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In this answer I've interpreted the x axis of the plot in the question (also shown below) labeled as $C_3$ ([characteristic energy]) to be the "excess $v^2$ above escape velocity from Earth", and so I'd written the relation.

$$C_3 = v^2 - v_{esc}^2.$$

Could I have some confirmation that this is a correct way to use the $C_3$ used in the plot? Am I missing a factor of 2 or are there other subtleties nor not-so-subtleties that I've overlooked?

The expression in @MarkAdler's answer is cast in terms of potential energy (per unit mass) but I've assumed conservation of total energy and applied it to kinetic energy (per unit mass).

The figure below is from the linked question, there is currently no original source listed for the image.

In this answer I've interpreted the x axis of the plot in the question (also shown below) labeled as $C_3$ (characteristic energy) to be the "excess $v^2$ above escape velocity from Earth", and so I'd written the relation.

$$C_3 = v^2 - v_{esc}^2.$$

Could I have some confirmation that this is a correct way to use the $C_3$ used in the plot? Am I missing a factor of 2 or are there other subtleties nor not-so-subtleties that I've overlooked?

The expression in @MarkAdler's answer is cast in terms of potential energy (per unit mass) but I've assumed conservation of total energy and applied it to kinetic energy (per unit mass).

The figure below is from the linked question, there is currently no original source listed for the image.

In this answer I've interpreted the x axis of the plot in the question (also shown below) labeled as $C_3$ ([characteristic energy]) to be the "excess $v^2$"above escape velocity from Earth, and so I'd written the relation.

$$C_3 = v^2 - v_{esc}^2.$$

Could I have some confirmation that this is what $C_3$ actually means in the plot? Am I missing a factor of 2 or are there other subtleties nor not-so-subtleties that I've overlooked?

The expression in @MarkAdler's answer is cast in terms of potential energy (per unit mass) but I've assumed conservation of total energy and applied it to kinetic energy (per unit mass).

The figure below is from the linked question, there is currently no original source listed for the image.

In this answer I've interpreted the x axis of the plot in the question (also shown below) labeled as $C_3$ ([characteristic energy]) to be the "excess $v^2$ above escape velocity from Earth", and so I'd written the relation.

$$C_3 = v^2 - v_{esc}^2.$$

Could I have some confirmation that this is a correct way to use the $C_3$ used in the plot? Am I missing a factor of 2 or are there other subtleties nor not-so-subtleties that I've overlooked?

The expression in @MarkAdler's answer is cast in terms of potential energy (per unit mass) but I've assumed conservation of total energy and applied it to kinetic energy (per unit mass).

The figure below is from the linked question, there is currently no original source listed for the image.

In this answer I've interpreted the x axis of the plot in the question (also shown below) labeled as $C_3$ ([characteristic energy]) to be the "excess $v^2$"above escape velocity from Earth, and so I'd written the relation.

$$C_3 = v^2 - v_{esc}^2.$$

Could I have some confirmation that this is what $C_3$ actually means in the plot? Am I missing a factor of 2 or are there other subtleties nor not-so-subtleties that I've overlooked?

The expression in @MarkAdler's answer is cast in terms of potential energy (per unit mass) but I've assumed conservation of total energy and applied it to kinetic energy (per unit mass).

In this answer I've interpreted the x axis of the plot in the question (also shown below) labeled as $C_3$ ([characteristic energy]) to be the "excess $v^2$"above escape velocity from Earth, and so I'd written the relation.

$$C_3 = v^2 - v_{esc}^2.$$

Could I have some confirmation that this is what $C_3$ actually means in the plot? Am I missing a factor of 2 or are there other subtleties nor not-so-subtleties that I've overlooked?

The expression in @MarkAdler's answer is cast in terms of potential energy (per unit mass) but I've assumed conservation of total energy and applied it to kinetic energy (per unit mass).

The figure below is from the linked question, there is currently no original source listed for the image.