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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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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.

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    $\begingroup$ 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. $\endgroup$
    – Russell Borogove
    Nov 25, 2015 at 16:05
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    $\begingroup$ 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. $\endgroup$
    – Brian Lynch
    Nov 25, 2015 at 16:10
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    $\begingroup$ 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. $\endgroup$
    – Russell Borogove
    Nov 25, 2015 at 16:22
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    $\begingroup$ 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. $\endgroup$
    – Russell Borogove
    Nov 25, 2015 at 17:28
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    $\begingroup$ @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! $\endgroup$
    – Brian Lynch
    Jan 31, 2016 at 12:46

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