Why doesn't New Horizons fly by nearer to Pluto so the gravitational pull would slingshot the probe to one of the next dwarf planets in the outer solar system (Eris, Haumea, Makemake,...)? I would think it could fly in sleep mode for another decade to visit another dwarf planet.
Are The bigger KBOs too far away? What is the limiting resource to the probe.
Why doesn't New Horizons fly by nearer to Pluto so the gravitational pull would slingshot the probe to one of the next dwarf planets in the outer solar system (Eris, Haumea, Makemake,...)?
There are two key reasons:
The reason it can't is because the delta V would be rather small even if New Horizons was to flyby Pluto as close as possible. Pluto's mass and orbital velocity are rather small and New Horizons' velocity is rather large. A Pluto flyby at 100 km above the surface with a v∞ of 13 km/sec will turn New Horizons' trajectory by about 1.4 degrees. That's not much. Even a flyby that is just above the surface at closest approach is essentially a straight-line trajectory. There is very little gravity assist to be had in a Pluto flyby at New Horizons' high velocity.
The reason it shouldn't is because that close flyby goes strongly against the grain of the primary mission, which is to observe Pluto. Suppose Pluto was orders of magnitude more massive than it is, thereby making a gravity assist feasible. Also suppose a large KBO was discovered after the New Horizons launch, and that this newly discovered KBO was reachable but only if the Pluto encounter was changed from its planned 10,000 km closest approach to a flyby that involves a very close approach. Should they do it?
The answer is no. The Pluto encounter is the primary goal of the mission. That radical change in trajectory would mean failure to achieve that primary goal. New Horizons' sensor package was designed with that 10,000 km closest approach in mind. The sensor package cannot handle a close flyby.
Suppose New Horizons does achieve its primary goal. That alone is enough to qualify the mission as a complete success. If it fails to achieve its primary goal, that is enough to qualify the mission as a failure. The post-Pluto portion of the mission has always been viewed as an optional bonus.
Details on "it can't"
One of the key parameters in determining the gravity assist the results from a flyby is the turn angle $\delta$. This is the angle by which the planet turns the satellite's hyperbolic trajectory (as viewed from a planet-centric inertial frame). This is given by
$$\delta = 2\arcsin\left( \frac 1 e \right)$$
where $e$ is the eccentricity of the hyperbolic trajectory. This can be computed from the planet's gravitational parameter $\mu = GM$, the satellite's periapsis distance $r_p = a(1-e^2)$, and the satellite's velocity at infinity $v_\infty = -\mu/a$:
$$e = \sqrt{1+\frac{r v_\infty}\mu}$$
The eccentricity is 227 in the case of a Pluto flyby with a v∞ of 13 km/sec and a closest approach of 10,000 km. This reduces to an eccentricity of 81 for a close-in flyby (100 km above the surface). Both of those eccentricities are too large to get a significant delta V from the flyby. The turn angle for the nominal mission is 0.5 degrees, 1.4 degrees for the close-in flyby.
That's pretty much what they're planning to do -- except that the planned target is a smaller Kuiper Belt Object, not a dwarf planet.
UPDATE : I overestimated the course change possible by using Pluto's gravity. See David Hammen's answer for the details.
After the Pluto flyby, the current plan is for New Horizons to continue on its path and fly by one or two Kuiper Belt objects (KBOs). Two such objects have been identified as possible targets. As far as I can tell a selection has not yet been made. One is estimated to be about 30-45 kilometers in diameter; the other is likely somewhat larger.
The choice of target is tightly constrained by available fuel for course adjustments. Before October 2014, there were no known bodies beyond the Pluto system that New Horizons would be able to reach. A search was conducted using ground telescopes, and later the Hubble Space Telescope.
Once a target is chosen, the flight path will be adjusted during the Pluto flyby to ensure that it passes near the selected object. Pluto's gravity will have to be taken into account when planning the course correction.
It's not clear that this would be a "slingshot". The goal is not necessarily to increase the spacecraft's velocity, as was done by the Jupiter flyby in 2006; rather it will select whatever course is needed to reach the target.
A flyby of a another dwarf planet after leaving the Pluto system is, unfortunately, not feasible. The choice of targets past Pluto is tightly constrained by fuel, distance, and time. The spacecraft has only a very limited amount of hydrazine propellant available for maneuvers, and it can't effectively return good science data past about 55 AU. This limits available targets to a cone less than 1 degree wide. There just aren't any dwarf planets past Pluto that New Horizons is able to reach. Only three objects past Pluto have been officially recognized as dwarf planets; a handful of others are likely.
This information is summarized from the Wikipedia article.
