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Related to this question about single-stage-to-orbit vehicles, the Atlas-B launch vehicle seems to have been the closest to a true SSTO solution. Although it did jettison booster engines, the vehicle was basically a single stage -- often referred to as "stage-and-a-half" configuration.

Could the Atlas-B have made it to a relatively stable orbit without jettisoning those engines? Even if it carried no payload? Note that this Astronautix page provides some numbers we can use for analysis and simulation.

I welcome any modifications to this question in regards to what defines a "relatively stable orbit", but let's say we just want to reach an orbit that won't decay for at least 24 hours.

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Brian Lynch
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Given the values on the linked Astronautix page, it looks like jettisoning those booster engines is necessary to get into orbit -- even without any payload. Splitting the flight into two $\Delta v$ calculations, we get this value for the "first stage" (i.e., the two booster engines and main engine firing in tandem, all pulling fuel out of the main tank):

$\Delta v_A = 4753 \frac{m}{s}$

If we jettison the booster engines, we lose 3050 kg and end up with a $\Delta v$ for the "second stage" of:

$\Delta v_B = 4469 \frac{m}{s}$

Now, if we retain that 3050 kg as empty mass for the second stage flight, we get a new value of:

$\Delta v_B = 3235 \frac{m}{s}$

So for the Atlas-B with no payload (which is only 70 kg nominal anyway), we have a total $\Delta v$ of about 9.2 km/s if the booster engines are jettisoned, and 8.0 km/s if they are not. Getting to low-Earth orbit usually requires at least around 8.7 km/s, so it would seem that jettisoning those engines was critical for a successful flight.

Brian Lynch
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    The situation is much the same for the other stage-and-a-half Atlases. SLV-3 has a slightly heavier booster module (~3342kg) and carries something like 800kg payload to LEO. – Russell Borogove Nov 25 '15 at 16:05
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    Thanks for pointing that one out too, these examples really emphasize the need for staging and just plain mass reduction in general. Then extend that to something like a Mars return vehicle! SSTO is so much harder than a lot of people imagine -- basically all of our favourite sci-fi relies on magically high Isp and engine thrust-to-weight ratios to send spacecraft to orbit without boosters or stages. – Brian Lynch Nov 25 '15 at 16:10
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    SSTO isn't in itself an important goal except for bragging rights; I think it's straightforward to do on a hydrogen engine. Doing so with a useful payload is prohibitive. – Russell Borogove Nov 25 '15 at 16:22
  • Good point, I suppose the only value is potential reusability as a trade-off for reduced payload capability. I guess no solutions have been proposed that can actually provide reusability and also deliver an actual useful payload as you mention. Perhaps single stage vehicles will only be useful for sub-orbital flight for now. – Brian Lynch Nov 25 '15 at 16:29
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    Yeah, SSTO is attractive if your goal is reusability because you only have to recover one thing. Hybrid air-breathers like Skylon may get around the rocket equation problem since they don't have to carry all their oxidizer mass, but it might be simpler to just accept that you're going to recover and reuse two separate stages. – Russell Borogove Nov 25 '15 at 17:28
  • @Brian Lynch, I've tried following your calculation. It looks to me as if there isn't enough information in the referenced web page to get the whole answer out. Would you mind putting in all the details please? I accept there will have to be some gross assumptions somewhere, I'd just be interested to follow the method. I think the bit that is missing is how the Isp varies from sea level to vacuum. – Puffin Jan 30 '16 at 20:53
  • @Puffin, everything you need is there, masses and Isp's. I will add more details as soon as I have a chance. This calculation doesn't account for varying Isp, you would need a full flight profile to do that -- not too hard but it shouldn't impact the conclusions much (both configurations will see a change in the same direction). – Brian Lynch Jan 31 '16 at 07:33
  • @Brian Lynch Ok, thank you. For when you get around to it, here's my thinking: A) one needs to know how much propellant is consumed by each engine as their Isp are very different and B) choose which Isp for each. The reported figures of Booster 282 (vac), 248 (sea level), Sustainer 309 (vac), 215 (sea level). Were you using the sea level figure for the booster and the vacuum figure for the sustainer? I get 9281 that way. – Puffin Jan 31 '16 at 12:31
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    @Puffin, you can apply the rocket equation knowing the mass change and the Isp. The Isp for the motors together can be computed by considering the thrust equation $T = \dot{m} g_0 I_{sp}$ and extending it to the motors in parallel $T = T_1 + T_2$ where mass flow is also additive $\dot{m} = \dot{m}_1 + \dot{m}_2$. All this effort for a comment when I should be updating the answer! – Brian Lynch Jan 31 '16 at 12:46
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    @Brian Lynch thanks, that's the approach I'd taken, hence leading to the query about which Isp – Puffin Jan 31 '16 at 13:21