyond the initial acceleration.

Isp	290
Ve	2842
Tankage	0.025
Thrust	750000
Burn rate	2586
Tower	50
			inc	inc		delivered	tower
G	Mf	Me	delta-V	time	G-loss	delta-V	clear
2	375000	146875	0	0	0	0	3.2
1.9	394737	147368	136	7	73	63	3.4
1.8	416667	147917	143	8	81	62	3.6
1.7	441176	148529	151	9	91	60	3.8
1.6	468750	149219	159	10	102	57	4.1
1.5	500000	150000	169	12	115	53	4.5
1.4	535714	150893	179	13	132	47	5.1
1.3	576923	151923	191	16	152	39	5.8
1.2	625000	153125	205	18	178	27	7.1
1.1	681818	154545	221	21	210	11	10.1
1	750000	156250	240	26	252	-12	N/A

You can see why the Saturn V initial acceleration was just 1.13 Gs. You can also see why launching the Saturn V in a strong wind could have been a problem: it takes a long time to get past the tower.

Now let's look at the other extreme, a solid rocket first stage, like the Titan 4 or the Stick proposal. (Because most of the Shuttle's liftoff thrust is from it's solids, it fits in this category too.) Here the Isp is a bit lower, but more importantly the tankage fraction is far higher: 12%.

Isp	242
Ve	2371.6
Tankage	0.12
Thrust	750000
Burn rate	3099
Tower	50
			inc	inc		delivered	tower
G	Mf	Me	delta-V	time	G-loss	delta-V	clear
2.5	300000	126000	0	0	0	0	2.6
2.4	312500	127500	69	4	35	34	2.7
2.3	326087	129130	71	4	38	33	2.8
2.2	340909	130909	73	4	41	32	2.9
2.1	357143	132857	75	5	45	30	3.0
2	375000	135000	78	5	50	28	3.2
1.9	394737	137368	80	6	55	25	3.4
1.8	416667	140000	83	6	61	22	3.6
1.7	441176	142941	86	7	68	18	3.8
1.6	468750	146250	90	8	77	13	4.1
1.5	500000	150000	93	9	87	6	4.5
1.4	535714	154286	97	10	99	-3	5.1

You can see why a Titan 4 gets off the pad with nearly 1.5 Gs of initial acceleration.

Solid rockets are a good match for first stage engines. It takes relatively little engineering work to produce a large amount of thrust from a solid motor, which is the biggest cost driver for first stage engines. But because the casing weighs so much, and also because solid propellants have lower Isp, the delta-V of a solid first stage is never going to be as good as a liquid-fuelled analog. That leaves more work for the second stage, which means thinner engineering margins and more desire for high-Isp propellants, like liquid hydrogen. And so, cheap solid rocket first stages drive more cost into the upper stage.